The dissipation rate of turbulent kinetic energy and its relation to pumping power in inline rotor-stator mixers

Chemical Engineering and Processing 115 (2017) 46–55

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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep

The dissipation rate of turbulent kinetic energy and its relation to pumping power in inline rotor-stator mixers Andreas Håkanssona,* , Fredrik Inningsb,c a b c

Kristianstad University, Food and Meal Science, Kristianstad, Sweden Tetra Pak Processing Systems AB, Lund, Sweden Lund University, Food Technology and Engineering, Lund, Sweden

A R T I C L E I N F O

Article history: Received 29 September 2016 Received in revised form 13 January 2017 Accepted 18 January 2017 Available online 23 January 2017 Keywords: Rotor-stator mixers High shear mixers Dissipation rate of turbulent kinetic energy Turbulence Emulsification Mixing Pumping power

A B S T R A C T

The theoretical understanding of inline rotor-stator mixer (RSM) efficiency, described in terms of the dissipation rate of turbulent kinetic energy as a function of mixer design and operation, is still poor. As opposed to the correlations for shaft power draw, where a substantial amount of experimental support for the suggested correlations exists, the previously suggested correlations for the dissipation rate of turbulent kinetic energy have not been experimentally validated based on primary hydrodynamic measurements. This study uses energy conservation to reformulate the previously suggested dissipation rate correlations in terms of pumping power which allows for empirical testing. The dimensionless pumping power of three investigated geometrically dissimilar inline RSMs were found to be qualitatively similar to that of centrifugal pumps and decrease linearly with the inline RSM flow number. The previously suggested models for turbulent dissipation in inline RSMs are inconsistent with this observation. Using this reformulation approach, the previously suggested correlation for power-draw is extended to a correlation for dissipation. A new model is suggested based on conservation of energy and angular momentum, and the empiric pumping power relationship. The new model compares well to CFD simulations of total dissiaption and show reasonable agreement to emulsification drop size scaling. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Rotor-stator mixers (RSMs) are used for mixing and dispersion of multiphase systems in many applications of chemical engineering – e.g. in food, pharmaceutical and cosmetic industry [1,2]. RSMs can be operated in batch, semi batch and inline (continuous) mode of operation, and there is an increased industrial interest in transitioning from batch RSMs to inline mode of operation due to the economic advantage of continuous production. However, application and design of inline RSMs suffers from a lack of theoretical understanding. Mixing and dispersion in batch RSM are already less theoretically developed than for other high intensity processing equipment such as impeller-systems or highpressure homogenizers [2,3]. Moreover, inline RSMs offer additional complications compared to batch mode of operation. The inline RSM can be seen as a hybrid between a centrifugal pump and a batch RSM. The rotor-stator device is typically of the same or similar design as in batch RSM, but is mounted in a narrow

* Corresponding author. E-mail address: [email protected] (A. Håkansson). http://dx.doi.org/10.1016/j.cep.2017.01.007 0255-2701/© 2017 Elsevier B.V. All rights reserved.

casing with inlet and outlet piping. Similar to a centrifugal pump, the inlet pipe is directed towards the central axis of the rotor, the fluid is accelerated radially by the torque of the rotor blades and redirected towards the peripheral outlet at increased static pressure. As for batch RSMs, high turbulence intensity is generated in or near the holes or slots of the stator screen. Different manufacturers offer different rotor-stator designs, e.g. in terms of number and size of stator holes, design of rotor blades and in the number of rotors and stators. During the last decade, significant advances have been made in understanding the inline RSM power draw, i.e. the shaft power required to obtain a desired rotor speed at a set volumetric flowrate. Under turbulent conditions, it has been suggested that the shaft power draw consists of three terms [4–8] Pshaft ¼ Prot þ Pflow þ PL ¼ NP0 rN3 D5 þ NP1 rQN2 D2 þ PL

ð1Þ

where NP0 and NP1 are design dependent constants. The first term in Eq. (1), P rot, is similar to the power draw of a batch RSM and describes the effect of rotor speed. The second term, P flow, describes the effect of increasing power draw with increasing flowrates and the third, P L, is a loss-term describes power lost in

A. Håkansson, F. Innings / Chemical Engineering and Processing 115 (2017) 46–55

Notation

Abbreviations CFD Computational fluid dynamics TKE Turbulent kinetic energy RSM Rotor-stator mixer Roman A Ahole Astat c1, c2 D h Ljet N NP NP0, NP1 NQ Pdiss PL Ppump Pshaft P’shaft Pthermal Q r1 r2 T U, U0 V Vdiss Vu, V0 u

Rotor blade area, m2 Total stator hole area, m2 Total stator screen area, m2 Empirical constants in Eq. (18) Rotor diameter, m Rotor blade height, m Characteristic length of stator hole exit jet, m Rotor revolution speed, Hz Batch RSM power number Inline RSM shaft power-draw constants Inline RSM flow number Total turbulent dissipation power, W Loss power, W Pumping power, W Shaft power draw, W Loss-corrected power-draw (Pshaft–PL) Total increase in fluid thermal energy, W Volumetric flow rate, m3 s1 Inner rotor blade radius, m Outer rotor blade radius, m Rotor blade torque, Nm Rotor blade tip-speeds (see Fig. 3), m s1 Volume, m3 Dissipation volume, m3 Angular components of the fluid velocity (see Fig. 3), m s1

Greek

a

b d DpD, DpS e hpump P flow, P rot, P L

Fraction of dissipation place in the effective region of emulsification Blade angle, rad Rotor-stator clearance, m Dynamic and static components of the pressure gradient over the RSM, Pa Dissipation rate of TKE, m2 s3 Pumping power efficiency Flow, rotation and loss terms in the shaft power draw expression of Eq. (1), W

bearings or due to vibration [8]. Note that the terms of the shaft power-draw correlation are denoted by Greek symbols (as opposed to the shaft power which is denoted by Latin P). This convention is followed throughout the text and will become important in distinguishing the correlation terms from the power derived from the energy balance in Section 2. Eq. (1) has obtained substantial experimental support through the last decade [4–9]. More recently, a correction to Eq. (1) has been suggested by Jasinska et al. [10] adding two additional terms. However, as noted by the authors (Ref. [10,p. 47]), this correction would only apply for flowrates that are typically below those used in technical operation. It should also be noted that RSM designs differ between manufacturers and the generalizability of Eq. (1) is still somewhat unclear since most of the experimental work are on Silverson or Tetra Pak RSM designs. Exceptions exists, it has

47

recently been found that Eq. (1) does not comply with the ContiTDS RSM design produced by Ystral [11]. Mixing and dispersion efficiency are influenced by the dissipation rate of turbulent kinetic energy (TKE) in the most turbulent region [12], which in turn is influence by the total amount of dissipated energy. It should be noted that the dissipation rate of TKE, or the total dissipated energy, is not equal to the power input for an inline mixer. Thus Eq. (1) cannot be used to calculate the energy available for dissipation without adding more assumptions. As opposed to batch RSM, there is still no generally accepted expression for estimating the dissipation rate of TKE in the effective region of mixing and dispersion at different operating and design parameters. For a batch mixer, the power available for turbulent dissipation, Pdiss, relates directly to the power-draw [1,13] Pdiss / Pshaft = NPrN3D5.

(2)

with a design dependent constant NP. However, the situation is more complicated for inline systems since some of the power input from the shaft is used for pumping the fluid. Experiments have also revealed that RSMs run in batch and inline differ in how they scale with regards to mixing efficiency [14] and emulsification efficiency [15]. This illustrates that the dissipation rate of TKE in inline RSM differs in a substantial way from that in a batch RSM, i.e. Pdiss is not equal or proportional to Pshaft for inline mode of operation. Moreover, this also implies that it is not obvious that the dissipation power is proportional to the rotation-term in the shaft power-draw correlation (P rot) as is often assumed, e.g. Refs. [6,10]. More investigations are needed to clarify the relation between the terms in the shaft power-draw correlation and the dissiaption. CFD studies have been conducted with the intention to shed light on the hydrodynamics of inline RSMs [10,16–18]. Especially with micromixing simulations, a good correlation between experimentally observed mixing and simulations based on local dissipation rate of TKE have been obtained [18]. This illustrates that the spatially resolved dissipation rates obtained from CFD are a fruitful starting point for theoretical discussions. In a recent study we have showed that the lower flow through the stator screen in inline RSMs, as compared to batch RSMs, results in a shift of the position where turbulent kinetic energy is dissipated, thus, suggesting a different mixing and breakup mechanism in the inline RSM than in the batch RSM [19]. CFD is useful for detailed investigations. However, it is highly time consuming and is therefore often complemented by empiric correlations, especially in discussing scale-up and interpreting emulsification results. Several suggestions for correlations describing how dissipation rate of TKE scales with operating and design parameters for inline RSMs have been presented [6,10,15,20,21]. However, several of these correlations are contradicting, and no consensus has yet been established even on such basic aspects as scaling with flowrate, rotor speed and diameter. Moreover, experimental studies trying to correlate emulsion drop diameters under different operating conditions to these previously suggested correlations of dissipation rate of TKE, generally conclude that there is poor fit between the two [6,15]. In summary, our current understanding of how mixing and dispersion efficiency depends on operating and design parameters needs to be improved. And an important step in this direction is to describe the scaling of dissipation rate of TKE in the effective region of mixing and dispersion with design and operating parameters. Of special importance is how the dissipation power relates to the terms in the shaft power draw correlation. The specific objective of this contribution is to compare the previously suggested models for dissipation rate of TKE in inline RSMs with special attention to

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the total turbulent dissipation, discuss underlying assumptions in the light of available experiments and simulations, and propose an improved correlation. 2. Modelling dissipation rate of TKE in inline RSMs 2.1. Previously suggested models If the power Pdiss is dissipated in a fluid of volume V with density

r, the mean dissipation rate of TKE is e¼

Pdiss : V r

ð3Þ

However, for the purpose of discussing mixing or dissipation efficiency, the local dissipation rate of TKE in the effective region of mixing or emulsification is a more relevant quantity to consider [10,12,15,20],

e¼

a  Pdiss : V diss  r

ð4Þ

where a is the proportion of energy dissipated in the effective region Vdiss. Modelling e, thus, generally requires modelling all three factors: Vdiss, a and Pdiss, all of which can depend on RSM geometry and operating condistions The dissipation volume (Vdiss) can be calculated from geometrical parameters once the effective region of emulsification has been identified. However, there is a disagreement in literature as to where the effective zone is located in inline RSMs. Hall et al. [6] suggest the rotor-swept volume, Jasinska et al. [10] the rotor-stator clearance, Kamiya et al. [21] the stator hole, Sparks [20] suggest both clearance and the stator hole, and Håkansson et al. [15] the jet volume downstream of the hole, see Fig. 1 for an illustration of these different regions. Some of this variation can be due to the geometrical differences between RSMs. However, interestingly, two recent CFD-studies find the highest dissipation rate of TKE in the rotor-stator clearance for two different designs [10,19], V diss ¼ phdD;

ð5Þ

at least when the dimensionless flowrate, henceforth referred to as the (inline RSM) flow number, NQ ¼

Q ND3

;

ð6Þ

is sufficiently low [19]. The proportionality constant in Eq. (4) (a) has traditionally been assumed to be constant [20]. However, recent investigations suggest that it depends on the flow number [10,19].

Judging from previously published literature, the total dissipation power (Pdiss) is the most controversial of the terms in Eq. (4). Although an agreement has now been reached on the form of the power draw correlation for several RSM designs, i.e. the total power input (see Eq. (1)), there is still considerable disagreement in literature with regard to how much of this power that is dissipated as turbulence. These previous suggestions can be divided in four classes (I–IV) based on what they assume on the relationship between the dissipation power (Pdiss) and the terms in the power draw correlation (P rot and P flow):  (I) Sparks [20] and Håkansson et al. [15] assume that the total dissipation power is equal to the flow-factor in the power draw correlation in Eq. (1), e.g. Pdiss = P flow.  (II) Hall et al. [6] investigate two different models, one where dissipation is assumed equal to the rotation part of the shaft power draw correlation (Pdiss = P rot). This assumption was also used by Jasinska et al. [10], however, it should be noted that Jasinska et al. do open up for the possibility of including a fraction of P flow by modifying what in the present study is denoted a.[10,p. 52]  (III) Hall et al. [6]. also explored a model where all shaft power (except losses) is assumed to be dissipated (Pdiss = P rot + P flow).  (IV) Kamiya et al. [21], finally, suggests that dissipation is the rotation term of the correlation subtracted by the flow-term (Pdiss = P rot  P flow). The assumptions on where the effective region of dissipation is, and how the dissipation relates to the terms in the power draw correlation, are the two principal dimensions describing the different previously suggested models for the dissipation rate of TKE in inline RSMs. Table 1 illustrates how the five previously mentioned correlations combine these two sets of assumptions. As seen in Table 1, there is at present a large variation between models. As previously stated, some of this variation could be due to geometrical differences between RSM designs. However, several of the models refer to the same mixer type, e.g. Ref. [6] and Ref. [10] or Ref. [15] and Ref. [21]. Hence, there is a need for finding the most suitable correlation among the many suggestions. No method for experimentally testing the validity of different dissipation rate of TKE correlations for inline RSMs has yet been proposed based on basic hydrodynamics. 2.2. Conservation of energy The total turbulent dissipation, Pdiss, is difficult to measure directly. However, the different assumptions on Pdiss (i.e. I–IV in Table 1) can be converted to more readily measured quantities by application of an energy balance. Fig. 2 displays a schematic illustration of the inline RSM, showing how shaft power will ultimately be converted to three different energy terms: thermal energy (i.e. an increase in temperature of the fluid), pumping energy (i.e. an increase of static and/or dynamic pressure of the fluid) and mechanical losses (e.g. vibrations and noise), cf. Refs. [8] and [20]. A differential energy balance over the mixer volume results in Pshaft = Pthermal + Ppump + PL

(7)

An increase in the thermal energy, Pthermal, can be the result of laminar viscous heating or of turbulent dissipation of TKE; and the dissipation Pdiss corresponds to the turbulent part of the temperature increase only. However, the laminar viscous heating is small for turbulent flows, and can be neglected. Under steadystate, the rate of turbulent energy dissipation equals the rate of production and the thermal energy increase is thus identical to the Fig. 1. The four previously suggested dissipation volumes.

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Table 1 Summary of the assumption made on the dissipation power and dissipation volume in inline RSMs from previously published studies. How does the total turbulent dissipation relate to the terms in the power-draw correlation? (I) Assumption Pdiss ¼ Prot Where does mixing and dispersion take place? Rotor-swept volume [6] Rotor-stator clearance volume [10] Stator hole volume Clearance and stator hole volume Jet volume Ppump ¼ Pf low Implied pumping power dependence (via Eq. 10)

turbulent dissipation-term, Pthermal = Pdiss. Consequently, the energy balance, Eq. (7), is Pshaft = Pdiss + Ppump + PL.

(8)

or P0shaft ¼ Pshaft  PL ¼ Pdiss þ Ppump

ð9Þ

Assuming that the loss-term in Eq. (1) is equal to the true power loss (PL = P L), the loss-free power draw (P0 shaft), and thus the lefthand-side of Eq. (9), can be calculated from Eq. (1) (as the sum of P rot and P flow). The total turbulent dissipation (Pdiss) and the pumping power (Ppump) are therefore related as a direct consequence of energy conservation: Pdiss = P rot + P flow  Ppump.

(10)

Each suggestion for Pdiss (each column in Table 1) can thus be expressed in terms of what assumptions it makes on Ppump. The last row of Table 1 describes what assumption each of the models for Pdiss implicitly makes on Ppump as a consequence of energy conservation (Eq. (10)). Since Ppump can be experimentally obtained, this also allows for experimentally testing the different models for the turbulent dissipation in Table 1. Before discussing measurements of pumping power, it should be noted that Ppump consists of two terms, representing the

(II)

(III)

(IV)

Pdiss ¼ Pf low

Pdiss ¼ Prot þ Pf low [6]

Pdiss ¼ Prot  Pf low

[21] [20] [15] Ppump ¼

NP0 1 NP1 NQ

Ppump ¼ 0

Ppump ¼ 2Pf low

dynamic and static pressure increase of the flow: Ppump ¼ Ppump;S þ Ppump;D ¼ Q DpS þ Q DpD

ð11Þ

where DpS (DpD) is the static (dynamic) pressure difference between the inline RSM outlet and inlet. Experimental studies often only report the static component, but DpD is generally assumed to be small (Ref. [8,p. 243]). 2.3. Conservation of angular momentum To the best of our knowledge, no general expression of how pumping power depends on operating parameters for inline RSMs have yet been suggested. However, the inline RSM is highly similar to a centrifugal pump, and ideal centrifugal pumping theory can therefore be used as a first step towards estimating how pumping power depends on operating parameters and design. Fig. 3 displays a schematic image of an angled RSM rotor blade. The same general energy balance as in Eq. (8) applies for the pump and thus generally Ppump < Pshaft. However, considering an ideal centrifugal pump where all shaft torque, T, is used for pumping: [22] Ppump = QDp  Pshaft = 2pNT .

(12)

The torque is the difference in angular momentum between the fluid at the lower and upper blade tip, [22] T = rQ(r2Vu  r1V0 u) .

(13) 0

Neglecting inlet fluid momentum at the blade entrance (V u = 0) and using the geometrical relationship between rotor tip speed, volumetric flowrate and the blade geometry (i.e. ‘the Euler triangle’

Fig. 2. Schematic illustration of the energy balance.

Fig. 3. Illustration of the angular momentum balance over the rotor blade. U and U0 denote the peripheral rotor speed at the tip and bottom of the blade respectively. Vu and V0 u denote the angular component of the fluid velocity in the two points.

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A. Håkansson, F. Innings / Chemical Engineering and Processing 115 (2017) 46–55 Table 2 Geometry and operating conditions of the three inline RSMs.

in pump theory) results in (Ref. [22,p. 719])

Dp D2 NQ ; ¼1 2 p  A  tanðbÞ rU

ð14Þ

Data source

where A is the total rotor blade area. Reformulated in terms of pumping power, this corresponds to ! D2  NQ Ppump ¼ rQN2 D2 p2  p  A  tanðbÞ ! 2 p D2 ¼ Pf low  NQ :  ð15Þ NP1 p  A  NP1  tanðbÞ

D [m] d [mm] Nholes [–] Aholes/Astator U [m/s] Q [m3/h]

A

B

C

[9]

[20]

[20]

0.20 1.0 420 50% 10–30 3–120

0.12 0.23 13 50% 13–19 0.1–18

0.12 2.4 18 50% 13–19 0.1–18

Also note that the proportionality constant in Eq. (16) is approximately equal to 1 (since NP1 is often found to be close to 10, e.g. Ref. [8]). Ideal radial mixers therefore has pumping power corresponding to model I in Table 1 (Ppump = P flow and consequently Pdiss = P rot). However, inline RSMs often have either sloped blades (b < 90 ), additional pumping blades mounted on top of the rotor (e.g. Ref. [21]) or toothed rotors (e.g. Ref. [20]). These are all cases where the assumptions underlying Eq. (16) are not valid. Thus, it is far from obvious that Eq. (16) would apply generally for real inline RSMs.

Sparks [20] measured shaft power draw and the pump curve (static pressure increase) as a function of flowrate and rotor tipspeeds for two principally different designs, both designed for lower volumetric flowrates than mixer A. The first design (a Silverson 452 LSM) is geometrically similar to mixer A but with four rotor blades (positioned with an offset to the shaft center) and a single row of stator holes, see Fig. 4B. This mixer will be referred to as mixer B. The second mixer design investigated by Sparks [20] has a circular toothed rotor and a stator screen with rectangular slots, see Fig. 4C. This was an in-house constructed design that Sparks describes as similar to “units that are typically produced by Swiss and German manufacturers (e.g. Ystral, IKA, Kinematica).” [20,p. 43] This mixer will be referred to as mixer C. The principal dimensions of the three designs, together with the range of investigated rotor tip-speeds and flowrates in the respective studies can be seen in Table 2. Further details of the experimental procedure can be found in the original studies: Ref. [9] for mixer A and Ref. [20] for mixers B and C.

3. Experimental and simulation data used in the analysis

3.2. Simulation of inline RSM hydrodynamics

3.1. Experimental data on power draw and pumping power

We have recently reported a CFD-simulation of mixer A run at a wide range of tip-speeds and flowrates (U = 10–31 m/s, Q = 3– 120 m3/h) [19]. The CFD model consisted of 79.6 million tetrahedral and hexagonal computational cells (34 cells across the diameter of each of the stator holes in Fig. 4A). Sliding mesh was used to model the rotor rotation and turbulence was modelled using a realizable k-e model. Simulations were carried out in FLUENT 16 (ANSYS, Canonsburg, PA). Further details together with

Eq. (15) describes the pumping power of an inline RSM without losses, where all of the torque on the rotor blades are converted to a static pressure increase. Note that for a mixer with radial blades (b = 90 ), Eq. (15) simplifies to Ppump ¼

p2 NP1

Pflow ;

ð16Þ

Lindahl [9] measured the pump curve (i.e. the static pressure increase measured between outlet and inlet) of a production-scale inline RSM (Tetra Pak Scanima design) run with water. A schematic view of the rotor-stator design can be seen in Fig. 4A. It consists of six-bladed rotor and a single stator screen with a total of 420 circular holes (70 holes in six rows). This mixer will be referred to as mixer A.

Fig. 4. Schematic drawings of the three different RSM designs under investigation. The upper pane for each mixer displays a cut through the stator screen and rotor. The lower pane for each mixer shows the stator screen design with circular holes (A and B) or rectangular slots (C). Geometrical dimension of mixers A, B and C can be found in Table 2.

A. Håkansson, F. Innings / Chemical Engineering and Processing 115 (2017) 46–55

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a mesh-sensitivity discussion can be found in the original study [19]. Two pieces of information were obtained from the CFD-model. First, the total dynamic and static pressure on the inlet and outlet boundaries were calculated in order to investigate the relative effect of the static and dynamic terms in Eq. (11) for mixer A. Secondly, the total dissipation, Pdiss, at each rotor tip-speed and flowrate was calculated from the CFD model and compared to the correlations. The total dissipation was calculated from CFD by integrating the local dissipation of TKE over the entire mixer volume Z Pdiss ¼ r edV: ð17Þ V

The volume-integration was performed using a built-in function in the CFD software. 3.3. Experimental investigation of emulsion drop diameter Chaudhry [23] reported emulsification experiments on mixer A. Oil-in-water emulsions were created by mixing rapeseed oil, tap water and non-ionic low-molecular emulsifier (polysorbate 20) and re-circulating over a tank connected to the inline mixer. Data was obtained for both low and high volume fraction emulsions (0.5%(v/v) and 52%(v/v)), across a range of rotor tip-speeds and flowrates (U = 10–31 m/s, Q = 5–10 m3/h). The emulsion was sampled after processing for a time representing an average of 24 passages and drop sizes measured using laser diffraction technique. Emulsification theory suggests that the obtained diameters of the largest drops scale with the dissipation rate of TKE in the efficient region of emulsification [24]. The size of large drops were quantified in terms of Dv,95, the limiting size of the 5% drop volume with the largest diameters. More details of the experiments can be found in the original publication [23]. The general methodology is also discussed in ref. [15]. 4. Results and discussion This section is organized as follows: Section 4.1 compares shaft power draw to Eq. (1). Section 4.2 discusses the importance of dynamic pressure in evaluating pumping power. Section 4.3 studies the experimental pumping power for mixers A–C and compares it to centrifugal pumping theory. Section 4.4 uses the experimentally obtained pumping power to test the validity of previously suggested correlations for Pdiss. Section 4.5 discusses the pumping and mixing efficiency of the mixers. Section 4.6 proposes a new model for Pdiss and compares it to CFD simulations and emulsification data. Finally, Section 4.7 summarizes the model and provides an outlook for continued investigations.

Fig. 5. Experimental shaft power (symbols) fitted to the power-draw correlation in Eq. (1) (lines). Mixer A: no symbols, solid line; Mixer B: grey circles, dashed line; Mixer C: white circles, dash-dotted line.

Table 3 Fitted parameter values in the power-draw model (Eq. (1)) with 95% confidence intervals and R2-values.

Mixer A Mixer B Mixer C

NP0

NP1

R2

0.11 0.10 [0.095, 0.10] 0.13 [0.11, 0.14]

9.2 6.4 [6.3, 6.5] 9.7 [9.3, 10]

*

100% 99%

* Not available since ref. [9] only reports the NP0 and NP1, not individual measurements.

4.2. Relative influence of dynamic pumping power As illustrated by Eq. (11), the pumping power, Ppump, can be divided in a static and a dynamic component. The dynamic pressure difference, and thus the dynamic pumping power, are typically not reported in the experimental datasets, i.e. average fluid velocities entering and exiting the mixer are not measured. However, the CFD simulations provides a means to estimate the importance of this factor. Fig. 6 displays the ratio between the

4.1. Shaft power draw Fig. 5 displays the experimental shaft power draw for mixers B and C together with the best fit to the power draw expression in Eq. (1) plotted as a function of the flow-number, NQ. The correlation in Eq. (1) accurately describes the three mixers, as has been seen with several other inline RSM designs [4–8]. Estimated values of NP0 and NP1 together with R2-values of the fit are given in Table 3. For mixer A, only the fitted model values NP0 and NP1, and not individual values of Pshaft, were reported by Lindahl [9]. The correlation for mixer A has been included in Fig. 5 as a comparison (solid line).

Fig. 6. Dynamic component of the pumping power for mixer A, expressed as a percentage of the static pumping power.

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A. Håkansson, F. Innings / Chemical Engineering and Processing 115 (2017) 46–55

dynamic and static terms of pumping power from the CFD model as a function of flow number. As seen in Fig. 6, the dynamic term increases somewhat in importance with increasing flow number but is always small, less than 2%. The dominance of the static pressure term is expected from the close similarity to a centrifugal pump, which is designed to transform as much as possible of the power input to static pressure. The dynamic pumping power is therefore neglected in the following discussion, assuming Ppump = Ppump,S.

Table 4 Fitted parameter values in the pumping power correlation (Eq. (18)) with 95% confidence intervals and R2-values.

Mixer A Mixer B* Mixer C *

c1

c2

R2

0.46 [0.45; 0.47] 0.44 [0.43; 0.45] 0.23 [0.22; 0.24]

3.8 [4.1; 3.5] 3.2 [3.4; 3.0] 2.4 [2.6; 2.3]

99% 98% 98%

The fitting was obtained for the linear range (NQ > 0.02).

by 4.3. Pumping power

    Ppump ¼ Pflow c1 þ c2 NQ ¼ NP1 rQN2 D2 c1 þ c2 NQ ;

Fig. 7 displays the measured pumping power as a function of flow number for the three investigated RSMs (markers). The pumping power has been rendered dimensionless by dividing it with the flow term in the shaft power draw correlation (P flow) in order to compare the experimental values to the ideal pump equation (Eq. (15)). First note that the pumping power ratio (Ppump/ P flow) decreases linearly with flow number for all three investigated RSM. This is similar to what Eq. (15) predicts for a centrifugal pump with backward-curved blades. However, the intercept at NQ = 0 is substantially lower (0.23–0.46) for the three mixers than for the ideal pump which according to Eq. (14) occurs at Ppump/P flow  1. This difference is expected since all shaft power is not used for pumping for an inline RSM. The corresponding curve for a typical centrifugal pump (LKH50, Alfa Laval, Sweden) has been included in Fig. 7 as a more realistic comparison to the inline RSMs. The centrifugal pump results in an intercept at 0.55 and a generally higher pumping power ratio than the inline RSM. This is expected since more of the torque input is transformed to turbulent dissipation in the RSM than in a centrifugal pump. The centrifugal pump also deviates from linear behavior; a well-known phenomenon for centrifugal pumps attributed to friction and recirculation losses, see Ref. [25,p. 78]. The linear decrease in Fig. 7 suggest the same general scaling of dimensionless pumping power as obtained from the angular momentum conservation for the ideal pump: the pumping power is the product of the flow term of the power draw correlation and a linear function of the flow number, but with different intercept and slopes than the ideal case. The three mixers can thus be described

with mixer specific, but rotor speed and flowrate independent, constants c1 and c2. Estimated values of the constants in Eq. (18) for mixers A, B and C are reported in Table 4, with confidence intervals. It should be noted that two of the mixers (A and B) have almost identical c1 and c2 parameters despite having different geometrical dimensions, number of rotor blades and stator screen hole arrangement. This suggest that c1 and c2 are independent of RSM scale. Mixer C, which have a circular toothed rotor design as compared to the blade-design in mixer A and B (see Fig. 1), display a lower pumping power but a similar linear dependence on the flow-number, showing that c1 and c2 do depend on rotor-stator design as is also the case for the power-draw correlation constants, NP0 and NP1 (see Table 3 and Ref. [4]). For low flow numbers (NQ < 0.02), mixer B shows pumping power ratios that deviate from the linear form. This can be due to hydraulic losses, similarly to the centrifugal pump. However the pressure readings in these points are low in comparison to the measurement uncertainty which makes it highly susceptible to measurement errors. In the light of previously reported deviations for low flow numbers [10], further investigations into this region is of interest.

Fig. 7. Dimensionless pumping power: pumping power divided by the flow-term in the power draw equation. Markers show experimental measurements (colors see Fig. 5). Numbered lines (I–IV) corresponds to the models in Table 1. The bold dotted line display the corresponding curve for a typical centrifugal pump (LKH50, Alfa Laval, Sweden).

ð18Þ

4.4. Comparison to previously suggested models for the total turbulent dissipation As discussed in Section 2.2, each of the previously suggested models for the total dissipated power found in literature can be transformed into a relationship between the pumping power and the flow term in the shaft power-draw correlation. The different assumptions (columns in Table 1) can therefore be inserted in Fig. 7 to obtain an experimental validation of the suggested models. First note that the linear decrease in Fig. 7 implies that the pumping power is significantly lower than P flow and decreases with flow number. This shows that neither of the three mixers behave as the ideal centrifugal pump in Eq. (16) and hence model I in Table 1 is not an appropriate description of the total dissiaption in any of these inline RSMs. The flow-number dependence of the ratio between pumping power and P flow also disproves model III and IV, since they imply either zero pumping power (model III) or a pumping power twice as large as the flow-term (model IV). The assumption that Pdiss = P pump (model II in Table 1) results in a decreasing ratio between Ppump and P flow as a function of flow number, and fits the experimental data better, at least for high flow numbers. However, the fit is poor for lower flow numbers. Therefore this suggestion can also be rejected based on the experimental data. In conclusion, neither of the proposed correlations for dissipation energy comply with the experimental data. 4.5. Pumping efficiency The pumping efficiency is defined as hpump = Ppump/P0 shaft. By combining Eqs. (1) and (18), the proportion of inline RSM power

A. Håkansson, F. Innings / Chemical Engineering and Processing 115 (2017) 46–55

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described by previous investigators [4] and the pumping-power constants (c1 and c2) are obtained from linear regression on the dimensionless pumping power and flow number, as described in Section 4.3. The expression in Eq. (20) can also be compared to the emulsification efficiency factor suggested by Jasinska et al. [10] which also includes a flow number dependence. However, the principal difference is that this previously suggested model assumes that the dissipation power scales with the rotational term in the shaft power-draw, i.e. it is classified as a case I-model in Table 1.

Fig. 8. Pumping efficiency of the three mixers. See Fig. 5 for symbols. The bold dotted line show the pumping efficiency of a typical centrifugal pump (LKH50, Alfa Laval, Sweden).

spent on pumping, the pumping efficiency, can be expressed as   c þc N N hpump ¼ 1 2 Q Q : ð19Þ NP0 =NP1 þ NQ The experimentally measured pumping efficiency for the three mixers are displayed in Fig. 8 (markers) as a function of flow number together with the model from Eq. (19) (lines). The model accurately describes the experimental data. The pumping efficiency of a typical centrifugal pump (LKH50, Alfa Laval, Sweden) has been inserted in Fig. 8 for comparison. The pumping efficiency of the three mixers, and the model in Eq. (19), are qualitatively similar to a centrifugal pump but show a lower pumping efficiency, as has been previously noted [20]. As opposed to a centrifugal pump, which is constructed to convert electrical power to static pressure, the inline RSM is designed to dissipate energy. An efficient inline RSM should have a low pumping efficiency since, by the principle of energy conservation, energy used to increase the static pressure of the fluid cannot simultaneously be used for generating turbulence. It can also be seen that the mixer that is least similar to a centrifugal pump (mixer C), has the lowest pumping efficiency. This illustrates how each inline RSM design must find a suitable balance between being an efficient pump and an efficient dissipator of TKE. As seen in Fig. 8, this balance also shift between different flow numbers, for the inline RSM as for the centrifugal pump. The inline RSM is therefore expected to have a lower dissipation efficiency at intermediate flow numbers (0.02–0.04) where pumping efficiency is at a local maximum.

4.6.1. Comparison to CFD simulations Fig. 9 compares the different models for the total TKE dissipation power expressed as a ratio of loss-corrected shaft power draw (Eq. (1)), for the settings of mixer A. In the figure, models I to IV refer to the previously suggested models and model V to the new suggestion in Eq. (20). The total dissipation of TKE from the CFD simulations (calculated from Eq. (17)), normalized by the total power input (calculated from the CFD calculated torque) has been inserted as a comparison (markers). The CFD simulated total dissipation complies well with model V (Eq. (20)) and deviate substantially from all the previously suggested models in Table 1. The finding that Eq. (20) compares well with CFD-results obtained over a range of tip-speeds (10–30 m/s) and flowrates (3– 120 m3/h) offers support for the validity of the new model. However, it should be kept in mind that CFD with two-equation turbulence models does not always predict absolute values of dissipation rate of TKE accurately [26]. Further validation is therefore of interest. 4.6.2. Comparison to emulsification experiments Emulsification experiment data offers an orthogonal comparison of the ability of the new model to describe the dissipation rate of TKE at different conditions. From basic emulsification theory [24], the diameter of the largest drops is expected to scale with the dissipation rate of TKE in the most intense region, which for a given mixer design is proportional to the total turbulent dissipation taking place in the efficient region of emulsification. Thus, if the expression for Pdiss as a function of flowrate and rotor speed is correctly specified, it should collapse measurements of maximum drop diameter on a single master curve.

4.6. A new model for total turbulent dissipation A new model for Pdiss based on energy and momentum conservation, and consistent with experimental data on pumping power, can be formulated from Eq. (9): Pdiss ¼ P0shaft  Ppump ¼ P0shaft  hpump P0shaft    ¼ Prot þ Pflow 1  hpump ;

ð20Þ

where P rot and P flow are defined in Eq. (1) and hpump in Eq. (19). This expression contains four constants that needs to be determined from experiments in order to apply it to a general inline RSM. The two power-draw parameters (NP0 and NP1) are obtained using torque measurements and regression analysis as

Fig. 9. Comparing the suggested models for Pdiss, based on the geometry of mixer in A. Models (I) to (IV) refer to Table 1. Model (V) is the new suggestion in Eq. (20). Markers are estimations of total dissipation from CFD (Eq. (17)).

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Fig. 10. Scaling between drop diameter (Dv,95) and dissipation power for two volume fractions of oil: ’D = 0.5% (o) and ’D = 52% (&). Error bars show mean value plus minus two standard errors estimated from replicate experiments at a single point. A: Based on the dissipation correlation originally used in Refs. [15,23], Pdiss,II. B: Based on the new correlation for dissipation power in the effective region of emulsification (aPdiss), as suggested in the present study.

Fig. 10A displays the drop diameter of the largest surviving drops (Dv,95) from ref. [23] plotted as a function of the total dissipated energy, Pdiss, according to the correlation originally used for interpreting the results in Ref. [23] (i.e. using model II in Table 1). First note that drop diameters are smaller for the higher volume fraction of dispersed phase. At a first glance this might seem counter-intuitive since increasing volume fractions of oil increases the rate of coalescence during emulsification. However, it also increases the emulsion viscosity, which in the turbulent viscous breakup regime greatly increases the fragmenting stress. This interpretation that the emulsification takes place in the turbulent viscous regime, is also supported by the tip-speed scaling and from comparisons of drop sizes with estimations of Kolmogorov length-scales [15,23]. It should also be noted that the measurements were obtained at different rotor tip-speeds and volumetric flow-rates but that the correlation from model II includes no flowrate or flow number dependence. Hence it suggests that there should be no systematic variation between measurements with the same rotor tip-speed but different flowrates. However, substantial variation between the points can be seen in Fig. 10A (these observations are indicated by arrows in Fig. 10A) For the steady-state drop-diameter, turbulent emulsification theory predicts a power-law dependence between drop-diameter of the largest drops (i.e. Dv,95) and the dissipation rate of TKE in the effective region of emulsification [24]. The best-fit power-law models for the two cases in Fig. 10A have been inserted as lines and have slopes 0.38 (R2 = 94%) for the low volume fraction and 0.56 (R2 = 84%) for the high volume fraction case. This could be compared to the 0.5 theoretical exponent of the turbulent viscous regime Fig. 10B displays the experimentally obtained drop sizes plotted versus the correlation for the dissipation in the effective region of emulsification (aPdiss) as suggested in this study. The experimental data complies better with the loglog-scaling suggested by theory when applying the new correlation, the R2-value increases from 94% to 96% and from 84% to 85% for the low and high volume fraction cases respectively. Moreover, the slopes are 0.46 and 0.65 for the low and high volume fraction cases respectively when using the new correlation. For the low volume fraction case this complies better with the theoretical scaling and for the high volume fraction case it complies equally well. Furthermore, for the low volume fraction case, the four measurement points with constant tip-speed, but varying flowrate (see arrow in Fig. 10B), the variation between points is described by

the new correlation. For the high volume fraction case, the experimental uncertainty is larger and it is more difficult to see any improvement using the new correlation. In conclusion, the data, provides some experimental indications of the improved validity in interpreting emulsification experiments using the new correlation, especially from the low volume fraction case. However, more investigations using data with higher experimental accuracy is needed to elaborate on more general improvements on emulsification interpretation. 4.7. A model for the dissipation rate of TKE and notes on future work In summary, the new model for calculating the dissipation rate of TKE in the region of emulsification is given by

e¼

a  Pdiss V diss  r

ð21Þ

with Pdiss as defined by Eq. (20). As seen from the discussion above, the critical step of formulating a correlation for the dissipation rate of TKE for an inline RSM is in the model for the total dissipation, Pdiss. The main finding here is that the previously suggested expressions perform poorly. The model proposed in this study (Eq. (20)) is based on previously published experimental data and complies with energy and angular momentum conservation. However, further experiments are needed to test the generalizability of the expression across flow-numbers and mixer designs. The deviations from the linear trend in Fig. 7 at low flow numbers also requires special attention, especially in the light of recent findings by Jasinska et al. [10] of deviating hydrodynamics of inline RSMs at low flow numbers. Moreover, the experimental comparisons in Fig. 10 should only be considered a first test. Testing with high quality experimental data will provide further validation of the suggested expression. 5. Conclusions By analyzing previously published data, it was concluded that the pumping power of three different inline mixers is proportional to the flow-term of the power draw correlation multiplied with a linear function of the flow number. By comparison to centrifugal pump theory, this type of linear scaling is expected based on conservation of angular momentum in the rotor. From energy conservation, pumping power and total turbulent dissipation power are related. Moreover, the empirically observed

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pumping power as a function of flow number is not in agreement with previously suggested models for the dissipation rate of TKE. A new correlation for the dissipated energy in an inline RSM was suggested based on two main propositions: First, that the dissipation is the loss-free shaft power subtracted with the pumping power (as suggested by energy conservation). And secondly, that there is a linear relationship between pumping power and the flow-term in the power draw correlation multiplied with a linear function of flow number (as suggested by angular momentum conservation). The proposed model compares well to CFD data. Comparisons to emulsification experiments suggest some improvements as compared to a previously used correlation. However, further mixing and emulsification experiments are needed to test the generalizability of this specific form, and how well it predicts mixing and emulsification performance. Acknowledgments The authors would like to thank the participants on Mixing XXV in Quebec June 2016 for valuable suggestions and comments on the topic of the study, in particularly Prof. Richard Calabrese is acknowledged for valuable discussions on the relationship between pumping efficiency and dissipation. This study was funded by The Knowledge Foundation (grant number 20150023) and Tetra Pak Processing Systems AB. References [1] V.A. Atiemo-Obeng, R. Calabrese, Rotor-Stator mixing devices, in: E.L. Paul, V.A. Atiemo-Obeng, S.M. Kresta (Eds.), Handbook of Industrial Mixing, Wiley, Hoboken, 2004, pp. 479–505. [2] J. Zhang, S. Xu, W. Li, High shear mixers: a review of typical applications and studies on power draw slot pattern, energy dissipation and transfer properties, Chem. Eng. Process. 57–58 (2012) 25–41. [3] V.A. Atiemo-Obeng, R. Calabrese, Rotor-stator mixing devices, in: S.M. Kresta, A.W. Etchells, D.S. Dickey, V.A. Atiemo-Obeng (Eds.), Advances in Industrial Mixing, Wiley, Hoboken, 2016, pp. 255–502. [4] M. Cooke, T.L. Rodgers, A.J. Kowalski, Power consumption characteristics of an in-line silverson high shear mixer, AIChE J. 58 (2012) 1683–1692. [5] J.A. Baldyga, A.J. Kowalski, M. Cooke, M. Jasinska, Investigations of micromixing in the rotor-stator mixer, Chem. Process Eng. 28 (2007) 867–877.

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