Ind. Eng. Chem. Res. 1994,33, 2253-2258
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Two-Score Years of the Metzner-Otto Correlation Deepak Doraiswamy,’ Richard K. Grenville, and Arthur W. Etchells I11 E. I. du Pont de Nemours and Company, Wilmington, Delaware 19880-0323 Work over the past four decades clearly indicates that the Metzner-Otto approach has been the single major practical technique for incorporating non-Newtonian effects in a variety of processes. It has mainly been used in the field of mixing for the correlation of results for non-Newtonian fluids relating to power requirement, heat transfer, and blend time. As anticipated, it has been most successful in situations where the process of interest is controlled by the fluid motion in the vicinity of the impeller. A major application has been in the development of unique rheometers for systems which are not readily handled by conventional devices. T h e role of viscoelasticity has not been definitively established and is an area for future work.
Introduction The most useful results are often the consequence of the simplest assumptions that capture the essential physics of a situation. An excellent illustration of this approach is the classic work of Metzner and Otto, begun in 1953and published in 1957, which was one of the first major contributions to address the power requirements for mixing shear thinning fluids. The correlation provides an engineering basis for design of mixing processes based on specifictransport related criteria. In reviewing the impact of this work, we have tried to be as exhaustive as possible in our literature search (and well over 200 references to this classical work were found in the Science Citation Index). However, the emphasis has been less on a comprehensive listing of citations to this work than on its physical implications. Thus we have attempted essentially to assess the impact this correlation has had in the field of mixing, establishing guidelines for proper application of this rule and providing directions for future work. The basic assumption of the Metzner-Otto work is that the fluid motion in the vicinity of the impeller can be characterized by an average shear rate, i.,which is linearly related to the rotational speed, N, of the impeller: +=BN where B is a constant independent of the tank-to-impeller diameter ratio and dependent only on the impeller geometry (e.g., B = 13 for flat-blade turbine impellers). This representative shear rate enables estimation of the effective viscosity for power prediction in inelastic nonNewtonian fluids. Earlier, Magnusson (1953) had indicated that the apparent viscosity of non-Newtonian fluids could be determined from the power number curves for a Newtonian fluid, but presented no method whereby such results could be used for engineering design. The values of B, as determined by Metzner and Otto (1957), were relatively constant for arange of impeller speeds and fluid properties, and an average value could be used in the prediction of non-Newtonian power consumption data. Suspensions of clay and (carboxymethy1)cellulosewere used in this study which enabled a variation in flow behavior indices from 0.2 to 0.4. Flat-bladed turbines were used in both unbaffled and baffled vessels;the impeller diameters were varied from 5 to 20 cm and the tank diameters from 15 to 56 cm. Measurements are usually made of the torque-speed characteristics of non-Newtonian fluids and the analogous power number-Reynolds number characteristics for Newtonian fluids. The Reynolds, Re, and power, Po, numbers
are defined as follows: Re = pD2N/p
P o = P/(pPDs) = 2irNT/(pPD6)
(3)
At a specified rotational speed, measurement of the torque enables calculation of the power number for the nonNewtonian fluid. The corresponding Reynolds number can be extracted from the Newtonian curve. This enables estimation of an apparent viscosity from the definition of the Reynolds number. The shear rate, i.,corresponding to this apparent viscosity is determined from the rheological curve for the non-Newtonian fluid and the mixer shear rate constant, B, is then determined from eq 1. The Metzner-Otto correlation has been extensivelycited in reviews as the preferred method to characterize the average shear rate of non-Newtonian fluids (e.g., Bourne, 1964; Nienow and Elson, 1988; Silvester, 1985; Su and Holland, 1968). It is routinely recommended as a standard procedure in mixing textbooks (e.g., Harnby et al.,1992). Various numerical analyses have confirmed or extended the validity of the concept of average shear rate which varies linearly with impeller speed (e.g., Carreau, et al., 1993; Hiraoka et al., 1979; Biardi et al., 1976; Williams, 1979). Many workers have measured the velocity profile of the discharge from various impellers in low viscosity fluids. Dyster et al. (1993) have extended these studies to the laminar region and shown that the mean velocity profile can be nondimensionalized with the impeller tip speed. This implies that the local velocity gradient in the impeller region varies linearly with the impeller speed and serves to further validate the Metzner-Otto average shear rate concept. Applications of the method have ranged from viscosity estimation for sizing relief vent valve area in polymerization reactors (Boyle, 1967) to estimating striation thicknesses in micromixing models in polmerization reactors (Antiquillah and Nauman, 1990) to providing a basis for calculation of shear rates of bubble column reactors (Allen and Robinson, 1991). The work has become a cornerstone of the field as indicated by its inclusion in the Soviet standards for design and selection of mixing equipment (Ushakov, 1975).
Applications Power Requirements. The Metzner-Otto equation was originally developed to enable power estimation for agitation of non-Newtonian fluids using flat-blade turbines from Newtonian data. Various modifications have been
OSSS-5SS5/94/2633-2253$04.50/0 0 1994 American Chemical Society
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suggested to incorporate effects of geometry and fluid rheology. Values of the Metzner-Otto constant, B , in various mixers have been tabulated extensively in the literature (e.g., Blasinski and Rzyski, 1976, 1978). It is worth noting that while Metzner and Otto did not work with close-clearanceimpellers, such as helical ribbons, screws, or anchors, they predicted that as the impeller diameter approached the tank diameter an effect of clearance on B would be found. This was confirmed for anchor impellers (Beckner and Smith, 1966)and for helical ribbon impellers (Ayazi-Shamlouand Edwards, 1985).For the latter,
B = 34 - 144(c/D)
(4)
The Metzner-Otto concept is a useful approach to the formulation of conditions which permit approximate hydrodynamic similarity on scale-up although it might not express the rheological properties of studied materials exactly (Wein et al., 1972). For most non-Newtonian fluids, a nontrivial scaling theory is not possible (Astarita, 1979). For purely mechanical constitutive equations, it can be shown that only two dimensional parameters are needed (although the number of dimensionless parameters may be unbounded). This situation may not exist except in simple circumstances which involve just one dimensionless group. An implication is that if one assumes that afluid does not exactly obey power law behavior, the "shear rate range" must be the same (this is equivalent to introducing a second parameter after the viscosity). Alternatively, one may assume power law behavior at all shear rates, which is never the case. The role of fluid rheology is discussed in more detail in the following section. It is worth noting here that, although most laboratory scale tests are carried out under laminar flow conditions, scale-up usually results in a change to a transitional or turbulent flow regime. Failure to realize this effect can result in misapplication of the Metzner-Otto rule. It cannot be overemphasized that the "average shear rate" concept is only applicable in the laminar region and the apparent value in the turbulent region would be significantly higher. This has been highlighted by a number of experimental and numerical studies (e.g., Fajner et al., 1982; Van't Riet and Smith, 1975; Kamiwano et al., 1990; Zeppenfield and Mersmann, 1988; Nagata et al., 1971; Pollard and Kantyka, 1969). The local energy dissipation can be written (in scalar form) in terms of the mean and the fluctuating components of these velocities: i = (p/p)[d(U+ u')/dxI2
(5)
This implies that the shear rate is proportional to the square root of the local energy dissipation rate (or power per unit mass) in the fully turbulent region. This relationship has been confirmed experimentally by Wichterle et al. (1985),who used an electrochemical technique to measure the shear rate on the blades of Rushton turbine impellers over a wide range of Reynolds numbers. Their correlation for the shear rate is
For a Newtonian fluid (n = l ) , this yields
(7) whichis to be expected theoretically. Hocker et al. (1981) and Bourne et al. (1981) have also proposed expressions which relate the shear rate in a transitional or turbulent fluid to the power input. This result is important in the
design of shear-sensitive mixing systems since the contribution of turbulence to shear is at least 1 order of magnitude greater than the contribution from the velocity profile. Thus estimating the shear rate in turbulent flow based on the Metzner-Otto concept will seriously underestimate its true value (as in the method presented by Bowen (1986)). Rheological Characterization and Role of Fluid Rheology. The Metzner-Otto definition of apparent viscosity has played a significant role in the rheological characterization of a variety of systems which might not be easily analyzed using conventional viscometric techniques. Shear stress vs shear rate data can be obtained from torque-impeller speed meaurements using agitators with complex geometries if it is assumed that the shear rate is directly proportional to the rotational speed of the agitator and independent of the rheological properties of the fluid. A significant advantage is that it enables the use of mixer-type rheometers to analyze systems which could not be handled otherwise because of special problems (e.g., settling, clogging). It has been used to define the rheological properties of fermentation broths and biological systems (e.g., Roels et al., 1974; Kemblowski and Kristiansen, 1986), foods (e.g., Mackey et al., 1987; Rao and Cooley, 19851, and slurries (e.g., Kernblowski et al., 1989; Sestak et al., 1982). It has even been used to characterize the rheology of concrete (Banfill, 1990) for product development and quality control. For dilatant fluids, the high shear rate near the impeller tends to be transmitted some distance away from the impeller; consequently, the impeller-to-tank diameter becomes an important influence upon the mean shear rate. Calderbank and Moo-Young (1959; 1961) tried to incorporate this effect using the following form of the MetznerOtto constant: = 38N(D/Tlo.'
Many workers have suggested that the Metzner-Otto constant might depend on the power law index (Calderbank and Moo-Young,l961;Beckner and Smith, 1966;Yap et al., 1979; Brito et al., 1991; Castellperez and Steffe, 1990) especially for dilatant fluids (Nienow and Elson, 1988). The recent results of Carreau et al. (1993) suggest that shear thinning influences the change in flow regime from laminar to transitional or turbulent rather than the effective shear rate. For fluids which are not strongly shears thinning (n > 0.25), the power consumption does not deviate from the corresponding Newtonian curve until the early transition regime; this behavior has also been reported by Allen and Robinson (1990). For highly shear thinning fluids (n < 0.2), on the other hand, the power number-Reynolds number curve shows shows a delayed transition corresponding to lower power consumption. There has been considerable ambiguity about the role of viscoelasticity on power consumption. In the laminar region, the role of fluid elasticity has not been definitively established, although some studies suggest that elastic effects are dominated by viscous effects and therefore play a negligible role (e.g., Ulbrect, 1974). I t is often suggested that the viscoelastic effects are expected to become less significant on scale-up. This is because the Deborah number (which is the ratio of the fluid relaxation time to the process time) generally decreases on scale-up in view of the increased process time (e.g., process time 1/N D for constant tip speed and process time D2J3for constant powerlunit volume). The presence of elasticity appears to result in lower values of the Metzner-Otto constant as suggested by Ducla et al. (1983). The recent
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Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2255 studies of Carreau (1993)on helical ribbon agitators suggest that the power number for shear-independent viscoelastic (Boger) fluids is represented by the Newtonian curve only for very low values of the Reynolds and Weissenberg numbers (ratioof elastic to inertial forces). With increasing Re, the power number for viscoelastic fluids is much larger than for Newtonian fluids. In view of the opposing effects of shear thinning and viscoelasticity, the role of viscoelasticity is less pronounced when both factors are involved. In general, viscoelasticity is a strong function of the flow geometry and this makes even qualitative extrapolation from one one geometry to another difficult (Ulbrecht and Carreau, 1985). It must be pointed out that lower power consumptions are to be expected in the turbulent regime because of dragreducing effects (Mashelkar et al., 1975;Kale et al., 1974; Kelkar et al., 1972). In general, therefore, it appears that the Metzner-Otto approach would result in overprediction of the power consumption in such systems in the turbulent regime (‘or which it is not intended) although there would be a considerable effect of mixer geometry. Blending. Blending is one of many processes in which the rate is governed by the fluid viscosity in laminar flow. In fully turbulent flow (typically, Re > lo4), the dimensionless blend time, NO, is a constant independent of Re. As viscosity increases and the flow regime becomes transitional (typically, 10 C Re C lo4)the dimensionless blend time becomes inversely proportional to Reynolds number (see, for example, Hoogendoorn and Den Hartog, 1967; Shiue and Wong , 1984):
Thus the blend time is proportional to the viscosity and the accuracy of blend time prediction is determined by the accuracy of the viscosity measurement. Several workers have studied blending in non-Newtonian fluids, but none have satisfactorily correlated their data with blend times measured in Newtonian fluids. Godleski and Smith (1962) measured blend times in shear thinning fluids and used the Metzner-Otto shear rate to estimate the viscosity of the fluids. They found that blend times were 10-50 times longer than predicted by an earlier correlation proposed by Norwood and Metzner (1960) for Newtonian fluids. Moo-Young et al. (1972) measured power and blend times, used the Calderbank and Moo-Young (1961) definition of shear rate to calculate Reynolds numbers, and plotted power numbers and NO against it for the impellers studied. They found that the Newtonian and shear thinning power data correlated well but the blend time data diverged, with the blend times being longer in the shear thinning fluids a t a given Reynolds number. The Metzner-Otto method gives a shear rate near the impeller where, in a shear thinning fluid, the viscosity is low and blending rates are fast. The viscosity will be higher near the wall of the vessel where the shear rates are low, and the blend time for the whole vessel will be determined by the blending in this region. Landau et al. (1963) measured blend times in various regions of the vessel in Newtonian fluids using a conductivity technique and found that blend times are longer near the wall than in the impeller region. Following this analysis, Grenville (1992) defined a shear rate at the wall using a torque balance and obtained good agreement between data taken in Newtonian and shear thinning fluids. Several workers have defined a lower limit for Reynolds number below which the performance of “nonproximity” impellers is drastically reduced by the viscosity of the fluid
and “proximity” impellers such as helical ribbons should be used. (The terms proximity and nonproximity agitators are often used to describe close clearance (anchor-type systems) and wide clearance (turbine-type systems), reepectively.) The value for Reynolds number at this limit is about 200 (Metzner et al., 1961; Hoogendoorn and Den Hartog, 1967). Johnson (1967) found that the dimensionless mixing time is inversely proportional to Re13 in the range 10 C Re C 100 for Rushton turbines. In general, it would appear therefore that the Metzner-Otto shear rate has not been used successfullyin the correlation of blending data for turbine-type impellers. Many workers have found that the dimensionless mixing time is a constant, independent of viscosity, for close proximity impellers in the laminar regime (Re C 10). These include Johnson (1967), Hoogendoorn and Den Hartog (19671, and Coyle et al. (1970). The recommendations that can be drawn from the literature on blending are the following: 1.Turbines should be used for Re > 200. For Re C 200 a close proximity impeller should be used. There may be situations where a turbine can be used in the laminar regime, but the blend time will be very long. 2. In a baffled vessel operating in the turbulent regime, the dimensionless blend time is a constant independent of viscosity. In the laminar regime the dimensionless blend time is a constant provided a proximity impeller, such as a helical ribbon, is used. If a nonproximity impeller, such as a turbine, is used, the dimensionless blend time is a strong function of viscosity. Heat Transfer. A region of frequent application of the Metzner-Otto principle is in the area of heat transfer to stirred vessels. The work can be roughly divided into the following categories. 1.Close-clearanceimpellers in laminar flow: jacket only. 2. Turbine impellers with jackets: laminar flow; transitional flow; turbulent flow. 3. Turbine impellers with coils: laminar flow; transitional flow; turbulent flow. Good general reviews of heat transfer for Newtonian fluids are given by Penney (1983) and Bondy and Lippa (1983) among others. Edwards and Wilksinson (1972) review both Newtonian and non-Newtonian conditions. The basic correlations obtained for Newtonian flow are given in the Seder-Tate form: Nu = K,ReaPrbRc
(10)
whereK1 is a function of geometry, clearance, and impeller type; the parameter R (=PI& is a viscosity correction factor. In the Nusselt number, the characteristic dimension depends on the geometry. For jackets it is the tank diameter while for coils it is the coil diameter. (a) Close-ClearanceImpellers: Newtonian Fluids. The two most studied close-clearanceimpellers are anchors and helical ribbons because these are commonly used for general mixing duties in viscous systems. In general such impellers are used for highly viscous liquids so that the entire contents of the vessel are in laminar flow. The extensive work of Uhl (e.g., Uhl(1970) and Harry and Uhl (1973)), which is not easily obtained, is summarized by Penney (1983). For Reynolds numbers less than about 12 the coefficients are a = 113
b = 113
c = 0.14
The correlation shows no effect of bulk viscosity as the viscosity terms in the Reynolds number and Prandtl number cancel out. Close-clearance impellers transfer heat by a surface renewal mechanism of bulk motion which
2256 Ind. Eng. Chem. Res., Vol. 33,No. 10, 1994
does not depend on the local viscosity. Effectively, material at the wall is heated and removed by the turning impellers and blended in with the bulk while fresh bulk material is placed near the wall for heating. Although viscosity does not affect heat transfer, the power required to turn the impeller is still proportional to viscosity and the wall viscosity correction ( R ) can be quite significant. In the transition regime, for 12 < Re < 100, a = 1/2
and at higher Reynolds numbers, a = 213 Thus for close-clearance impellers the effect of nonNewtonian viscosities is only important in transitional and turbulent flow where the viscosities are usually low. (b) Close-Clearance Impellers: Non-Newtonian Fluids, Heinlein and Sandell (1972) , Kuriyama et al. (1983) , and others found that using a Calderbank-MooYoung form for the Metzner-Otto coefficient gave good correlation of heat transfer in the transition region for a wide range of shear thinning and yield stress materials. (c) Turbine Impellers: Newtonian Fluids. Researchers studying turbine systems get a variety of coefficients on the Reynolds number and Prandtl number. Typically, a = 0.56-0.66 with 0.66 most common b = 0.3-0.4 with 0.33 most common c = 0.14-0.5 with 0.14 most common
The variety of coefficients may be due to the wide range of Reynolds numbers used including data in the transition regime. Unlike the work on close-clearanceimpellers,there is no generally agreed or recognized transition between turbulent and fully laminar flow regimes. The greatest variety and differences come in describing the effects of geometry. The inclusion of the viscosity ratio was originally taken from pipe experiments to describe local properties near the wall in contrast to the bulk properties that go into the Reynolds and Prandtl numbers. It seems to work and is logical. The turbine correlations are of the form h
-
k-1f3
(11)
Thus even in fully developed turbulent flow the heat transfer coefficient is sensitive to the viscosity of the fluid because this process is taking place near the wall where the viscous film plays an important part. (d) Turbine Impellers: Non-Newtonian Fluids. With non-Newtonian fluids it becomes very important to know which shear rate to use to calculate the effective viscosity in the correlations. The shear rate of interest will be that local to the heat transfer surface whether coil or jacket. While the MetznerOtto correlation was strictly developed for power measurements, it has often been used to estimate shear rates near heat transfer surfaces. A number of researchers have attacked the problem by using a non-Newtonian Reynolds number which explicitly contains the shear thinning index. Skelland and Dimmick (19691, Edwards and Edney (19761, and Desplanches et al. (1980)developed correlations for propellors and various turbines with a close helical coil and found that within experimental error shear thinning fluids followed the same relation as Newtonian fluids using an apparent viscosity
based on the Metzner Otto shear rate. (Reynolds number range 300-20 000). A few workers looking a t jacket heat transfer have tried to define a shear rate at the wall but in the end used a shear rate defined by power measurements and therefore only accurate near the impellers. Desplanches et al. (1982) show significant discrepancies in using this method. Steiff et al. (1981)looked at the effect of multiple phases (liquidliquid, solid-liquid and gas-liquid) and used the impeller shear rate as needed. Close-clearance impeller systems seem to be well understood. A topic occasionally mentioned and worth further study is the effect of poor bulk mixing. At very high viscosities the bulk takes a long time to come to thermal uniformity and the gradients would be expected to adversely affect heat transfer. Ayazi-Shamlou and Edwards (1986) mention this effect. The bulk mixing of anchors is much poorer than of helical ribbons. The correlations however show little difference. In the turbine systems there is still much confusion particularly with jackets where the Metzner-Otto shear rate does a poor job predicting local shear rates. It would be expected that some of the newer local shear rate forms (e.g., Grenville, 1992) being developed for mixing time studies would also work for heat transfer. In addition more study of the transition to laminar flow near the heat transfer surface would be useful. Mass Transfer. Several researchers have tried to use Metzner-Otto concepts with mass transfer from gases and solids to viscous liquids. (a) Gas-Liquid. Gas-liquid mixing in viscous nonNewtonian fluids is a growing area of interest driven primarily by biotechnology businesses involving growth of microorganisms in fermenters. Transport of oxygen and carbon dioxide across the fluid is often detemined by the viscosity of the fermentation broth because of its effect on the mass transfer coefficient. Mass transfer takes place at the surface of the gas bubbles and liquid phase mixing is required to transport the dissolved gas species into the bulk of the liquid. A major issue is the location of the rate-limiting step and the viscosity in this region. Herbst et al. (1992) have measured oxygen mass transfer rates in xanthan fermentation and compared their results with those predicted by several correlations for mass transfer coefficients from the literature using different definitions of the shear rate such as the Metzner-Otto correlation. They found great discrepancies between their measured values and the predictions and concluded there is scope for considerable work in this area. Henzler (1980) and HBcker et al. (1980) used a representative viscosity based on a Metzner-Otto constant of 11.5 in their Schmidt and Reynolds numbers and found good agreement with Newtonian data. This is surprising and may be due to the weak functionality of the viscosity where discrepancies might be lost in the scatter of the data. (b) Liquid-Solid. Rai et al. (1989) review many alternate forms of local shear rate to the Metzner-Otto form. They find the Metzner-Otto relation correlates the data as well as the more complex forms though none is ideal.
Conclusions and Recommendations for Future Work In summary the Metzner-Otto principle works in the following situations. 1.Close-clearance impellers for low Reynolds numbers (
