Performance Evaluation of Helical Separators Applied to Olive Oil–Water Two-Phase Flows at Low Reynolds Numbers

: One of the unit operations involved in the production of olive oil is the separation of liquid– liquid systems (and other multiphase ﬂows) in their fundamental phases. The use of helical separators could be an alternative to be considered for that task in order to reduce energy consumption and improve the quality of products in olive oil mills (‘almazaras’). In this work, four models of helical separators have been built and tested in order to manage olive oil and water two-phase ﬂows (with olive oil as the majority phase). Separation yields were analyzed from a dimensional analysis perspective, considering variables such as density and viscosity, ﬂow rate, head losses, or the water concentration in the ﬂows studied. The best separation yields (of the order of 80% to 100%) were obtained for olive oil–water two-phase ﬂows in which the water concentrations could be higher, in some cases, than 5–10% for Reynolds numbers of below 60.


Introduction
Among the main objectives linked to food industry operations is the transformation or adaptation of incoming raw materials into manufactured products that meet the needs and wishes of consumers. Focusing our interest on a group of these industries, specifically those devoted to the production of olive oil (oil mills), one founds that it is a common problem to have to work with liquid-liquid systems (unstable suspensions or emulsions), liquid-solid or solid-liquid systems (suspensions and pastes) that have to be separated in their fundamental phases. In conventional practice, the aforementioned separations are usually carried out with techniques such as decantation (centrifugal or gravitational), sieving, or filtration, each of them having advantages and disadvantages with certain controversy [1][2][3][4]. In this sense, one could highlight that centrifugal decantation is usually considered as being a very fast, clean operation, but its energy cost can be high and the volume of wastewaters generated ('alpechin') can also be important. Gravity settling (using decantation tanks) is energetically less costly than centrifugation, but often provides slower separations, which can lead to sensory defects in the oil. With respect to sieving and filtration, it should be noted that these operations are complementary to the previous ones, as they are aimed at removing the small amounts of solids not retained in a previous decantation.
In this context, the use of helical separators (Figure 1), employed experimentally to treat water-oil systems associated with the petrochemical industry [5], could be an alternative to be considered in the olive oil mills area. It is worth noting that these devices do not require any moving parts, and they can easily be set up by using a cylinder shape in which a tube is coiled. Energetically they have fewer requirements than the centrifugal machines, and their residence (separation) times are low, of the order of seconds [6]. Regarding their internal dynamics (the behavior of the flow inside them), it can be pointed out that the use of helical tubes has been extensively studied for one-phase flows under laminar regime [7][8][9][10] or under turbulent flow [11][12][13]. It is a characteristic of these types of streams that, together with the main movement of the longitudinal flow (along the axis of the pipe), two transverse vortices are present. These structures gradually lose their influence on the stream as the Reynolds number increases (as the flow transits from laminar to turbulent regime). Due to the action of the centrifugal force, the velocity profiles are not symmetrical in a given section, which signifies a further complication in understanding the behavior of this type of flows (see Figure 2 as an example).   In helical two-phase flows, a large part of research has been orientated towards those cases in which liquid-gas systems are analyzed [8,[14][15][16][17], because the latter are frequently involved in industrial applications (such as power generation, refrigeration, heat recovery systems, etc.). Taking into account that liquids and gases show comparatively broad differences in their physical properties, it is reasonable to assume that the results of these studies may have limited value for other systems such as those formed by two liquids [18]. Studies made on water and oil two-phase flows in helical separators, are not numerous, which means that there is room for improving the design and operating conditions that could be of use for achieving acceptable separations with this type of technology. In this context, one reference work is that of Zhang et al. [19], in which, starting from tubes with an internal diameter d = 25 mm (with helix radius R = 150 mm or R = 400 mm), separation yields of close to 80% were obtained with water concentrations of around 40%. According to these authors, the separation efficiency of helical pipes depends on variables like: the helix radius, density difference between phases, residence time, pressure at the inlet and outlet or liquid droplet diameter. In this regard, for a given device, and thanks to the density difference between phases, when the flow rate is increased (by operating, for instance, the inlet pressure) or the helix radius is reduced, it is possible to obtain a higher centrifugal force. This usually leads to obtaining better separations if the turbulence disturbances are not too strong. This could be considered to be the key point of the performance of these devices: to achieve the highest centrifugal force with the lowest turbulence level. The above authors [19] also found that for low flow speeds, of the order (or less) of 1 m/s, the gravity force could have some influence on the distribution of water and oil within the duct, which allowed the obtainment of acceptable phase separations by making small discharge holes at the bottom of the helical tube outlet.
Other works like those of Zhao et al. [5,20] and Niu et al. [21] reported separation yields of close to 91% (for Reynolds numbers of the order of 7600) using a wide helix radius (R = 500 mm) with tubes of diameter d = 25.4 mm. In these studies, the maximum water concentration in the flow did not exceed 0.4%. From another viewpoint, by using pipes and helixes of a small diameter (d = 4 mm and R = 13.15 mm or R = 38.95 mm), Vallesquino and Molina [6] managed to obtain separation yields of close to 100% in water-oil two-phase flows, in which the oil concentrations were below 2% and the Reynolds numbers were less than 2000. With this in mind, and considering the usefulness and potential offered by this new alternative for the phase separation of liquid-liquid systems in any type of industry (especially in those devoted to olive oil production), this experimental work has aimed to study the behavior of these devices when they are applied to olive oil-water systems at different concentrations (water will be the minority phase). Separation yields will be analyzed, considering variables like the Reynolds number, head losses or water concentrations in the flow, and some recommendations will be given that could be useful for future studies.

Materials and Methods
This section details the material and methods used in the experiments performed in this work. To this end, the elements that are part of the separators, and of the hydraulic system to which they were coupled, are presented. Then, the methodology applied throughout the experiments, in which these components were used, is described.

Separators
Taking into account the results and guidelines proposed by Vallesquino and Molina [6], together with the available resources, it was decided to build four helical separators using polyethylene pipe with a nominal diameter of 10 mm (inner diameter d = 9.45 mm plus 2 mm thickness). Two of these separators had a helix radius R = 34.05 mm, and the other two were constructed with R = 41.75 mm. In turn, for each value of R, two possible lengths (L) of tube were taken: L = 4 m and L = 7 m. In each separator, a length of 1 m was used for coupling the device to the hydraulic installation employed, leaving the rest of the tube (L-1 m) to construct the helical part of the device ( Figure 3). In each separator, the dispersion (or emulsion) to be treated was introduced under pressure from the top (see 1 in Figure 3), and at the lower end (4), at atmospheric pressure, the flow was split into two streams (q in and q out ) by dividing the original pipe section into two equal semicircles, which were separated by the wall of the coil mold (that was 0.3 mm thick). The flows obtained were driven towards two separate funnels (6), and then into two different plastic vessels (7). With this approach, it was sought to disturb the streamlines at the end of the pipe as little as possible to prevent a water and oil remix. Figure 4 schematically depicts the hydraulic set-up used to test the separators cited above, in which the 19 mm PE pipe was generally used (except in elements (6) and (11)). In carrying out the experiments, a series of steps was followed with frequent repetitions. Next, the general procedure applied during the experiments is described taking as a reference the abovementioned Figure 4: firstly, and maintaining all the valves closed, a 0.05 m 3 regulation tank (1) located at a level of about 7.5 m was filled with olive oil. Meanwhile, by gravity, the pipe stretch (1)-(2) was filled up with the oil from (1). After this, and through a removable coupling (5), water was supplied manually into the system by using a funnel that was momentarily coupled to a special stretch of 40 mm diameter pipe 1.5 m length (6). Subsequently, the rest of the system was filled up with olive oil by opening valves (3) and (3 ). By means of a valve (4) it was possible to simultaneously purge the air that could be occluded in the ducts, thus eliminating any difficulty that might appear in this regard. This operation could be performed whenever necessary. Once olive oil and water have been fed into the hydraulic system by opening valves (3 ) and (7), with a larger or smaller aperture, different concentrations of both fluids could be injected at the mixing zone (8). At point (9), a digital manometer was placed (with a maximum operating pressure of 20 m.W.C and accuracy of ±2%) to measure the pressure head at the inlet of the separator (11). In this context, numerous tests (not less than 20 per separator) were carried out at different pressure heads and outflow rates (as will be shown later) to analyze the hydraulic performance and separation yields of these devices under different conditions. In each experiment, the samples from the outer and inner sides of a given separator ( Figure 5 as an example), were collected in coded and tared plastic vessels for their subsequent, efficient management. To enable the measurement of the volumes of olive oil and water contained in each sample, the following procedure was applied: samples taken in each experiment were stored and protected for further study. Then, they were weighed by using a laboratory scale with a precision of ±0.1 g. After this, the samples were placed in an oven at 105 • C for several days until the water was completely evaporated (checking the weights regularly until the final weight loss was verified). From the value of the last weighing, and the difference in the readings with respect to the pre-drying measurement taken, the weight of the oil and water content present in each sample could be estimated in each case. From this information, the corresponding volumes, of water and olive oil in each sample, could be computed by taking into account the density of each fluid. Accordingly, water density (ρ w ) was taken from the values tabulated in Singh and Heldman [22]; olive oil density (ρ o ) was estimated from the following experimental function [23], determined particularly for the olive oil used in these tests (in which T is the oil temperature in • C, ρ o is computed in kg/m 3 , and S E = 2.040 kg/m 3 is the standard error):

Hydraulic System and Test Procedure
ρ o = 932.262−0.977 T ± S E (1) Once the volumes of olive oil and water were estimated in each sample, it was possible to determine the flow rates linked to them, since each sample was taken at a given time that was measured in each experiment using a digital chronometer (with an accuracy of ± 0.01 s). To ensure that all the experiments were conducted under similar conditions, the flow temperature was measured regularly in the tank (1), and at the separator outlets, with a digital thermometer (with an error of ± 1 • C). As all tests were performed under a stable environment, the flow temperature remained at around 25 • C in all cases.

Results and Discussion
Next, the results of the tests described in the previous section are presented and analyzed, on a theoretical basis assuming stationary flow, in order to ascertain whether helical separators are suitable for treating olive oil-water two-phase flows with adequate separation yields (η). Thus, it is reasonable to assume that, among very many other factors, physical properties such as fluid densities ρ w and ρ o , and dynamic fluid viscosities µ w and µ o , are variables that could be preliminarily taken to address the corresponding dimensional analysis of this problem. Note that subscripts "w" and "o" in each variable mean, respectively, that that variable is related to water or to olive oil. Beside the former, operating parameters like: the flow rate q, the pressure head h, temperature T, phase (water) concentration C (as a dimensionless fraction or percentage) and the inlet droplet diameter d p (of the dispersed fluid) could also be integrated into this discussion. Moreover, geometrical variables like d (inner pipe diameter), R (helix radius), L (separator length) or Z (helix pitch) should not be forgotten. Finally, gravitational (g) or surface tension (γ) effects could also be accounted for, which could imply, overall, that the problem raised initially would signify managing 16 variables (including η).
As the previous approach could be difficult to handle, despite it being possible to reduce the number of variables by applying dimensional analysis methods, at this point, some assumptions need to be made to reduce the complexity of this question (leaving for future works those issues that could improve this study): (1) Since all the experiments have been developed under a stable flow temperature, variable T can be excluded in this case. (2) As the separators were built using the same helix pitch (Z was equal to the outer pipe diameter, see Figure 3), the effect of this variable on η cannot be evaluated, so this variable can also be excluded. Likewise, surface tension (γ) effects will not be studied because the type (composition) of oil, water and pipe was not changed in the experiments, so that it becomes difficult to make any comparison. (3) The density and viscosity of the two-phase flows treated in the experiments can vary along each separator, because both parameters may depend on the water and oil concentrations, and on the type of mixture that may exist between them: homogeneous or heterogeneous, dispersion or emulsion, etc. So, as an approximation, the characteristic density and viscosity of a given flow (q) will be those linked to its majority phase (this issue will be discussed later). Considering this, ρ w and µ w will not be used in the dimensional analysis. (4) Given the material available in this study, there was no direct method for measuring the value of the inlet droplet diameter d p . However, and due to the design of the hydraulic system used (see Figure 4), in practice it could be visually observed in the tests (for d p sizes of the order of 1 mm, or even less), that the droplets of the dispersed fluid (water) depend on the ratio C/q (or equivalently, on C/h): the inlet droplet diameter tended to be small when q was high (elevated pressure in the system) and the water concentration C was reduced. Therefore, d p could be indirectly accounted for, in this case, by the ratio C/q.
Applying the above, the following relation could be stated: Knowing that the variables included in the previous equation are expressible in terms of the 3 basic units of mass, length and time, if the Vaschy-Buckingham πtheorem principles are applied [24], can be obtained. According to the above expression, the separation yield η could depend on: -Geometrical relationships such as d/(2R) and L/(2R).

-
On the Reynolds number: -On the Euler number: -On the phase (water) concentration: C. -On the dimensionless factor t r *: A version of the Fourier number [24,25], adapted to a stationary momentum transfer context, that is related to the residence time t r : -And finally, on the Froude number: In view of the above, and taking into account the experimental design presented in Section 2, the results of the tests carried out will be analyzed by comparing the performance of the available separators (four different geometries), when the Reynolds, Euler and Froude numbers, as well as C and t r *, are considered. Moreover, some other relationships, that could help to understand the behavior of these devices, will also be studied.

Reynolds Number vs. Total Flow Rate
As the flows treated in this study are composed of two phases (olive oil and water), it is not easy to estimate with any accuracy the actual Reynolds number which may exist in any flow section (or stretch), since the density and viscosity of these flows can be variable along the duct. In order to solve this problem, one can roughly assume that, through an apparent value of the Reynolds number, the information available can be analyzed to extract useful considerations [6]. That apparent Reynolds number (Re a ) can be estimated by means of Equation (4), also assuming that q = q o (the flow is characterized by the values related to its majority phase, in this case olive oil): where the subscript "o" in each variable, remember, is related to olive oil, and ν and U are, respectively, the kinematic viscosity and the average flow velocity. In this work, the value of ρ o (in kg/m 3 ) was determined by applying Equation (1) (see Section 2.2) and the value of µ o (in Pa·s) was estimated from an experimental function [23], that was obtained for the olive oil used in these tests, and in which T represents the oil temperature (expressed in • C) and S E = 0.002 Pa·s is the standard error: Obviously, the value of the olive oil kinematic viscosity (ν o ) can be deduced from ρ o and µ o considering that ν o = µ o /ρ o . In addition, the formulation of Equation (9) has been established taking into account the relationship that can be formulated between the average flow velocity (U) and the flow rate (q) for any incompressible fluid: in which A is the area of the pipe section. According to Equations (9) and (11), the value of U o must also be taken as an apparent value, since it has been estimated from a flow rate (q o ) that does not fully fill the pipe section through which it flows (please note that water is also present in the two-phase flow). In the scientific literature, any U value calculated in this way is usually known as superficial velocity [6,26]. Besides the Reynolds number, in this manuscript section the total flow (q t ), which circulates through each separator in any experiment, has been considered to be a variable of interest because it serves to indicate the processing capacity of the separators built (on scale). This flow can be calculated as being the sum of the olive oil flow (q o ) plus the water flow (q w ) which are present in each test (q t = q o + q w ). From another perspective, the flow q t can also be equally estimated as: q t = q in + q out . In this case, q in and q out are, respectively, the flow rates collected from the inner and the outer sides at the end of the separator (see Figure 3). In Figure 6 the values of total flow q t (in l/h) are shown as a function of the apparent Reynolds number Re a for the four different geometries used (remember that their characteristics are detailed in Section 2.1). In general, from the results obtained, it is worth highlighting that the four separators follow a similar behavior, clearly observing how, as the apparent Reynolds number increases, the total flow rate (less than 140 l/h in all cases) augments almost linearly. Given the viscosity of the majority phase (see Equation (10)), Re a values in all tests have been very low (within a range from 0 to 60). This fact allows one to assume in advance-in the absence of other results that will be shown later-that the experiments have been conducted under laminar flow if the criteria established by Srinivasan et al. [27], for single-phase flows, is roughly accepted: where Re crit is the critical Reynolds number in which transition starts and δ = r/R is a factor given by the ratio between the pipe radius (r) and the helix radius (R). In this study, Re crit could be of the order of 10,000 in all separators, a much higher value than Re a = 60. At this point, what deserves a mention is that, despite the experiments presented in this research having been run on scale, they could be of use for other works, because in food processing it is very usual to handle low Reynolds numbers (as food fluids usually have large viscosities). As an example, for an olive oil production of 3000 l/h -a common flow rate treated by the centrifugal separators in the 'almazaras'- [6,28] Re a could, in many cases, be of the order of 200. This Re a value is not very far from those presented in Figure 6. With a different approach, Zhao et al. [5,20] or Niu et al. [21] proposed using helical separators for values of Re a just above Re crit , but, as will be discussed later, this solution may not be the most suitable one for reaching good separation yields for water concentrations like those addressed here. In this respect, and using water as a majority phase, Vallesquino and Molina [6] observed that for Re a > 2000 it was not possible to achieve good separations, regardless of the oil concentration values, due to the agitation (turbulence) present in the flow.

Reynolds Number vs. Pressure Head and Euler Number
In the dimensional analysis made above, a generic pressure head term (h) was included as one of the operating parameters that could have some influence on the separation yield η. This term could be represented by the pressured head at the inlet (or the outlet) of the separator, or by the pressure head difference between both ends. As the discharge of these devices was at an atmospheric pressure, the following could be accepted: in which h fk are the head losses occurring along the separator, H i is the pressure head at its inlet and ∆z is the difference in height between the inlet and outlet points. As an approximation, velocity heads have been neglected because they are small and are mutually offset between the separator ends. According to this, Figure 7 displays the values of head losses h fk in each separator, as a function of Re a , to establish (on scale) the energy requirements of these devices. Results displayed in that figure show a roughly linear trend in all the separators, which indicates a direct (practically proportional) relationship between the energy losses and the Reynolds number Re a (which will be used to understand other results analyzed later). Moreover, some scattering is observed in the values shown in Figure 7, which could be explained by the difference in water concentrations in the tests, and by the mixing degree in the olive oil and water. As is usual in this context, the results of h fk can also be expressed as a function of a friction factor. If the Euler number included in Equation (3) is multiplied by the geometric factor d/L (a linear combination of terms d/(2R) and L/(2R) from Equation (3)), the following could be deduced, taking into consideration the Darcy-Weisbach equation [24]: in which f a has to be taken as an apparent friction factor because it is based on the olive oil flow q o (that also implies that the Euler number is computed through an apparent value Eu a ). It should be noted that Equation (14) has been formulated under the hypothesis that h fk has the same value (from Equation (13)) for q o and q t (as the pressure head for both flows was the same at the inlet or at the outlet of each separator). On the basis of the above, Figure 8 shows the experimental values of f a (obtained from Equation (14)) together with a rough estimation of this factor (f aI ) that can be made from an adaptation of the Ito equation, that is valid for helical single-phase laminar flows [13]: where De a is the apparent Dean number that can be calculated from Re a : In analyzing Figure 8, worth a mention is the approximation that exists between the theoretical and the experimental data of the friction factor f a , despite the Ito equation being originally established for helical single-phase flows. This fact reinforces the idea that the flow regime in the experiments was predominantly laminar and it was dominated by the main phase, which also had a higher viscosity than the minority phase. This circumstance confirms what was already commented on in the previous section in relation to the low Re a values recorded and the performance of q t . In this sense, it could be verified that between the head losses h fk and the flow rate q t there is also an increasing linear relationship (with some degree of scattering), that is compatible with the existence of the laminar flow.
The preceding results could be of interest for future works in this field, since in the literature of helical separators [5,6,[19][20][21] the Darcy-Weisbach friction factor is usually not calculated or analyzed. In helical two-phase flows, focused on gas-liquid systems [8], deviations have been reported of the order of, or less than, 30% (between single-phase and two-phase flow friction factors) for volumetric water concentrations of below 0.3%. These deviations are of the same order of the ones addressed here, but in this study the water concentrations applied were higher (as presented next in Sections 3.3 and 3.5). Figure 9 displays the values of water concentration C (as a percentage) as a function of the Reynolds number (Re a ) in each experiment. The values of C are obtained from the ratio between the water flow rate q w and the total flow rate q t : C (%) = 100·q w /q t . It is noteworthy that these flow rates can be obtained by simple arithmetic from the samples taken, in a given time and in each test, once the volumes of oil and water have been determined according to the procedure already described in Section 2.2. In observing Figure 9, it is notable that, in general, C is less than 10% in most of the experiments, but there are some cases in which this variable can reach values of the order of 30-40%. Comparatively, reference could be made to the work of Vallesquino and Molina [6], which described water-olive oil systems with olive oil concentrations of less than 12% (water was the majority phase with C > 88%). In other studies [5,20,21], related to watermineral oil systems, this problem has been mainly tackled from a theoreticalsimulation perspective by presenting some laboratory experiment results mainly linked to an oil concentration of 0.4% (C ≈ 99.6%). Other authors [19], with a similar approach, presented the results of four tests, in which water concentrations seem to vary from 39.8-64.4%. In view of the above, it is reasonable to assume that, in the field of the food industry (and especially in applications related to edible oils), the present study could be of use for future works interested on helical separator technology.

Reynolds Number vs. Water Concentration, t r * and Froude Number
Regarding the distribution of C in Figure 9, it should be noted that this factor seems to vary in each separator in an apparently erratic way, with no relationship with the Re a values. Since the injection of the water into the system was not carried out by following an established flow or pressure pattern, it is normal for no relationship to be observed between these variables. However, as was stated at the beginning of Section 3, since the droplets of the dispersed fluid (water) depend on the ratio C/q, that zone in Figure 9, where Re a is high and C is small, could probably be related to small droplet size experiments (note that q t is proportional to Re a according to Figure 6). On the contrary, where C is high and Re a is small, the droplet size expected could probably be larger. Considering this, Figure 9 shows that the tests could have been run with a great variety of droplet sizes.
On the other hand, if Figures 7 and 9 are observed together, a certain relationship seems to be established between the variability of C and the scattering of h fk . In this sense, the data from the separator with R = 41.75 mm-L = 7 m, and from the case with R = 41.75 mm-L = 4 m, are worth highlighting, because the former has the most dispersed values of C and h fk , and the latter shows the least dispersed values of those variables. Viewing this, it is reasonable to suppose that the more the C values differ in the tests, the more the variability in the viscosity present in the flows, which will result in a wider scattering of h fk . Likewise, the degree of mixing (or emulsification) in the flow can also influence this variability, a fact that could explain the scattering that h fk shows, for Re a > 20 and C < 2%, in the separator with R = 41.75 mm-L = 7 m (a great dispersion of h fk is observed in Figure 7, despite the fact that C changes very little in Figure 9 for the referred range of Re a ).
As for the relationship between the Reynolds number and t r * (Equation (6)), it should be noted that this dimensionless variable depends on q = q t (or in q = q o if it is decided to compute the apparent value t r * a ) and on some other factors that, for a given separator and fluid (olive oil), are constant. In this situation, and as q t = f (Re a ) (see Figure 6), it follows that t r * is a simple function that depends on 1/Re a , and no extensive analysis is required, because no relevant results would be obtained. A similar approach can be made with the Froude number Fr (Equation (8)), which ultimately could be expressed, in this case, as a simple function of (Re a ) 2 .

Reynolds Number vs. Separation Yield
Once the different terms included in the second member of Equation (3) have been studied, it is time to take into consideration the separation yield (η) and its relationship with these terms. In this respect, it is necessary to establish the following expression to determine the relative separation yield (η) associated with a given test: where q wout is the water flow rate collected from the outer side of the separator and q w , according to Section 3.1, is the water flow rate entering the separator (q w = q win + q wout , with q win as the water flow rate from the inner side of the separator). As an example, Figure 10 shows an image of one of the tests performed, in which the main part of the water flow rate is present on the outer side of the separator, and the oil flow rate is basically placed on the inner side. As expressed in Equation (17), a relative separation yield is equal to 100% when q wout = q w (the water is collected in its totality from the outer side and q win = 0). If q wout = 0 it so happens that q w = q win , and the separation yield η will be null, which implies that the water is completely extracted from the inner side of the separator (an undesired situation because the water would be expected to be collected from the outer side as it is a denser fluid than oil). These two cases are extreme, but they help to interpret the results that are presented below. To this end, Figure 11 shows the values of η as a function of Re a for the separators tested. On the whole, from the cited figure it is highlightable that, from a given Re a value, the separation yields improve, reaching values of between 80% and 100% in most cases. Specifically, for approximately Re a > 15, acceptable η values can be observed in all the separators, so that it can be inferred that these devices comply with the purpose for which they were designed. On these lines, also notable is the performance of the separator with R = 41.75 mm-L = 4 m, which presents values of above 80% in practically all the experiments (even for Re a < 15). However, the fact that for Re a > 15 there may be other cases in which η < 80% suggests that Re a is not the only variable to take into account for establishing the best operating conditions. That is why, in the next sections, more relationships between parameters will be explored to verify which variables could be the most significant ones to consider.  Apart from the above, and directly related to the contents of Section 3.1, it is also notable, see Figure 11, that, to achieve an acceptable separation yield, it is not necessary to reach high Reynolds numbers. On the contrary, and according to Vallesquino and Molina [6], it is desirable to work under Re a < 2000, a value that is usually much lower than Re crit (see Equation (12)). In this context, Zhang et al. [19] discuss a case in which the separation yield was of the order of 80% for a flow rate that could be linked to Re a ≈ 300 (if the criterion established in Equation (9) is accepted). Other studies [5,20,21] proposed working with flow rates that, accepting Equation (9) used here, would be associated with Re a ≈ 7600. That value (just above Re crit in those studies) could, in many cases, not be conducive to achieving good separation yields [6], which adds some controversy to this field and justifies the need for more research.

Water Concentration vs. Separation Yield
Similarly to the above section, the separation yields η are represented in Figure 12 as a function of water concentrations C (%), taking into account that C does not exceed 40% in any of the cases. It can be seen that, at very low water concentrations, the variability of separation yields η is high, which suggests that the water concentration C, by itself, is not the only variable to be considered in the search for adequate separations. However, the best performance values (η > 80%) seem to generally be linked to a working area in which C < 5%. From this value (C > 5%), there seems to be some stability of η, tending to be in the range of 60-80%. This working area is not deficient, but it could be improved, which sets a context in which advancement in the design or handling of these devices could be studied. For its part, the separator with R = 41.75 mm and L = 4 m seems to maintain good results (η > 80%) even for C > 5-10%, which underscores this idea. As was remarked in Section 3.3, there are few studies addressing the relationship between phase concentrations and separation yields in helical devices. According to Vallesquino and Molina [6], it is not recommended to work with oil concentrations of above 2% in water-olive oil systems. In other studies [5,20,21], in the field of water-mineral oil systems, very acceptable separation yields could be obtained for oil concentrations of the order of 0.4%. Other authors [19], report one case (in mineral oil-water systems) in which, for a water concentration of close to 47%, it was possible to reach a separation yield of the order of 80%. Analyzing these references and the data presented herein, it seems reasonable to consider that η, in oil-water systems, could be greater at higher concentrations of the minority phase (C mp ) than in water-oil systems, which could be due to the viscosity of the majority phase. As any oil is usually more viscous than water, the flow tends to be more stable in oil-water systems (there is less turbulence) than in water-oil systems for a given C mp value. This agrees with the assumption that the separation phenomena are normally promoted in those systems less affected by disturbances [28].

Ratio Re a /C vs. Separation Yield
From Sections 3.4 and 3.5 it is deduced that there are two conditions for obtaining the best separation yields. On one hand, it has been verified that, from a certain value of Re a (variable directly linked to the energy available for the separation), the η yields reached could be considered as being acceptable. Regarding water concentration, it would seem that the separation yields are not adequate for large C values, but this condition does not imply that for small C values η could be appropriate. This section has studied how the ratio Re a /C can have an influence on the yield η. To this effect, if the considerations expounded at the beginning of Section 3 are respected, as well as the relationship between q t and Re a (see Figure 6), it follows that the factor Re a /C is directly related to the inlet droplet diameter of the dispersed fluid (visually it was observed how, as Re a /C was increased, d p tended to decrease under values that could be of the order of 1 mm or even less). On the other hand, it should be noted that Re a /C is also linked to the energy level available in each experiment (as h fk is proportional to Re a because the flow is predominantly laminar, see Figures 7 and  8).
For those reasons, in Figure 13 the values of η are displayed as a function of those of Re a /C (taking C as a percentage) from certain threshold values that seem to be characteristic in each separator. Considering these results, it is reasonable to assert that, with enough energy and adequate water concentrations (Re a /C higher than 5-10), the separators constructed can provide adequate separation yields. Since Re a values have been limited in this work (given the means available), the separation yields (in general) have not been satisfactory for water concentrations C > 5%. However, according to the results presented in Figure 13, it seems possible to guess the likelihood of obtaining better separation yields, for water concentrations C > 5%, if the energy available were to be greater (maintaining Re a /C > 5-10). This opens up an area for improvement that should be considered in future works. The fact that C has to be higher than a certain threshold value (of the order of 0.4-0.9% depending on each separator) might be explained by the mixing or emulsion degree reached by the flow. In this respect, it must be noted that after re-using the same olive oil on successive occasions with the same separator, part of it was intimately emulsified with the water used (in these cases, d p could probably be of the order of some microns), and that phase was hardly separable. Hence, below a certain water concentration C (not controllable) the separation yields were not as expected and the head losses h fk were randomly scattered in some tests (remember the information in Section 3.3). In the works directly related to this study [5,6,[19][20][21], the influence of Re a /C on η has not been properly studied. Despite this, it could be argued that, in water-olive oil systems [6], the best separations yields were obtained for 500 < Re a /C < 22,000 (with Re a < 2000, C < 2%). In the works of Zhao et al. [5,20] and Niu et al. [21], the experimental results analyzed show that for Re a /C ≈ 19,000 (with Re a ≈ 7600) it was possible to obtain good separation yields in water-oil systems. In the study of Zhang et al. [19] on oil-water systems, the best separation yield was found for Re a /C ≈ 6.6 and Re a ≈ 300. On the whole, the values of Re a /C presented here are in the smaller range of those in which acceptable separation yields have been reported. This again makes it possible to expect (as was indicated above) that in olive oil-water systems the ratio Re a /C could be higher to reach adequate η values for water concentrations C > 5%.

Separation Yield and Other Terms
In order to gain a better understanding of the minimum energy level required to achieve adequate separations in the models built, the head losses h fk and the separation yields η have now been analyzed comparatively. The relationship between these variables is shown in Figure 14, in which it stands out that, from a certain level of h fk (of the order of 2-3 m depending on each device), the separation yields can be quite appropriate, even next to 100%. However, for this energy level there are also low η values, which could be explained by an inappropriate water concentration (too high to reach a ratio Re a /C > 10, see previous Section 3.6). Additionally, it has been seen that, when h fk is below 2-3 m, the separation yields generally begin to worsen. Again, it has been observed that when the available energy is low, the separations are often not good ones, because there is a lack of centrifugal force in the flow. With regard to the relationship between the Euler number (Eu) and the separation yield (η), this is shown in Figure 15, taking as a reference the definition of the apparent value Eu a just defined in Equation (14). In general, the conclusions that can be extracted from this figure are similar to those obtained from Figure 11, in which Re a and η were addressed. Again, the performance of the separator with R = 41.75 mm-L = 4 m is striking, and, for a certain range, the studied devices offer acceptable separation yields (though in that zone there are also points with bad separations). The only difference to be pointed out is that the data distribution in the relationships between Re a and η, and between Eu a and η (Figures 11 and 15), follow opposing linear trends. The explanation to this can be supported by the relationship that can be established between Re a and Eu a = f a · L/d (see Equation (14) and Figure 8). Since f a (or Eu a ) is inversely proportional to Re a , the trends that could be present in these figures should be inverse of each other. By the same logic, Figures 16 and 17 display the results of η as a function of the dimensionless time t r * (Equation (6)) and the Froude number Fr (Equation (8)) considering, in both cases, that q = q t . It is characteristic in these figures for the distributions of the points represented to be very similar to those given in Figures 11 and 15, respectively, since t r * = f (1/Re a ) and Fr = f (Re a 2 ), as was stated in Section 3.3. This fact allows one to surmise (in this particular case) that the possible influence of t r * or Fr, on the yield η, can be accounted for through Re a , which could help simplify the analysis of this question in cases similar to the ones dealt with in this study.  As an example of the above, and taking into account the designs shown in Figures 1  and 3, it is obvious that, in order to achieve an adequate separation yield, the water flow in any test should be placed at the outer zone of the pipe section (not at the bottom part of the duct). This situation can only be reached if the centrifugal force is more important than the gravitational one, which implies that the Reynolds number (or the Froude number) must be higher than a given value. For this reason, in Figures 11 and 17, and in practically all the separators, it can be observed an increasing trend before η reaches the zone of 80-100%. In this context, by only using the information in Figure 11 could have been enough for making a decision, which could thus have simplified this analysis. These considerations could make it possible to reformulate Equation (3). Thus, if the premises established in Section 3 can be accepted, and the flow developed is laminar with a clear dominance of the main phase over the friction in the pipe (Equation (15) is roughly fulfilled), it could be stated that: which could allow future works to handle the design of these devices more easily. Regarding the analyses made in similar studies to this one [5,6,[19][20][21], in the terms discussed in this section, it is worth mentioning that no dimensional analysis was carried out in them, so that it is difficult to make any further comparisons. Moreover, in [5,[19][20][21] there was no discussion on the head losses h fk (or the Euler number), the residence time t r or the Froude number bound to each separator. For its part, in [6] the variables h fk and t r have been determined for the separators tested there, which allows to summarily show their more representative values in Table 1. Likewise, in that table, other characteristic values of the separation tests done in [5,[19][20][21] are given. For these studies, t r and Re a have been estimated, respectively, from Equations (7) and (9), taking into account the information reported in those works. Viewing the tabulated data, it is remarkable that there seems to be two different design proposals for helical separators: in references [5,[19][20][21] wide helix radiuses (of 400 and 500 mm) are used, while in this study and in [6] narrow helix radiuses (of the order of 40 mm) are applied. In all the cases, t r is less than 1-2 min, a much shorter time than that needed for a gravitational decanter (usually well above 12-24 h [4]). Table 1. Characteristic values of the separation tests conducted in some works directly related to this study (C mp refers to the concentration of the minority phase in each case).

Parameter
Ref. [6] Refs. [ In general, from the values shown, it could be inferred that narrow helix designs enable one to reach better yields, but the flow rates applied must be lower than in the wide radius cases. In water-oil systems [5,6,20,21], with olive oil or mineral oil as minority phases, the narrow helix design [6] allows the obtainment of better separation yields at oil concentrations of below 2%. In oil-water systems (this study and [19]), the narrow helix approach reaches better η values for water concentrations of up to 13.3%. Head losses h fk and separation yields η are almost of the same order in [6] and in this study, but here the concentrations of the minority phase are higher. This fact makes it possible to consider the narrow helix design as being a valid approach to managing olive oil-water systems, which supports the idea that other alternatives (to the wide helix configuration) are possible in designing these devices.

Conclusions
Currently, one of the most important costs in industry is that related to energy consumption, so that the development of new equipment which could reduce this input is a field of research that still deserves to be addressed. In this work, the behavior of an alternative type of separator, orientated towards treating substances in two-phase flows, has been studied from a dimensional analysis viewpoint (an approach not applied in other studies directly related to this one). Specifically, four models of helical separators have been built and tested in order to handle olive oil and water two-phase flows. From the tests carried out with these devices, some results and questions are worth highlighting: -Due to the device's design and the characteristics of the hydraulic system used, the working flow rates have been below 140 l/h, which has resulted in Reynolds number values (Re a ) of below 60 in all cases. This circumstance has enabled all the tests to be performed under a laminar flow regime. -By analyzing the relationship between variables like the flow rate, the Reynolds number, the head losses and the water concentration present in the flows observed, it has been possible to establish a suitable operating area (not considered in other studies) that facilitates the existence of acceptable separation yields. -Specifically, it was observed that, for Re a values > 15, it is feasible to obtain separation yields of the order of 80% to 100% in all the separators tested. Likewise, the energy required (head losses h fk ) in these cases has been in the range of 2 to 7 m. -Due to the experimental design applied, there was no direct relationship between the water concentration provided in each test with the flow rate processed. However, it was observed that the best separation yields η (of the order of 80% to 100%) are given, in general, for values of C < 5%. A particularly significant case was the behavior of the separator with R = 41.75 mm and L = 4 m, which seems to maintain η above 80% even for C > 5-10%. However, and since the number of devices tested was small (only four) it is premature to assign a given relationship between the geometrical parameters used and the separation yields obtained.
-On making a careful study of the values of the ratio Re a /C, it follows that it is possible to obtain adequate separations at moderate water concentrations (C < 5%) if the level of energy available is high enough (Re a /C > 5-10). For higher water concentrations (C > 5%), it is inferred, from the data recorded, that adequate separations could be reached if the ratio Re a /C > 40-70. - The influence of some dimensionless terms, like the Euler number, tr* or the Froude number, on the separation yield η can be considered (in this study) as being of secondorder or computable through the Reynolds number Re a . This fact means that, in cases like those presented in this work, these devices could be designed more easily. -Considering the results obtained as a whole, it can be concluded that the separators built have served their purpose on a reduced scale, but more studies are understood to be required in order to improve their design and performance if it is desired to comply with the demands that any food industry could make on them. In particular, operating flow rates must be scaled to higher values while separation yield η and head losses h fk should be reasonably maintained. This will probably affect the future selection of geometrical parameters such as d, L,or R in a design context based on a narrow helix configuration (as an alternative to the wide helix designs proposed by other authors).