Non-Newtonian flow behaviour in narrow annular gap reactors

Chemical Engineering Science 56 (2001) 3347–3363

www.elsevier.nl/locate/ces

Non-Newtonian "ow behaviour in narrow annular gap reactors M. Stranzingera; ∗ , K. Feiglb , E. Windhaba a Swiss

Federal Institute of Technology Zurich, Institut fur Lebensmittelwissenschaften, LFO E 12.1, ETH Zentrum, CH-8092 Zurich, Switzerland b Michigan Technological University, Department of Mathematical Sciences, 1400 Townsend Drive, 311 Fisher Hall, Houghton, MI 49931-1295, USA Received 26 May 2000; received in revised form 18 October 2000; accepted 24 October 2000

Abstract In this paper local "ow investigations under isothermal conditions have been established for a narrow annular gap reactor (NAGR, given by a rotor=stator system with a radius ratio of ri =ro = 0:8) including two wall scraper blades of di7erent geometry. Two-dimensional laminar "ow 9elds are considered (with Reynolds numbers below Re ¡ 82:6), based on numerical "ow simulations, where validations with experimental velocity measurements are in good agreement. Comparisons of the macroscopic "ow structuring behaviour are shown for Newtonian and inelastic shear-thinning "uids based on velocity pro9les, secondary "ows, hydrodynamic pressure contours, shear stress and energy dissipation, by varying the rotor velocity (described in terms of a characteristic rotational Reynolds number Re), the scraper blade angle ( ) and the non-Newtonian "ow behaviour (power-law exponent n). ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fluid mechanics; Simulation; Imaging; Numerical analysis; Laminar "ow; Non-Newtonian "uids; Scraped surface heat exchanger (SSHE); Narrow annular gap reactor (NAGR)

1. Introduction Annular gaps are the basis of a variety of technical apparatus, such as scraped surface heat exchangers (SSHE), shear crystallizers (SC) and colloid mills, used in the processing of multiphase "uid systems. In general, the inner cylinder or rotor moves together with scraper blades, pins or other tools attached to the rotor and, for continuous processes, an axial "ow is additionally superimposed. This leads to a biaxial deformation of the "uid. The outer cylinder, or stator, is stationary and is usually cooled or heated. In particular, for crystallizing processes the scraper blades remove crystal nuclei (e.g. fat-containing food systems) from the stator and thus keep the stator wall clean. The "uids mainly treated in such apparatus are multiphase systems which undergo combined mechanical and thermal treatment (tempering, dispersing, mixing and=or phase transition processes). The "ow behaviour of these "uid systems is generally non-Newtonian. The ∗ Corresponding author. Present address: Sandrainstr. 83, CH-3007 Bern, Switzerland. Tel.: +41-031-371-3351. E-mail address: [email protected] (M. Stranzinger).

width of the gaps used in industry and considered here is small (so-called narrow gaps with rotor=stator ratios of ri =ro ¿0:8) in order to generate the high shear stresses necessary for eCcient e7ects in dispersing, heat transfer and=or phase transition (e.g. shear crystallization). For these narrow gap geometries, local values of the velocity 9eld, pressure values, shear stresses, energy dissipation and other physical properties are generally not measurable. Research in the area of SSHE has been primarily concerned with the experimental investigation of heat transfer and the e7ect of various quantities on the heat transfer coeCcient. The in"uence of controlled experimental quantities such as "ow rate, mean residence times, rotational speed, mean input and output temperature, number of scraper blades and the gap width have been studied, as well as the viscosity and other properties of the "uid (Huggins, 1931; Houlton, 1944; Skelland, 1958; Trommelen & Beek, 1971a,b; Trommelen, Beek, & Van de Westelaken, 1971; Weisser, 1972; Maingonnat & Corrieu, 1983; Yamamoto, Itoh, Taneya, & Sogo, 1987; HIarrIod, 1987,

0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 0 3 9 - 2

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1988; de Goede, 1988; Alcairo & Zuritz, 1990; Lee & Singh, 1993; Russell, Burmester, & Winch, 1997). Attention has also been given to the "ow patterns observed in SSHE, particularly the transition from laminar "ow to vortical "ow. The conditions leading to this transition and the e7ect of vortical "ow on mixing have been studied (Sykora, Navratil, & Karasek, 1968; Trommelen & Beek, 1971a,b; Weisser, 1972; HIarrIod, 1987, 1988, 1990). On the theoretical side, researchers have developed models for predicting heat transfer based on data measured during the experiment (Skelland, 1958; Trommelen & Beek, 1971a,b; Weisser, 1972; Trommelen et al., 1971; Cuevas, Cheryan, & Porter, 1982; Yamamoto et al., 1987; HIarrIod, 1988; de Goede, 1988). In practice, it is only possible to determine integral (or average) values, such as power or energy input, mean input and output temperature, and average residence times. There is a wide 9eld of “structured” (macromolecular and=or dispersed) multiphase "uid systems (Windhab, 2000a) for which knowledge of the local "ow behaviour is important for the optimization of combined mechanical treatment (generation of multiphase structures, Windhab, 2000b) and heat treatment (microbial safety; aroma reactions). To collect local or discrete properties of SSHE process "ows, the key approaches presently are combinations of analytical and modelling methods, non-invasive experimental methods which do not in"uence the process "ow behaviour and, increasingly, two- and three-dimensional numerical simulations. In a series of papers comprising his Ph.D. dissertation in 1988, HIarrIod studied SSHE using both a Newtonian "uid (water) and non-Newtonian, inelastic "uids (starch pastes). One of the key purposes of the papers was to model the heat transfer for both laminar and vortical rotational "ow. Non-linear regression techniques were used to 9t the parameters in the model, and comparisons between model predictions and measured data were made. The e7ect of the following quantities were investigated: "ow rate, rotational speed, output temperature, number of blades, radius ratio, heating or cooling, heat transfer direction, and the "ow properties. In addition, the transition between laminar and vortical "ow was examined. In the same year, de Goede studied the heat transfer properties of a SSHE for paraxylene and developed a theoretical model for the heat transfer coeCcient. In relation to this work, the most notable aspect of de Goede’s work was his two-dimensional numerical simulations of the bench-scale SSHE. He solved the Navier–Stokes equations for Newtonian "ow using the PHOENICS computer program (Markatos, Tatchell, Cross, & Rhodes, 1986). The calculated velocity 9eld showed the formation of a vortex between the scrapers. However, his calculations were restricted to Newtonian "ows, with a simpli9ed SSHE geometry.

Other numerical simulations of processes involving SSHE include the work of SkjIoldebrand and Ohlsson (1993), who have calculated the time-dependent temperature of "uids and particles for viscous "uids containing discrete solid particles. Corbett, Phillips, Kauten, and McCarthy (1995) reported a non-invasive technique of nuclear magnetic resonance imaging (MRI) to investigate concentration and velocity pro9les of pure "uids and solid suspensions in rotating geometries. Choosing coaxial rotating cylinders, as well as coaxial cylinders in which a straight "ight rotates with the inner cylinder and a single screw extruder, Corbett et al. (1995), showed that concentration pro9les in"uence the particle migration from high shear to low shear regions in the concentric cylinder apparatus and the extruder. No concentration gradients across the gap in the straight-"ight cylinder were exhibited, indicating the importance of mixing in that geometry and thus the need for investigations of the local "ow behaviour. The work of Corbett et al. (1995), is restricted to a Newtonian "uid, where only a single scraper blade incidence in the radial direction of the annulus was used to model the SSHE "ow behaviour. Furthermore, no literature has been found, investigating the local annular gap "ow behaviour, comparing Newtonian and non-Newtonian "uids. The work presented here focuses on a narrow annular gap reactor (NAGR, developed at the Laboratory of Food Process Engineering at ETH ZIurich) as a new reactor device applied to crystallization processes of fat-containing food systems. The aims of narrow mechanical stress and “thin layer” heat and mass transfer are approached, focusing on the "ow pattern investigations of Newtonian and inelastic non-Newtonian "ows (modelled with power-law exponents of n¿0:65) generated in such NAGR devices. Thus, in a 9rst approach "ow investigations inside a narrow annular gap reactor (NAGR) are performed. Two-dimensional numerical simulations are used in the investigations with experiments being used to verify the calculated "ow 9elds. Fig. 1 shows a characteristic two-dimensional axial cross-sectional plane (2D-model) of a NAGR geometry. The rotor (inner cylinder) moves clockwise at a constant rotational velocity ! together with the scraper blades (attachments to rotor omitted in the 2D-model). The scraper blade angle (which in"uences the "ow incidence) is de9ned as the angle between the gap edge (close to the inner cylinder) and the leading edge of the scraper blade (de9ned with respect to the rotational direction of !). The angle ◦ ◦ has been varied from = 30 to 150 (bottom picture in Fig. 1). The ratio of the width of the gap between the inner cylinder and scraper blade hgap and the annular gap width h0 is rs = hgap =h0 = 0:25. The radius ratio of the inner cylinder (rotor, radius ri ) to the outer cylinder (stator, radius ro ) ri =ro is 0.8 (“narrow gap”). Because of the two-dimensional "ow investigations in the NAGR, three-dimensional e7ects like Taylor

M. Stranzinger et al. / Chemical Engineering Science 56 (2001) 3347–3363

Fig. 1. A horizontal axial cross-section geometry model of a narrow annular gap reactor (NAGR). The top picture depicts the rotor=stator con9guration (annular gap) including two scraper blades. The bottom picture indicates the variation of the scraper blade angle ("ow ◦ ◦ incidence 30 6 6150 ) at a constant scraper blade gap ratio of rs = 0:25.

vortices or turbulence cannot be considered. To ensure simulations in the Taylor vortex free and laminar "ow regime, which is of main interest for applications at high "uid viscosities, the critical rotational Reynolds number of Recrit = 82:6 (based on Taylor’s theory, (Taylor, 1923, assuming a Taylor–Couette "ow with a characteristic annular gap width h0 ) was not exceeded. The de9nitions for the Reynolds and Taylor number characterizing the NAGR model "ow process are given in Eqs. (1) and (2), respectively 0 u0 h0 ; (1) Reh = 0


Tah = Reh

h0 : ri

(2)

The characteristic parameters (index 0) of the Reynolds and Taylor numbers are the density of the "uid 0 , the circumferential speed of the rotor u0 = !ri at the rotor surface with radius ri , the annular gap width h0 between the inner cylinder (radius ri ) and the outer cylinder (radius ro ) and 0 , the characteristic viscosity of the "uid (i.e. the zero-shear-rate viscosity of the shear-thinning model "uids). For the remainder of this paper the characteristic rotational Reynolds number Reh is referred to as Re. As the numerical tool a 9nite volume code (FVM) has been used, which is capable of solving steady and unsteady "ow problems for incompressible Newtonian and

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non-Newtonian "uids (without elastic properties) in twoand three-dimensional space. The program (FVM code) has been successfully used in various applications (Dandy & Dwyer, 1990; Nirschl, Dwyer, & Denk, 1993; Kerschl, Dwyer, Nirschl, & Denk, 1993; Denk, Dwyer, & Nirschl, 1993; Nirschl, 1994; Delgado & Nirschl, 1997; Hartmann & Delgado, 2000). The experimental "ow visualizations and velocity measurements were performed with a digital-particle image velocimetry method (D-PIV) initially developed by Herrmann (Herrmann, Tylli, & Kaiktsis, 1998). Herrmann successfully applied the D-PIV method to a backward-facing step "ow within the laminar "ow regime, obtaining good agreement between the measured velocity 9eld and calculated two-dimensional velocity 9elds. Based on the collected "ow 9elds within narrow annular gap reactors (NAGR), for various process parameters Re; and n, the macroscopic "ow structuring mechanisms of Newtonian and shear-thinning viscous "uid systems are discussed. These include the hydrodynamic pressure, secondary "ows, non-Newtonian viscosity, shear stress and energy dissipation. 2. Materials and methods 2.1. Rheological model One main goal of this work is the investigation of the rheological response (e.g. shear stress  and viscosity  as a function of the shear rate ) ˙ of process "uids (e.g. milk-fat and chocolate "ows under precrystallization conditions, see Breitschuh (1998) and Bolliger, Breitschuh, Stranzinger, Wagner, and Windhab (1998), respectively, and shear e7ects on whey protein gels by WalkenstrIom and Hermannsson (1998) and (WalkenstrIom, Windhab, & Hermannsson 1998a; WalkenstrIom, Panighetti, Windhab, & Hermannsson, 1998b) which show Newtonian and non-Newtonian "ow behaviour during treatment inside NAGR. To enable the "ow investigations based on a numerical approach (see below), a rheological model is necessary which simulates the Newtonian and inelastic non-Newtonian "ow behaviour under time-independent and isothermal conditions. The empirical 9ve-parameter model of Carreau and Yasuda (CYL) has been chosen to consider inelastic non-Newtonian "uids, as described in Table 1. The CYL model is most "exible in comparison to other rheological models (e.g. power-law model). Unlike the power-law model, the CYL model gives better performance in the transition regions of the viscosity curves and describes both upper and lower viscosity plateaus. The established rheological measurements for the food process "uids of interest (e.g. milk-fat, see Breitschuh, 1998) did not indicate signi9cant yield values with shear rates approaching zero. Therefore, yield stress models

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Table 1 Carreau–Yasuda law model (CYL) describing the viscosity  as a function of the shear rate ˙ under isothermal conditions (Bird et al., 1987) Equation −∞ 0 −∞

= [1 +

Parameters () ˙ a ](n−1)=a

0 ∞ a

the zero-shear-rate viscosity (Pa s) the in9nite-shear-rate viscosity (Pa s) dimensionless parameter, describing the transition region between zero-shear-rate viscosity and power-law region (dimensionless) the power-law exponent (dimensionless) a time constant, describing the onset of the shear-thinning region in the viscosity curve (s)

n 

were not considered for the annular gap "ow investigations. The CYL model reduces to the power-law model at higher shear rates (with ∞ = 0), where the consistency index  of the power-law corresponds to 0 n−1 (Bird, Armstrong, & Hassager, 1987). Thus, the power-law exponent n also represents the characteristic "ow behaviour of the CYL model. If n = 1, the model shows Newtonian "ow behaviour. With n ¡ 1, the "uid is shear thinning (decreasing viscosity with increasing shear rate) and with n ¿ 1 the "uid is dilatant or shear thickening (increasing viscosity with increasing shear rate). The upper shear rate viscosity ∞ is small enough to be neglected so that a simpli9ed rheological CYL model which is implemented into the numerical algorithm is given by ˙ a ](n−1)=a : () ˙ = 0 [1 + ()

(3)

Along with having a shear-thinning behaviour with a 9nite 0 , another requirement for our model "uid is that it be transparent. This is necessary because of the non-intrusive optical "ow measurement technique we use. Furthermore, we desire a "uid without signi9cant elastic properties. Fig. 2 shows the storage and loss moduli G  ; G  for 2% and 5.7% of carboxymethyl-cellulose (CMC) in water together with a highly viscoelastic reference "uid, a 0.5% Praestol-30% sugar in water solution, measured with a cone and plate geometry (torque controlled rheometer, DSR, Rheometrics Europe GmbH, Frankfurt, Germany). Stress-sweep tests (varying the stress based on the minimal and maximal torque of the rheometer applicable to the measured "uid) were performed at a frequency of 1 Hz to determine an average stress linear occurring in the linear viscoelastic regime. These stresses linear are indicated in the parentheses of the legend of Fig. 2. All "uids ◦ were measured at a room temperature of 27 C. Under linear viscoelastic conditions of the oscillatory shear measurements, inelastic "uids show higher values of the loss modulus G  (representing the viscous "uid behaviour) compared to the storage modulus G  (representing the elastic "uid behaviour) throughout the frequency range. For the 2% concentration of the CMC sodium

Fig. 2. Oscillatory measurements (cone and plate geometry (DSR)) ◦ at room temperature (27 C) of two carboxymethyl–cellulose–water solutions compared to a highly viscoelastic 0.5% Praestol-30% sugar–water solution, represented by the storage modulus G  and the loss modulus G  as a function of the frequency under linear viscoelastic conditions (applied stress linear of linear regime indicated in parentheses).

in water, G  dominates throughout the investigated frequency sweep range (see Fig. 2). Whereas the aqueous solution of 5.7% CMC and the 0.5% Praestol-30% sugar in water solution depict dominating elastic properties (G  ) with increasing frequency and thus a typical viscoelastic behaviour. As one can see in Fig. 2, the Praestol-sugar in water solution shows a higher storage modulus already at very low frequencies compared to the 5.7% CMC water solution; the elastic properties dominate throughout the whole frequency sweep. As a summary it can be said that the Praestol-sugar in water solution depicts highest viscoelastic behaviour, followed by the 5.7% CMC water solution, whereas the aqueous solutions of CMC with concentrations up to 2% behave mainly inelastically. Based on the viscoelastic comparisons in Fig. 2, the aqueous solutions of polyethylene-glycol (PEG) and carboxymethyl-cellulose (CMC) were chosen as model Newtonian and shear-thinning "uids, respectively. Moreover, these "uids are easy to handle since the solid powders of PEG and CMC disperse, swell and separate completely in water. Using a least-squares 9tting (Microsoft Excel), the parameters for the CYL model in Eq. (3) can be found by

M. Stranzinger et al. / Chemical Engineering Science 56 (2001) 3347–3363

Fig. 3. Shear-thinning behaviour of a 2% wt=wt carboxymethyl– ◦ cellulose (CMC) –water solution at 27 C, depicted with the apparent viscosity and the shear stress as a function of the shear rate. A Carreau–Yasuda law model (CYL) has been 9tted (solid and dashed lines) to the measured data points.

maximizing the con9dence value R2 . Fig. 3 shows a typical shear-thinning behaviour of a CMC–water solution ◦ (2% wt=wt concentration at 27 C). The apparent viscosity  and the shear stress  are depicted as a function of the shear rate , ˙ measured with the DSR rheometer using a Searle type geometry together with the 9tted curves based on Eq. (3). Since the annular gap process operates at room temperature, the rheological models were adjusted with respect to the process temperatures Tprocess measured during the test sequences. Table 2 summarizes the chosen "uid compositions (model "uids) and their parameters (rheological data) found from the 9tting the CYL model (Eq. (3)) to the measured data.

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(Dwyer, 1991; Nirschl et al., 1993). The conservation laws of mass and momentum are solved with an alternating direction implicit (ADI) method (Anderson, Tannehill, & Pletcher, 1984). The physical model assumes steady state, incompressible Newtonian and inelastic non-Newtonian "ow under isothermal conditions (Stranzinger, 1999). In most expected applications of the NAGR principle, the circumferential velocity of the inner cylinder and the scraper blade will be much higher than the axial "ow through the gap in continuous processing. Consequently, the two-dimensional description of the velocity 9eld in the cross-sectional gap area is representative for many applications. Thus, the numerical method is adapted to an axial cross-section of the annular gap process (NAGR) as represented with Fig. 4. The scraper blade, which is attached to the inner cylinder (attachment omitted in the calculations) rotates with the same constant, angular velocity  as the inner cylinder. The outer cylinder, which is scraped by the blades, is stationary. To solve this unsteady problem and to avoid remeshing (see mesh generation of Fig. 4), we perform a coordinate transformation and solve the governing equations (continuity and momentum equation) in non-dimensional form in a rotating coordinate system. In the rotating frame of reference, the "ow is assumed steady. The relationship between the velocity v in the 9xed frame of reference and the velocity w in the rotating frame of reference is given with the transformation rule v = w + ( × r);

2.2. Numerical method A 9nite volume method (FVM) is used as a numerical tool to predict the "ow. The numerical scheme applied is a cell centred FVM for structured quadrilateral grids

(4)

where r is the position vector and  is the rotational velocity. Thus, the momentum equation (5) can be derived in the rotating frame of reference. This coordinate transformation produces additional forces. Besides the

Table 2 Chosen model "uids and 9tted parameters of the Carreau–Yasuda law (CYL) with respect to the measured annular gap process temperature Tprocess Model "uid composition

Parameters (rheological data) ◦

16.7% polyethylene–glycol (PEG) water solution at Tprocess = 17 C (Newtonian "uid without elastic or shear-thinning behaviour within investigated shear rate region) ◦

1% carboxymethyl-cellulose (CMC) water solution at Tprocess = 18 C (shear-thinning "uid)

◦

2% carboxymethyl-cellulose (CMC)water solution at Tprocess = 27 C (shear-thinning "uid)

0 = 0:09 (Pa s) n = 1:0 (dimensionless) 0 = 0:04 (Pa s) ∞ = 0 (Pa s) a = 0:9 (dimensionless)  = 0:017 (s) n = 0:8 (dimensionless) 0 = 0:3166 (Pa s) ∞ = 0 (Pa s) a = 1:01 (dimensionless)  = 0:044 (s) n = 0:6516 (dimensionless)

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frame, respective w in the rotating frame of reference, is non-dimensionalized with the circumferential speed of the rotor u0 . The pressure p is calibrated relative to a characteristic pressure value p0 at the rotor surface and made dimensionless with the stagnation pressure 0 u02 . The stress tensor
(the viscous stress tensor  being part of
) is also made dimensionless with the stagnation pressure x y v p − p0 ; x ∗ = ; y ∗ = ; v ∗ = ; p∗ = h0 h0 u0 0 u02
∗ =

Fig. 4. Mesh and scraper blade node de9nitions of the two-dimensional ◦ annular gap geometry with a scraper blade at an angle = 150 . The mesh lines into the  direction are distributed with a smooth mesh re9nement close to the cylinder walls. The lines into the  direction are distributed with a smooth mesh re9nement within the scraper blade region. A total number of 35 × 372 mesh nodes is used. The nodes in (A) de9ne edge nodes for the scraper blade 9nite volumes. The nodes in (B) correspond to 9nite volume centre nodes and the triangles of (C) de9ne the gap edge corner nodes. These scraper blade nodes are summarized as fringe nodes, which are used for the 9nite volume discretization and the boundary conditions close to the scraper blades (see upper picture).

convective term w · ∇w of Eq. (5), the Coriolis force 2 × w and the centrifugal force  × ( × r) appear, @w + w · ∇w + 2 × w +  × ( × r) @t = − ∇p +

1 2 ∇ w: Re

(5)

The explicit description of the body forces (Coriolis and centrifugal force of Eq. (5)) conserves the de9nition of the hydrodynamical pressure p, as described for the 9xed frame of reference. Thus, the pressure variable p has the same meaning in both frames of reference. Since the numerical results for velocity are compared with experimental data sets found in the 9xed frame of reference, the calculated velocity values are transformed back to the 9xed frame of reference in a postprocessing step. We consider non-dimensional forms of the governing equations. The geometrical and physical properties used for the non-dimensional equations of motion are shown below. The non-dimensional coordinates are given with x∗ and y∗ . The velocity v in the 9xed


; 0 u02

t∗ =

tu0 ; h0

∗

=

˙h0 ; u0

˙∗ =

h ˙ 0 : u0

Further variables are the non-dimensional time t ∗ , the elongational rate ˙∗ and the shear rate ˙∗ . Other non-dimensional properties used in this paper can be derived from the given de9nitions shown above. To generalize the governing equations for our inelastic non-Newtonian model "ow conditions, the dynamic viscosity  becomes a function of the shear rate . ˙ An incompressible generalized Newtonian "uid can be de9ned by replacing the constant viscosity  of Newton’s law with () ˙ (Bird et al., 1987): () ˙ = () ˙ ; ˙

(6)

where the rate-of-strain tensor is ˙ = ∇v + (∇v)T ;

(7)

∇v is the velocity gradient and ˙ is the shear rate de   1

9ned as ˙ = 2 i j ˙ij ˙ji ; the magnitude of the rate-of-strain tensor . ˙ In our investigations the CYL model is used for (), ˙ as given with Eq. (3). Replacing () ˙ of Eq. (6) with the CYL model, the generalized equations of motion can be formulated. In the non-dimensional form of the governing equations, the viscosity () ˙ is made dimensionless by the Reynolds number Re, which becomes a function of the shear rate . ˙ Thus, a non-dimensional CYL model can be de9ned, as ˙ a ](1−n)=a ; Re() ˙ = Rea [1 + ()

(8)

where Rea is the Reynolds number at constant viscosity 0 . In relation to the numerical solution procedure, Eq. (8) describes a local Reynolds number where the shear rate ˙ depends on the mesh discretization. Therefore, in each 9nite volume of the computational mesh, a shear rate and thus the local Reynolds number is found. For our numerical process investigations the rotor velocity ! is adjusted in terms of Rea (given as Re for the remainder of this paper). 2.3. Experimental method To validate the numerical "ow simulations (FVM), experimental "ow visualizations and velocity measurements

M. Stranzinger et al. / Chemical Engineering Science 56 (2001) 3347–3363

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Fig. 5. Sketch of set-up (top picture) of the transparent narrow annular gap reactor (NAGR) and the image acquisition components (laser, lenses and CCD-camera). Top view sketch (bottom picture) of the enlightened scraper blade area, as seen from the CCD camera.

have been performed, using the digital-particle image velocimetry method (D-PIV). Fig. 5 shows the experimental set-up (NAGR) used to visualize the "ow. A continuously working argon-ion laser (6300 mW) produces coherent light in the TEM00 mode. The laser beam of about 1 mm in diameter emitted by the argon-ion laser is captured by two cylindrical lenses to produce a light sheet, which has a de9ned width and focal point inside the annular gap region to visualize the "ow. The 9rst lens has a negative focal point of 100 mm (plane-concave lens), where the light spreads at an angle, and a second lens, which has a focal point of 750 mm (plane-convex lens), directs the light sheet inside the annular gap, where the sheet is focused to a thickness dla of about 0:45 mm and a width wla of about 30 mm. A 12-bit black and white CCD-camera is positioned perpendicular to the light sheet plane underneath the NAGR (see Fig. 5), where observation holes enable a view into the annular gap. The particle images are captured with a macro-zoom lens mounted onto the

CCD-camera. To ensure reproducible two-dimensional "ow images, the light sheet was 9xed at two di7erent axial positions (z-direction), between the lower scraper attachment and the bottom of the scraper (for detailed description, see Stranzinger, 1999). To visualize two-dimensional planar "ow patterns, an axial cross-section laser-light sheet (as shown in the bottom picture of Fig. 5) has been positioned in the centre part of the rotating scraper blades, which enlightens tracer particles following the "ow. The photos are then grabbed by a computer and the velocity is measured using auto correlation analysis of VISIFLOW. Comparisons to the FVM simulations have been established for a constant ◦ scraper blade angle of = 147 by varying Re. The numerical and experimental results are compared, by calibration of the velocity data, based on the given coordinate meshes used in the numerical (FVM) and experimental (D-PIV) method. The experimental velocity data are 9tted onto the numerical grid (bilinear interpolation).

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Fig. 6. Velocity pro9les at a time instant (Re = 10, Newtonian "uid (n = 1)) in an annular gap section (left picture) and in the scraper blade ◦ region (right picture) with a blade angle of = 150 .

3. Results 3.1. Velocity pattern An illustration of the "ow 9eld computed with the FVM program (for a Reynolds number of Re = 10, a ◦ scraper blade angle of = 150 and a Newtonian "uid with a power-law exponent of n = 1) in the 9xed frame of reference (xv ; yv ) is shown in Fig. 6. The velocity pro9les away from the scraper blade region (left picture of Fig. 6) show asymmetric parabolic pro9les relative to the gap width h0 with almost uniform behaviour in azimuthal direction, which indicates a superposition of pure pressure driven "ow (due to moving scraper blades) and shear driven "ow (due to the rotating inner cylinder). A more complex and nonuniform "ow behaviour occurs close to the scraper blades (right picture in Fig. 6). This area is of main interest for the process "ow investigations and related macroscopic "ow structuring behaviour in such "ow 9elds. Comparisons of the velocity magnitude (|v|=vrot [-]) as a function of the relative radial gap position s=smax in a cross-section of the scraper blade region (with s=smax = 0 indicating the stator position with zero velocity and s=smax = 1 the rotor position with constant rotor speed vrot ) using three di7erent mesh sizes are shown in Fig. 7. Using a coarse mesh (with 21 × 172 mesh nodes), a medium mesh (with 35 × 372 mesh nodes) and a 9ne mesh (with 43 × 472 mesh nodes), the re9nement shows good agreement between the medium and the 9ne mesh within 5% tolerance (indicated by the error bars in Fig. 7). Equally good agreements between the medium and 9ne mesh are found for other cross-sections. For all reported results in this work, no signi9cant di7erences were found between the medium and 9ne meshes. Typical cross-section comparisons between the numerical calculations (FVM) and the experimental results

Fig. 7. Comparison of the non-dimensional velocity magnitude |v|=vrot (in the scraper blade region) as a function of the relative radial gap ◦ position s=smax for a Newtonian "ow case (Re = 10; = 147 ; n = 1) using three di7erent computational mesh sizes. The error bars represent ± 5% variations with respect to the numerical results of the medium mesh (35 × 372 nodes).

(D-PIV) on both sides of the scraper blade (using an ◦ incidence of = 147 ) are shown in Fig. 8. D-PIV measurements with the non-Newtonian "uid were established within 1 h to ensure the same shear-thinning "ow behaviour throughout the time of a measurement with constant model "uid parameters (see Table 2). Sample "uids of the NAGR process were rheometrically measured before the D-PIV measurements and thereafter, to ensure reproducible curves for the viscosity as a function of the shear rate (with variations below 5%). Comparisons of a Newtonian "uid (Re = 9:6, top picture in Fig. 8) and a shear-thinning "uid with a power-law exponent of n = 0:8 (Re = 21:1, bottom picture in Fig. 8) are shown. In general, the predicted velocity pro9les of the numerical method (solid lines) agree very well with the experimental measurements (symbols) with deviations below 10%. (For graphical clarity, alternating diamond and asterisk symbols are used to represent

M. Stranzinger et al. / Chemical Engineering Science 56 (2001) 3347–3363

Fig. 8. Cross-section comparisons of numerical predictions (solid lines) and experimental velocity measurements (alternating diamond and asterisk symbols) of the velocity pro9les for a Newtonian "uid (Re = 9:6, upper picture) and a shear-thinning "uid (Re = 21:1, with a power-law exponent of n = 0:8, lower picture) at a blade angle of ◦

= 147 .

the experimental data along neighbouring cross-sections.) Higher local variations of the experimental data are due to the absence of tracer particles, scattering noise or weak light intensity (for a detailed description of the experimental method, see Stranzinger, 1999). A dark zone is given between the gap edge of the scraper blade and the rotor, where no "ow measurements were possible. The same reason caused a lack of information close to the leading edge of the scraper blade. Independent of the Reynolds number and the "uid type the good agreement of these comparisons was reproduced with up to three experimental evaluations. In summary, there is good reproducible agreement between the numerical and experimental velocity 9elds ◦ for an angle of = 147 for both Newtonian and shear-thinning "uids. Independent of the "uid type or the chosen Reynolds number (Re680), the velocity distributions depict similar pro9les (with a rotor velocity (at the inner cylinder, as shown in Fig. 8) normalized to |vrot |=u0 = 1, independent of Re). These small di7erences in the "ow behaviour are con9rmed by comparisons of the secondary mass "ow rates as described below. Further investigations of pressure distributions, apparent viscosity, shear stress and dissipative energy emphasize the di7erences between Newtonian and non-Newtonian "uids treated inside of NAGR (see later). Hence, to give a detailed description of the "ow behaviour for Newtonian and inelastic shear-thinning "uids inside of NAGR, numerical simulations with a computational mesh of 35 × 372 nodes (FVM method) have been established for di7erent annular gap geometries and "ow conditions.

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Fig. 9. Secondary "ow behaviour in the annular gap (with respect to the rotating frame of reference (xw ; yw )) using scraper blades at an ◦ angle of = 150 . The Reynolds number is Re = 10 and the "uid is Newtonian (n = 1).

3.2. Secondary =ow investigations An example of the typical secondary "ow patterns is illustrated by the streamline patterns in Fig. 9. Due to steady-state boundary conditions in the (xw ; yw ) coordinate system, the streamline patterns correspond to the particle tracks which enable us to visualize secondary "ows, such as vortices. In the 9xed frame of reference (xv ; yv ) the streamlines represent a time instant of the "ow dominated by the azimuthal direction. Secondary "ow patterns with respect to the rotating frame of reference (xw ; yw ) developed for every scraper blade angle investigated. Fig. 9 shows an example for a Newtonian "uid (n = 1) ◦ with scraper blades at an angle of = 150 . Since the "ow is periodically formed due to the use of two scraper ◦ blades positioned with a displacement of 180 , the secondary "ow phenomena can be divided into two main parts. Between the scraper blades, closed streamline loops or vortices are formed, which span the whole region between the blades. On the other hand a “secondary "ow exchange” takes place between the leading and the trailing edge of the scraper blades along the outer cylinder wall. A total of 12 streamfunction values are shown in Fig. 9. These values are equidistant with an increment of 9.09% based on a maximum mass "ow rate of 100% with respect to the secondary "ow occurring in the annular gap. Thus, the maximum occurring mass "ow rate of the secondary "ow (in the two-dimensional plane) within the annular gap reduces to about 46% which passes the scraper blade region between the leading and the trailing edge (“secondary "ow exchange”, V gap ), whereas the remaining mass "ow V vortex stays between the scraper blades, depicted with the vortex streamlines of Fig. 9. This mass "ow separation of 46 –54%

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Fig. 11. Pressure contours of a Newtonian "uid (n = 1) using scraper ◦ blades at an angle of = 150 . The Reynolds number is Re = 10.

3.3. Hydrodynamic pressure investigations

Fig. 10. Dimensionless mass "ow rate max (triangles) and non-dimensional scraper blade gap "ow rate V gap (circles) of the secondary "ow (with respect to the rotating frame of reference (xw ; yw )) for Newtonian and shear-thinning "ow (n = 1 and 0.65, respectively), as a function of (at Re = 10, top graph) and of the ◦ Reynolds number with constant incidence angle ( = 150 , bottom graph).

remains about constant independent of Re; and n within the laminar "ow regime (Re682:6). The top graph in Fig. 10 shows the scraper blade angle ( ) dependent maximum non-dimensional mass "ow rate max of the secondary "ow in the annular gap (with respect to the rotating frame of reference (xw ; yw )) and V gap as the remaining secondary "ow through the scraper blade gap width hgap (“secondary "ow exchange”) at a constant rotational Reynolds number of Re = 10. The variations for max and V gap are 1%; 9%; 3% and 9% with respect to ◦ ◦ ◦ ◦

= 30 ; 70 ; 90 and 110 , respectively, based on the ref◦ erence case with an angle of = 150 for the Newtonian "uid with n = 1 (see Fig. 9). No signi9cant changes occur for the shear-thinning "uid at n = 0:65: max and V gap as ◦ a function of Re at = 150 in the bottom graph of Fig. 10 increase with 4%; 9% and 13% at Re of 40; 60 and 80, respectively. No di7erences are found for Re = 20; ◦ again comparing with the Re = 10; = 150 ; n = 1 case as reference. For n = 0:65 the changes are 4%; 9% and 14% at Re of 40; 60 and 80; respectively, again without variations for the Re = 20 case. Nevertheless, the ratio of 46=54% between V gap and V vortex remains constant, independent of Re; or n.

The orientation and strength of the "ow structuring behaviour (i.e. extension and compressional behaviour) can be investigated by looking at the hydrodynamic pressure contours (Stranzinger, 1999). De9ning a range for minimum pressure p0 and maximum pressure p1 which covers the pressure range of all "ow cases investigated, the pressure contours have been calibrated. The annular gap "ow can be separated into pressure and shear driven "ow components. Assuming pure pressure driven "ow, in which the scraper blades push the "ow forward, the contours would be straight lines in the radial direction. Pure azimuthal contours appear in pure shear driven "ow, produced by rotating the inner cylinder without scraper blades, due to centrifugal forces which increase towards the outer cylinder wall. For a Reynolds number of Re = 10 the "ow is mainly caused by pressure driven "ow. As one determines in ◦ the reference case Re = 10; = 150 ; n = 1 of Fig. 11, the contours are primarily oriented in the radial direction. This behaviour was found regardless of the scraper blade angle and "uid rheology. Flows with Reynolds numbers above Re = 10 in contrast, exhibit contours of azimuthal dominance (Stranzinger, 1999). As a summary one can state that pressure is primarily dominated by shear driven "ow at Reynolds numbers of Re ¿ 10; while the pressure driven "ow (due to the motion of the scraper blades) dominates for Reynolds numbers of Re610: Fig. 12 shows the maximum occurring pressure pmax and the pressure ratio pmin =pmax of individual "ow cases as a function of and Re (from top to bottom in Fig. 12). The pressure ratio pmin =pmax (minimum pressure pmin at trailing edge divided by maximum pressure pmax at leading edge, always appearing close to the scraping edge of the blade) remains at a value of about

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ratio pmin =pmax as a function of Re increases by a factor of about 2 (from 0.23 to 0.48 for the shear-thinning "uid), if Re is increased by a factor of 4 (from 20 ◦ to 80) for a scraper blade angle of = 150 . In other words, higher Reynolds numbers reduce the structuring contribution of pressure signi9cantly, since the dimensionless pressure drop 1 − pmin =pmax is reduced by a factor of 2 as the rotor velocity increases from Re = 20 to 80. 3.4. Apparent viscosity and shear stress

Fig. 12. Maximum pressure pmax (triangles) and pressure ratio pmin =pmax (circles) for Newtonian and shear-thinning "ow (n = 1 and 0.65, respectively), as a function of the "ow incidence (at Re = 10, ◦ top graph) and as a function of the Reynolds number Re for = 150 (bottom graph).

0.155 (shear-thinning "uid with n = 0:65) for angles be◦ ◦ tween = 70 and 110 and decreases to values below ◦ 0.14 for angles smaller than = 70 and higher than ◦

= 110 (top graph in Fig. 12 at Re = 10). Consequently, the dimensionless pressure drop, de9ned as 1 − pmin =pmax is highest for the minimum and maximum scraper blade angles investigated. As shown in the top graph of Fig. 12, the behaviour is nearly symmetric or independent of the "ow direction. Instead incidences with angles tending into the azimuthal "ow direction cause higher pressure drops between leading and trailing edge of the scraper blades and thus cause higher structuring forces with respect to "ow pressure. On the other hand, a smoother "ow structuring contribution due to the hydrodynamic pressure is caused with scraper blades within angles between ◦ ◦

= 70 and 110 . As stated before, annular gap "ows with Reynolds numbers above Re = 10 are dominated by the shear driven "ow component. Centrifugal forces dominate the pressure driven "ow (due to the moving scraper blades) above Re = 10 and thus cause an almost radial increase of the pressure towards the outer annular gap cylinder for the highest Reynolds number (Re = 80) considered (Stranzinger, 1999). As demonstrated in the bottom graph of Fig. 12, the pressure

The relation between steady-state viscosity  and shear rate ˙ is given by the CYL model as described with Eq. (3). The shear rate ˙ is found by the magnitude of the rate-of-strain tensor , ˙ as de9ned previously. Fig. 13 shows the local apparent viscosity for the shear-thinning "uid (n = 0:65), at a Reynolds number ◦ of Re = 10 and a scraper blade angle of = 150 . The viscosity values  have been normalized by the maximum viscosity, i.e. zero-shear-rate viscosity 0 . As one can see, the lowest viscosity and consequently the highest shear rates occur at the scraping edge of the blade (outer cylinder wall, darkest grey levels in Fig. 13) on both sides, the leading and trailing edge. Compared to low shear rate regions (where the viscosity is maximum) the viscosity decreases to about 15% of the maximum viscosity 0 in regions with highest shear rates (refer to grey level scale in Fig. 13). An annular layer along the rotor (light grey close to the inner cylinder) re"ects low shear rate regions and thus the zero-shear-rate viscosity region. The "ow we consider is a superposition of Couette "ow (shear driven "ow) and pressure driven "ow as described above. The shear stress  is not constant (as it would be for plane Couette "ow), but increases towards the outer cylinder or stator (refer to Fig. 14). More complex stress variations occur close to the scraper blade region (see close-up view in Fig. 14). For any "ow case investigated, independent of the scraper blade angle, Reynolds number or "uid type, the maximum shear stress max appears at the scraping edge (outer cylinder) on the leading and trailing edges and reduces to 20% of max within a short distance. This layer region of high stress is mainly responsible for heating the "uid due to viscous dissipation of energy E˙ diss which is discussed in the next subsection. The scraper blade angle dependent variations of the minimum apparent viscosity min and the maximum shear stress max for a Reynolds number of Re = 10 are shown in the top graph of Fig. 15. The apparent viscosity =0 oscillates with min at scraper blade an◦ ◦ gles of = 70 and 110 and maximum values max at

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◦

Fig. 13. Normalized apparent viscosity =0 in the annular gap using scraper blades at an angle of = 150 . The Reynolds number is set to Re = 10. The "ow behaviour is shear thinning (n = 0:65). The upper picture shows a close-up view of the scraper blade area and the lower picture shows the full annular gap. The grey level table indicates the normalized apparent viscosity values (=0 ); the black level represents the minimum viscosity and the lightest grey level represents the maximum viscosity.

◦

◦

= 30 and 150 . With respect to the viscosity value at ◦ an angle of = 90 , the trend shows a qualitative symmetry for the angles investigated. The same symmetric but “inverted” behaviour is found for the shear stress trend. The maximum shear stress values max occur at angles of ◦ ◦

= 70 and 110 . The stress drop between the Newtonian and the shear-thinning cases varies between 16.4% ◦ ◦ ( = 150 ) and 18.4% ( = 70 ) relative to the Newtonian "uid. For both "uid cases (n = 1 and 0.65) the shear stress max in the top graph of Fig. 15 shows the opposite behaviour of the maximum pressure pmax as shown in the top graph of Fig. 12. Thus, "ow incidences which tend

into the azimuthal direction reduce max and thus decrease the shear structuring contribution of the NAGR ◦ process, whereas scraper blade angles between = 70 ◦ and 110 have highest max at a constant Reynolds number of Re = 10. The bottom graph of Fig. 15 shows the minimum apparent viscosity min for the shear-thinning "uid (n = 0:65) and the maximum shear stress max for the Newtonian (n = 1) and shear-thinning "uid (n = 0:65) ◦ as a function of the Reynolds number Re for = 150 . The minimum apparent viscosity min decreases by about 3.5% (of the 0 plateau) if the Reynolds number is increased from Re = 20 to 80. In contrast, the

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◦

Fig. 14. Shear stress  in the annular gap using scraper blades at an angle of = 150 . The Reynolds number is set to Re = 10. The "uid is shear thinning (n = 0:65). The upper picture shows a close-up view of the scraper blade area and the lower picture is the full annular gap. The colour table contains the non-dimensional values of  (left column) and a percentage (right column) where the black colour represents 0.2% of maximum stress and the red colour represents the maximum stress or 100%.

maximum shear stress max increases by about 18.5% for the Newtonian "uid (n = 1) and 28% for the shear-thinning "uid (n = 0:65) as Re increases from 20 to 80. The di7erence in the shear stress between the Newtonian and shear-thinning "uids varies in the range of 8.9 –15:6% relative to the Newtonian values. For Reynolds numbers above 40, the shear stress increases signi9cantly stronger for the shear-thinning "uid. As mentioned above, increasing Reynolds number improves the structuring contribution of the shear "ow components. Thus, in contrast to the pressure reduction of pmax as shown in the bottom graph of Fig. 12, the shear stress max improves signi9cantly with higher Re. In particular, the shear-thinning

"uid (n = 0:65) shows the strongest increase for max above Re = 40, as depicted in the bottom graph of Fig. 15, due to its shear-thinning behaviour at locations with higher stresses. 3.5. Viscous dissipation of energy The main source for temperature rise inside the annular gap is energy dissipation E˙ diss , or viscous friction. Neglecting the 9rst two terms on the right-hand side of the entropy equation, q  1 1 Ds i = − @i − 2 qi @i T + ij @j vi ; (9)  Dt T T T

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Fig. 15. Minimum apparent viscosity min (triangles) as a function of the scraper blade angle (top graph) at Re = 10 and as a function of ◦ Re (bottom graph) at = 150 for the shear-thinning "uid (n = 0:65); maximum shear stress max (circles) as a function of (top graph) ◦ at Re = 10 and as a function of Re (bottom graph) at = 150 for the Newtonian case (n = 1) and the shear-thinning case (n = 0:65).

an irreversible process is given as a result of viscous forces acting within the bulk of the "uid (Eq. (10)). This “friction” inside the "uid is transformed into heat, which causes the temperature rise. A description of the viscous dissipation process is given by Panton (1984), Ds = ij @j vi : (10) T Dt The dissipative term on the right-hand side of Eq. (10) is formulated in terms of the stress ij and the velocity gradient @j vi , where summation on repeated indices is implied. This dissipative term de9nes the energy dissipation density rate, or the energy "ow per "uid volume V . For our discussion, we use the energy dissipation E˙ diss , which is formulated as a function of the shear viscosity  (given with the CYL model, Eq. (3)), the shear rate ˙ and multiplied with a "uid volume V and thus becomes ˙ ˙2 V: E˙ diss = ()

(11)

Fig. 16 represents the local viscous dissipation of energy E˙ diss . The maximum dissipation (red colour = 100%) of energy E˙ diss; max is located in very small regions close to the scraping edges of the blades (visible in the close-up view at the outer cylinder wall). For the shear-thinning

"uid (n = 0:65), a sudden drop of more than 80% occurs close to these peak areas (light blue colour). The other contours throughout the annular gap (dark green) only re"ect around 4% of E˙ diss; max . The top graph of Fig. 17 shows the maximum shear stress max and the maximum energy dissipation E˙ diss; max as a function of the scraper blade angle at Re = 10. The same behaviour for all four curves is visible with max◦ ◦ imum peak values at 70 and 110 . The drop in the ˙ energy dissipation E diss; max from the Newtonian to the ◦ shear-thinning case varies between 15:5% (150 ) and ◦ 17:6% (90 ) with respect to the Newtonian "ow. The shear stress drop is slightly higher. As shown in Eq. (10) one has to expect increased dissipative e7ects in "ows where higher shear stresses max act. Therefore, the improved structuring contributions are always accompanied by an increased dissipative heating of the "ow as the shear rate ˙ increases. A comparison of maximum shear stress max and the maximum energy dissipation E˙ diss; max as a function of the Reynolds number Re is shown in the bottom graph of Fig. 17. Since E˙ diss increases quadratically as a function of ˙ (see Eq. (11)), whereas max increases almost linearly with , ˙ the dissipative e7ects in terms of E˙ diss; max contribute more pronouncely to the "uid "ow than max (structuring "ow contribution) as the Reynolds number increases. As depicted in the bottom graph of Fig. 17 the curves for E˙ diss; max show a higher slope in comparison to the slopes of max . Furthermore, a crossing over of the energy dissipation curves between Re = 60 and 80 takes place. This e7ect is not visible for the maximum shear stress curves as a function of the Reynolds number. Thus, the highest stress locations max in the annular gap close to the scraping edge of the blade (see Fig. 14) are associated with higher shear rates for the shear-thinning "uid (n = 0:65) and this may cause the crossing over and the highest E˙ diss; max between Re = 60 and 80 with n = 0:65. Switching from the Newtonian to the shear-thinning "uid, the maximum dissipated energy E˙ diss; max drops by about 14.8% for a Reynolds number of Re = 20 and reduces to 2.6% for Re = 60 with respect to the Newtonian "uid. At Re = 80; E˙ diss; max for the shear-thinning "uid is 5.4% higher than for the Newtonian "uid.

4. Conclusions Two-dimensional numerical simulations (FVM) have been performed for a narrow annular gap reactor (NAGR) using two scraper blades of di7erent orientations. The two-dimensional velocity 9eld in the cross-section gap plane is representative for real processing with such apparatus because the axial "ow component is negligible at typical high rotational velocities. Various "ow

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◦ Fig. 16. Energy dissipation E˙ diss in the annular gap using a scraper blade with an angle of = 150 at a Reynolds number of Re = 10. The pictures show the behaviour of the shear-thinning "uid (n = 0:65) in a close-up view (upper picture) and the full annular gap (lower picture). The colour table contains the non-dimensional values of E˙ diss (left column) and a percentage (right column), with 0% representing minimum dissipation, and 100% representing maximum dissipation.

incidences as a function of the scraper blade angle and rotor velocities u0 , described with the rotational Reynolds number Re, have been considered for both Newtonian and shear-thinning "uids, with constant annular gap ratio of 1 : 4 between the scraper blade gap width hgap and the rotor=stator gap width h0 . Experimental data sets of velocity 9eld measurements of two "ow cases (using a Newtonian (n = 1) and a shear-thinning model "uid (n = 0:8)) have been collected using digital-particle image velocimetry (D-PIV) in the laminar "ow regime (Re682:6) with a reference scraper ◦ blade angle of = 147 . Comparisons with the numerical simulations are in good agreement.

Two-dimensional annular gap "ow velocity 9elds including two scraper blades show strong dependency on the rotor velocity (in the laminar "ow regime). For Reynolds numbers of Re610, the "ow is mainly pressure driven, caused by the movement of the scraper blades. For Reynolds numbers of Re ¿ 10, the "ow is dominated by the shear stresses applied by the rotating inner cylinder. Above a Reynolds number of Re = 60, the pressure ratio (minimum pressure at trailing edge divided by maximum pressure at leading edge) does not signi9cantly increase further. Macroscopic "ow structuring mechanisms of Newtonian and shear-thinning viscous "uid systems of

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at low rotor velocities (i.e. for Re = 10) combined with ◦ ◦ increased "ow incidences (i.e. from = 90 to 150 ) should be preferred. Notation E˙ diss h n p ri ro rs Reh t Tah u0 ; vrot v Fig. 17. Maximum shear stress max (triangles) and maximum energy dissipation E˙ diss; max (circles) for Newtonian and shear thinning "ow (n = 1 and 0.65, respectively), as a function of the "ow incidence

(at Re = 10, top graph) and as a function of the Reynolds number ◦ Re for = 150 (bottom graph).

food applications have been described and discussed. These include the velocity pro9les, secondary "ows, the hydrodynamic pressure, non-Newtonian viscosity, shear stress and energy dissipation. In particular, the variation of the "ow incidence enables the adaptation of the NAGR process to food "ow applications. If an enhanced shear structuring "ow behaviour is desired (to reach a critical shear stress crit , which enables e.g. the dispersing of crystal nuclei of fat crystalliz◦ ing "ow) our investigations showed that a = 90 "ow incidence is most appropriate for improved shear struc◦ turing "ows. Reduced scraper blade angles at = 30 or below decrease the shear structuring behaviour of NAGR process "ows. Improved "ow structuring behaviour (enhanced shear rates) cause higher energy dissipation in the "ow process which leads to increased inhomogeneous temperature distributions. Our investigations depicted signi9cantly ampli9ed dissipation regions at the scraping edge of the blades (outer cylinder of the annular gap), which cause local temperature peaks. For example, for fat crystallizing food processes these temperature peaks locally change the nucleation eCciency and growth, which cannot be controlled by the rotor velocity u0 . Therefore, to combine improved homogeneous temperature 9elds (or optimized “thin layer” heat transfer) with e7ective "ow structuring behaviour, smooth "ow deformation changes

V w

non-dimensional energy dissipation, dimensionless annular gap width, m power-law exponent, dimensionless pressure, Pa inner radius, m outer radius, m scraper blade gap ratio, dimensionless characteristic Reynolds number with respect to h, dimensionless time, s characteristic Taylor number with respect to h, dimensionless rotor speed, m s−1 velocity with respect to 9xed frame of reference, m s−1 "uid volume, m3 velocity with respect to rotating frame of reference, m s−1

Greek letters

˙  
 

scraper blade angle, deg shear rate, 1 s−1 dynamic viscosity, Pa s density, kg m−3 stress, Pa shear stress, Pa streamfunction, dimensionless rotational speed, 1 s−1

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