1. Introduction
A fully continuous mode of production is often desired in most types of industrial processing. Continuous production decreases the per unit cost of production and reduces the risk of quality differences between batches. However, batch processing is still commonly employed in many production lines, especially in the food, pharmaceutical, and cosmetics sectors.
Continuous production does introduce some additional difficulties. The first requirement is that each unit operation in the production process can be achieved in a continuous setup. Second, converting a given batch process into a continuous one requires an understanding of what (if any) differences there are between the batch and continuous versions of each unit operation. This second difficulty also applies to product development projects for continuous mode production, since laboratory testing is almost always performed in batch.
This review focus on liquid processing in rotor-stator mixers (RSMs), also known as high-shear mixers. RSMs, together with high-pressure homogenizers, are considered the standard tool for mixing and emulsification of liquid dispersions. High-pressure homogenizers are generally used for low to intermediate viscosity products and RSMs for products with higher viscosities [
1]. Whereas high-pressure homogenization is an inherently continuous operation, RSMs can be operated in either batch or continuous mode. The same rotor-stator head is often used in both batch and continuous mode of operation, see illustrations in
Figure 1 and
Figure 2. Continuous mode RSMs are sometimes referred to as inline (or in-line) RSMs.
From an industrial perspective, it is important to understand how the characteristics of a given batch mode RSM compare to those of a given continuous mode RSM. This understanding is crucial for both converting an existing batch production process to continuous production, and for generalizing results from laboratory (batch) experiments to pilot and production scale in a product development process. Since the rotor-stator heads are often very similar between batch and continuous modes of operation, it is tempting to assume that converting between the modes is straight-forward, both in terms of production economy (i.e., power draw of the rotor shaft) and in terms of obtained product quality (mixing or dispersing efficiency). However, as has become apparent in the scientific literature, this is not obviously the case [
2,
3]. Great care must therefore be taken when comparing RSMs run in batch to RSMs run in continuous mode of operation.
RSMs are used in many different applications, particularly in the food, pharmaceutical, and cosmetic processing industries. However, as pointed out in an editorial from 2001, despite their wide use, there has been a lack of fundamental understanding [
4]. During the last 16 years there has been an increasing number of scientific research projects aimed at characterizing and understanding RSMs. Three major reviews have been published since then, summarizing many of these advances. In 2004, Atiemo-Obeng and Calabrese provided a comprehensive review of the mechanical designs of RSMs, focusing on power draw, flow profiles, and scale-up [
5]. This review was also updated in 2016 [
6]. Another fairly recent review has been provided by Zhang et al. [
7], focusing on power draw and flow fields, but also providing an overview of the proposed emulsification scaling-laws and the mass and energy transfer correlations. However, none of these previous reviews provide a comprehensive discussion on the difference between batch and continuous mode of operation. Moreover, there has been a number of relevant studies on this in the last couple of years, after these reviews were written.
The objective of this contribution is to provide a more specific review on what is known about the similarities and differences between RSMs operated in batch and continuous mode, including the most recent advances. The intention is to provide an overview, both for engineering professionals struggling with the transition from batch to continuous rotor-stator mixing, and for the research community utilizing or studying RSMs. After a brief description of RSMs, this review will focus on four topics: The shaft power draw (power requirements) of batch and continuous mode RSMs (
Section 3), the flowrate and pumping capacity of RSMs (
Section 4), the pumping and turbulent dissipation efficiency (
Section 5), and the implications of these differences on emulsion processing (
Section 6). A summary of recommendations for further studies to resolve remaining issues is provided in
Section 7, and the review is concluded in
Section 8. Although RSMs can be operated under both laminar and turbulent conditions, this review will focus on turbulent RSMs, which are the most common RSMs employed in industrial applications.
2. The Rotor-Stator Mixer
The term RSM does not refer to a specific design but a range of mixer geometries [
5]. RSMs are produced by several different manufacturers, and each have their own design, or more often several different designs for use with different applications, see References [
5,
7] for detailed overviews of different RSM geometries.
The common denominator of these RSMs is that they all consist of one or several high velocity rotors and one or several static stator screens separated by a short distance, the rotor-stator clearance, δ. The rotor accelerates the fluid tangentially and redirects it radially through the stator holes or slots. This gives rise to steep velocity gradients in the stator slot, or in the direct proximity of it, and creates a narrow region of high intensity hydrodynamics stresses [
8,
9,
10,
11,
12,
13,
14,
15], which give rise to high mixing and dispersing efficiency, characteristic of RSMs [
14,
15].
Although there are many different rotor-stator designs, they can broadly be classified into two groups based on the rotor: teeth-designs and blade-designs. A schematic view of the two design principles can be seen in
Figure 3. As seen in the figure, the blade-design uses a rotor similar to that found in a centrifugal pump, either extending all the way from the shaft (as in the figure) or with shorter blades mounted on a plate attached to the rotor. The teeth-design uses a circular plate-mounted rotor, as seen in
Figure 3. Both blade- and teeth-designs can use different stator screens (differing in the shape and size of the holes). Many rotor-stator heads also have multiple (i.e., 2–3) sets of concentric rotors and stators.
Figure 4 displays velocity profiles calculated with computational fluid dynamics (CFD) from two recently published investigations; one on blade-design [
16] and one on teeth-design [
11]. Looking at the general outline of the flow, there are many similarities. We can see how the fluid obtains a high tangential velocity in the rotor-stator clearance region, and how it is accelerated into a turbulent jet as it enters the (outer) stator slot. Note that the jet attaches to the leading edge of the stator and that the jet only fills a small portion of the slot [
8,
11,
12]. This gives rise to a re-circulation region in the slots and, consequently, a “back-flow” of fluid that re-enters the slot from the bulk without passing the rotor [
5,
8,
12,
15,
17,
18].
The same rotor-stator heads are often used in both batch and continuous RSMs [
3,
19], the difference is primarily in how they are mounted, and how the product flow is subjected to the rotor-stator. In batch operation, the rotor-stator is mounted inside a mixing tank, either as an integrated part of the bottom of the tank as in
Figure 1 (as is often the case in production-scale batch RSMs) or mounted on an impeller shaft lowered into the tank (more common for laboratory-scale batch RSMs). When used for continuous production, the rotor-stator head is mounted inside a narrow casing, similar to a centrifugal pump, with an inlet directing fluid towards the center of rotation and an outlet mounted at the periphery, see
Figure 2.
4. RSM Flowrate and Pumping
The net flowrate passing through the stator holes,
Q, can be described in terms of a flow-number,
NQ, via:
This applies for both batch [
8,
9,
11,
12,
38] and continuous modes of operation. However, it should be noted that the interpretation and controllability of this value differ substantially between the two modes. For a continuous RSM, flowrate is an externally set and easily measured parameter. Flowrate, and consequently
NQ, can be adjusted by varying what is referred to as the system curve in pump design; the total pressure loss of the system the mixer is connected to. In practice,
NQ can be decreased by using a valve downstream of the mixer or increased by adding a separate feed pump placed in series with the mixer.
When the mixer is operated in batch mode, however,
NQ is a constant that depends on the geometry of the mixing head [
8,
9,
12,
17,
38], and to some degree on the tank geometry. The
NQ parameter is important for batch RSMs since it determines how fast the liquid is mixed. More specifically the expectation value for the time a fluid element spends in the tank between two passages of the rotor-stator head is [
3,
19,
29]:
where
VT is the tank liquid volume.
Another difference between the two modes of operation is that
Q (and consequently
NQ) is difficult to measure for batch RSMs; it requires a non-intrusive experimental technique for measuring fluid velocities inside of and just outside of the stator slots, such as laser Doppler anemometry (LDA) [
11,
12,
38] or particle image velocimetry (PIV) [
17]. Alternatively, it can be determined by a CFD model that has been validated by one of the above-mentioned experimental techniques [
17].
Table 2 compiles values of
NQ for a number of different batch RSMs and compares them to the
NQ span resulting from operating some different continuous mode RSMs under technically relevant flowrates [
3,
16,
27,
34,
39]. As seen in
Table 2, flow numbers are between 0.1 and 0.3 for batch RSMs. Systematic investigations are scarce, but based on a recent PIV investigation, it has been suggested that
NQ decreases with increasing stator slot width. This phenomenon has been linked to the increase in backflow obtained when increasing the slot width [
9].
For continuous RSMs,
NQ is substantially lower (
NQ < 0.1); it is not uncommon that continuous mode RSMs are operated at a flow number one or several decades below that of batch RSMs. This implies that the flowrates through the mixer are substantially lower for continuous mixers compared to those through batch mixers. Using the same rotor-stator head and operating it at the same rotor speed will therefore result in much lower radial velocities in the rotor-stator region. This difference can also be explained using a centrifugal pump analogy. The flowrate is determined by the properties of the pump (what in pump-theory is referred to as a pump curve) and the properties of the system (the system curve) [
16]. Since tanks used with batch RSMs provide a much lower flow resistance than the pipes used for continuous RSMs, the flowrate becomes substantially higher.
6. Implications for Emulsification
From an industrial perspective, the most important question with regards transitioning from batch to continuous RSM is how to operate a continuous mode RSM in a way such that it results in the same product quality as batch RSM. In an emulsification context, this corresponds to the question of how to predict the resulting drop diameters in batch and continuous operation RSMs, and to the question of whether there are mechanistic differences between the two products.
Turbulent drop breakup is often explained in terms of Kolmogorov–Hinze theory [
44,
45,
46,
47,
48], which suggests scaling relations between the largest drop diameter that can survive a given turbulent field and the dissipation rate of TKE of that field. Depending on the size of this limiting drop in relation to the size of the smallest turbulent structures (the Kolmogorov length-scale), different explicit scaling laws have been suggested, see References [
7,
41,
47] for comprehensive reviews and some different explicit formulations.
However, as shown for impeller mixers [
49] and high-pressure homogenizers [
50], in order for this approach to be satisfactory, the dissipation rate of turbulent kinetic energy should be the local value in the most intense region (where breakup takes place). However, since the local dissipation rate of TKE is highly challenging to measure [
49], practical application of Kolmogorov–Hinze theory to RSMs are often based on externally measurable quantities such as rotor speed [
31,
39,
51], the total dissipation power (
Pdiss) [
52,
53], and the globally defined Reynolds and Weber numbers [
41]. The global Weber number is defined as:
where
σ is the interfacial tension of the drop. The Kolmogorov–Hinze theory is very general, and not specific to design or mode of operation. However, there is some disagreement in the scientific literature when it comes to the question of if there are mechanistic differences between emulsification in the two modes of operation, and hence, if the same scaling law expressions can be used for both modes of operation.
Experimental studies have reported some systematic differences. Emulsions passed
n times through a batch RSM do not always show the same drop size as an emulsion processed for a time
t = nτ, despite the fact that this would result in the same the number of passages though the RSM, at least in terms of expectation number [
2,
3]. Three different standpoints discussing this discrepancy and the emulsification implications can be found in the RSM literature.
6.1. Flowrate and Its Influence on Turbulence
As previously mentioned, there is a decisive difference in flowrate (and thus in
NQ) between RSMs operated in batch and continuous mode; batch mode RSMs show approximately ten times higher flowrates (
Table 2). The difference between the two systems can be investigated by understanding the effect of flowrate on emulsification. Hall et al. [
39] undertook a large systematic investigation of emulsification in continuous mode RSMs and suggested that the flowrate-based Reynold number:
where
d is the slot diameter and
Atot is the total flow-through area of the stator, has a small but significant effect on the resulting drop size (when tip-speed is kept constant). When keeping the geometry constant, it can be shown that Re
Q is proportional to the product between flow number and Reynolds number [
54]:
This would suggest that NQRe would be an appropriate scaling law when comparing emulsification results from batch to inline mode of operation. However, it should be kept in mind that the variations in flowrate seen in this type of experiment are much smaller than that between continuous and batch RSMs.
6.2. Radial Flow and Dissiaption Profile Scaling
RSM mixing and emulsification ultimately depends on the hydrodynamic conditions created in the rotor-stator region. Thus, it is interesting to investigate if there are any differences to the flow fields between batch and continuous modes of operation for the same rotor speed and rotor-stator head geometry. Unfortunately, no such experimental studies have been reported. However, a recent CFD investigation might be used to shed some light on the situation [
54]. The study [
54] reports flow fields obtained with CFD for a continuous mode RSM run at different flowrates and rotor speeds. The lower flowrates correspond to those generally obtained for continuous mode of operation (
NQ = 0.007) and the higher to those obtained in batch mode of operation (
NQ = 0.082). The radial velocity profiles in the stator holes were compared and it was found that neither Re
Q nor
NQRe
Q were appropriate scaling laws, in contrast to what was suggested in
Section 6.1. Instead, it was found that both radial and tangential velocities (appropriately scaled with rotor speed) were determined by the flow number [
54]. This has an important implication on the difference between the two modes of operation. Since transitioning from batch RSM to a continuous mode RSM decreases
NQ, the velocity profile in the rotor-stator head will undergo a substantial change. This change is not merely a scaling due to the reduction in flowrate but a shift into a fundamentally different turbulent flow [
54]. Most notably the position of the highest local dissipation rate of TKE shifts from the turbulent jet formed downstream of the slot in the batch RSM flow number, to the rotor-stator clearance for the continuous RSM flow numbers [
54]. This effect is illustrated in
Figure 6.
A shift in the position of highest local dissipation rate of TKE suggests a mechanistic difference between the two modes of operation. As seen in
Figure 6, the dissipation volume is smaller for the low-
NQ (continuous mode) case than in the high-
NQ (batch mode) case. This also suggests that the average dissipation rate in the two regions will scale with different parameters: with the clearance length-scale for continuous RSMs and with the slot diameter for the batch RSM. However, this has not yet been experimentally verified.
Due to the lack of experimentally measured flow fields, this difference in flow pattern between modes of operation has not yet been experimentally verified, but a validation study has shown that the CFD model employed is able to capture the position of high intensity local dissipation at least for the batch RSM [
55].
Further insight can be obtained by single drop breakup visualizations. However, only one such study on RSMs has yet been reported [
42]. This study was conducted for a batch RSM (
NQ = 0.11) [
8] and showed that drops are deformed and subsequently broken up just downstream of the stator hole [
42], as suggested by the CFD simulations for the
NQ-values found in batch RSM (i.e., in
Figure 6). However, no corresponding investigations on continuous RSMs (or low
NQ-systems) have yet been reported.
6.3. A Purely Stochastic Effect
An altogether different, but highly promising approach to describe the previously reported differences between batch and continuous modes of operation on emulsification results has recently been reported. Carrillo De Hert and Rodgers [
19] suggest that the differences only apply if the wrong scaling is employed when comparing data from different modes of operation.
In the first step of their study, they conclude that the mode drop diameter,
d0, resulting from processing an emulsion n times through their continuous mode RSM at flowrate
Q and rotor speed
N is given by [
19]:
Note the −6/5 exponent, that corresponds to the scaling expected from Kolmogorov–Hinze breakup in the turbulent viscous regime [
7,
47].
The authors then continue by suggesting that the only difference between continuous and batch modes of operation is in the stochastic effect of the rotor-stator head passage [
19]. After processing an emulsion for a time
t in a batch system, the expectation number of the number of passages is:
However, for each volume element of the emulsion, the actual number of passages is a stochastic property following a Poisson distribution. By assuming that Equation (16) (the model for continuous mode of operation) applies each time a volume element passes the rotor-stator head, they conclude that the corresponding model for a batch system after being processed for a time
t would be [
19]:
Moreover, for
t/
τ > 2, Equation (18) converges to [
19]:
which was found to accurately describe their data for a wide range of properties [
19].
These results suggest that there is no mechanistic difference between the modes of operation and that (at least when processing times are fairly large) emulsification results can be translated directly using Equation (17); one continuous mode RSM passage would then correspond directly to processing for
t/
τ in a batch RSM. However, this is not completely general. In a previous study on emulsification of mayonnaise, it was concluded that this scaling was inadequate to describe the experimental differences [
3]. The reason behind this discrepancy is still not understood, but it is hypothesized that it is related to the higher volume fraction of oil in Reference [
3] which increases the complexity of the process.
In summary, there is as of yet no consensus in literature on which (if any) of these theories (
Section 6.1,
Section 6.2 and
Section 6.3) best describes the differences between emulsification efficiency in batch and continuous modes of operation. Some of the confusion can be explained by postulating that the underlying differences between batch and continuous modes—i.e., the higher flowrate in batch systems—are the different hydrodynamic effects of the rotor-stator head in RSMs with different designs. It might be that the Re
Q-scaling is appropriate for the design investigated by Hall et al. [
39] (a pilot scale Silverson dual blade design), that the
NQ-scaling is appropriate for the design investigated by Håkansson et al. [
54] (a production scale Tetra Pak blade design), and that there is no mechanistic difference for the design investigated by Carrillo De Hert and Rodgers [
19] (a laboratory scale Silverson blade design). However, it is not clear why these different behaviors would occur, and how they could be linked to the design differences. Further investigations are needed in order to draw any definite conclusions on this matter.
8. Summary and Conclusions
The objective of this contribution was to review the current scientific based understanding of the differences between RSMs in batch or continuous mode of operation.
Section 3 showed that correlations for shaft power draw are available for both modes of operation, allowing for accurate prediction of process economy in terms of energy expenditure. In
Section 4, it was seen that the flow number (
NQ), and consequently the flow through the stator screen,
Q, is considerably lower for continuous mode of operation (compared to batch mode) when using the same rotor-stator head and operating it at the same rotor speed.
Section 5 showed that, in general, a much higher proportion of the energy fed to the shaft is converted into turbulence in the high-intensity region where mixing and emulsification takes place for a batch RSM than for an RSM operated in continuous mode. For a continuous mode RSM, more of the energy is used for pumping (i.e., increasing the head of the flow).
Section 6 discussed what this implies when comparing emulsification efficiencies between the two modes of operation. Several different theories have been suggested, but there is of yet no clear consensus in the literature for how continuous mode RSMs should be operated in order to give the same emulsion as in a batch RSM.