Metzner-Reed Reynolds number

The Metzner-Reed Reynolds number, Rcmr is given by equation (3.28b) ... 

Otherwise, bear in mind that very often the range of shear rates to which a process fluid is exposed in pip>e flow is often cjuite limited, and because of this it is often possible to adecjuately represent the flow behavior of a process fluid over a limited range of shear rates by the power law or Ostwald de Waele model, and the Metzner-Reed Reynolds number, which we shall shortly discuss. Meanwhile, in the absence of any lab equipment to provide you with shear-stress versus shear-rate data, we suggest that you vary the flow rate to provide several flow rates, and run pressure drop surveys over the pipeline in question. Tabulate the data and use this to develop a "power law" relationship (see following sections) as a first approximation for your pijjeUne and process fluid. Most likely you will not need to perform any more elaborate study than that... 

The Metzner-Reed Reynolds number is the generalized Reynolds number to use when working with the power law. In order to make comparison with other texts and charts easier for you, I will write it out in its usual form please bear with me, it really is not complicated. 

The Metzner-Reed Reynolds number can also be used in conjunction with the familiar expressions for frictional pressure loss (APf) and head loss (hf), just as with Newtonian flow as seen earlier in this chapter ... 

In general, non-Newtonian flow behavior (or rheology) is of dominant importance in laminar flow because it influences viscous shear. The transition from laminar to turbulent flow is much less sharply defined for non-Newtonian fluids than it is for Newtonian, and it is difficult to predict with any degree of accuracy. However, a value of the Metzner-Reed Reynolds number (Romr) of 2000 is still a reasonable estimate for the transition from laminar to turbulent flow. 

There is also a friction factor chart drawn by Dodge and Metzner plotting the Fanning friction factor versus the Metzner-Reed Reynolds number (although this has been subsequently redrawn for convenience with (j), see ref. 4). This chart shows that the friction factor decreases with n in turbulent flow—that is, it decreases as shear-thinning character increases. If you are studying this chart, be aware that the only experimentally verified data for turbulent flow is that given for m = 1.0 to n = 0.4. 

The Metzner and Reed Reynolds number Re K may be expressed in terms of nr and A. From equation 3.140, derived for a power-law fluid ... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

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 ... 

Metzner and Reed Reynolds number for power-law fluid... 

The generahzed approach of Metzner and Reed (AIChF J., 1, 434 [1955]) for time-independent non-Newtouiau fluids defines a modified Reynolds number as... 

Figure 3.40. Metzner and Reed correlation of friction factor and Reynolds number...

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

Use of this generalized Reynolds number was suggested by Metzner and Reed (1955). For Newtonian behaviour, K = n and n = 1 so that the generalized Reynolds number reduces to the normal Reynolds number. 

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 ... 

Re, the Metzner-Reed modified Reynolds number, is defined as... 

Dodge and Metzner (16) presented an extensive theoretical and experimental study on the turbulent flow of non-Newtonian fluids in smooth pipes. They extended von Karman s (17) work on turbulent flow friction factors to include the power law non-Newtonian fluids. The following implicit expression for the friction factor was derived in terms of the Metzner-Reed modified Reynolds number and the power law index ... 

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