Power Dissipation and Power Number Correlations for a Retreat

Article pubs.acs.org/IECR

Power Dissipation and Power Number Correlations for a RetreatBlade Impeller under Different Baffling Conditions Chadakarn Sirasitthichoke and Piero M. Armenante* Otto H. York Department of Chemical, Biological and Pharmaceutical Engineering, New Jersey Institute of Technology, 323 M. L. King Boulevard, Newark, New Jersey 07102-1982, United States ABSTRACT: Glass-lined tanks/reactors provided with a torispherical (dish) bottom and equipped with a retreat-blade impeller (RBI) are commonly used in the industrial syntheses of Active Pharmaceutical Ingredients (APIs). The power, P, dissipated by the RBI in the liquid in the tank is critical to many mixing processes, especially since the power per unit liquid volume, P/V, controls many mixing phenomena. Despite their common industrial use, limited information on P for RBIs and the corresponding nondimensional Power number, Po, is available. Here, P values for an RBI in liquids of very different viscosities at different agitation speeds and for different baffling configurations, i.e., unbaffled, partially baffled (with a single “beavertail” baffle), and fully baffled (four rectangular baffles), were measured experimentally in a scaled-down version of a typical reactor for API synthesis. The corresponding Po values were obtained for a large range of the Reynolds number, Re (1 < Re < 400,000). Po was found to vary significantly with baffling type and Re. Equations correlating Po with Re and baffling type are presented.

1. INTRODUCTION In the pharmaceutical industry, the majority of mixing and reaction operations involving liquids and multiphase systems, such as those commonly used for the manufacturing of Active Pharmaceutical Ingredients (APIs), are conducted in mechanically stirred, glass-lined tanks and reactors.1 Although new glass-lined system designs are now available, the most common type of industrial glass-lined reactors still in use consists of a cylindrical vessel with a torispherical bottom equipped with a single baffle and a retreat-blade impeller (RBI) mounted close to the vessel bottom.1 The use of glass lining is critical to provide corrosion resistance and ease of cleanliness and to reduce product contamination, but it requires coating with glass not only just the internal wall of the tank but also all the components that may come in contact with the liquid contents, including the agitation system (i.e., the impeller and the shaft), baffles, and all other internals. This requirement has typically resulted in the use of simple impellers that can be easily glass coated, such as the commonly used three-bladed, retreat-blade impeller (“Pfaudler-type” impeller or “De Dietrich-type” impeller).1,2 The typical baffling configuration found in most of the tanks and reactors used in the chemical industry consists of four vertical plates having a width equal to 8 to 10% of the tank diameter, i.e., equal to T/12 to T/10, and mounted at the tank wall.3−5 Without baffling or with insufficient baffling, the fluid moves in a swirling motion in the tank creating a central vortex.3,6−9 Installing baffles eliminates such a swirling motion and improves axial mixing. However, glass lining pharmaceutical tanks makes the insertion of wall-mounted baffles quite difficult in most cases, especially in older systems.1,10 Furthermore, the use of wall baffles has the additional drawback that cleaning becomes more difficult, especially behind the © XXXX American Chemical Society

baffles. Clearly, this can be an issue in pharmaceutical reactors where the removal of all material residues from previous manufacturing campaigns is a critical requirement for the manufacturing of APIs and their intermediates. As a result, some pharmaceutical tanks are actually unbaffled. However, much more common is the use of a single baffle, also glass coated, mounted through a nozzle on the roof of the vessel and placed between the shaft and the tank wall. This arrangement typically results in adequate baffling effects. Several single-baffle designs exist (beavertail, finger baffle, h-baffle, etc.)1,2 although in most pharmaceutical reactors a simple, flat, “beavertail” baffle is often found. Placing the baffle in a more central position in the tank additionally requires that the impeller be located low enough in the vessel (i.e., with a low impeller clearance off the tank bottom) in order to clear the baffle. Such an impeller placement has the additional advantage of facilitating the suspension of sinking solids in the liquid−a quite common occurrence. Finally, most industrial glass-lined tank reactors are provided with a torispherical tank bottom (dish bottom), which is different from the flat bottom most commonly used in mixing studies.2,10 In all fluid mixing systems, a sufficient level of the mechanical agitation system, typically achieved by rotating the impeller and thus dissipating mechanical energy in the liquid, must be maintained in order to achieve the desired process goals. A significant body of knowledge exists on mixing and power dissipation in fully baffled tanks for different systems.11−19 Received: Revised: Accepted: Published: A

June 27, 2017 August 9, 2017 August 11, 2017 August 11, 2017 DOI: 10.1021/acs.iecr.7b02634 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research However, despite their common use in the pharmaceutical industry, only limited information is available on power dissipation and the corresponding Power number, Po, in glass-lined, torispherical-bottomed vessels, equipped with an RBI and partially baffled. Even more difficult is to determine from the literature the effect of different types of baffling in these systems, especially over a wide range or impeller Reynolds numbers (Re). In his extensive work, Nagata8 generated a large number of Po data for many types of impellers and baffling configurations. Although he did not investigate the pharmaceutical system of interest here, he generated semiempirical equations incorporating a very extensive number of empirical coefficients. Some of these correlations can possibly be extended also to RBIs. Campolo et al.20 and Campolo and Soldati21 studied two RBI systems, a large-scale and a small-scale one, in both cases provided with two baffles. They reported computational results for Po at a few discrete Re values and showed good agreement with the experimental data that they additionally obtained. Dickey et al.10 presented Po-vs-Re curves for RBIs using the percentage of effective baffle area as a parameter. However, this information came from the unpublished work of another investigator and contained no experimental data. Hemrajani and Tatterson2 also reported a Po-vs-Re plot for an RBI in a tank equipped with a finger baffle. However, even in this case, the curve contained no experimental data and was of unknown origin. More recently, a Japanese group (Kato et al.22 and, later, Furukawa et al.23) reported Po results for an RBI in a small flatbottom tank under baffled and unbaffled conditions. This review shows that there is still a need to obtain accurate and extensive power dissipation and Power number data for the RBI in a partially baffled system, as well as with other baffling configurations of industrial relevance, over a wide range of Re. Therefore, the primary objective of this work was to determine experimentally the power dissipation P and hence the impeller Power number Po in a scaled-down system geometrically similar to the tanks and reactors used in the pharmaceutical industry. This was achieved here by measuring the power dissipated under different baffling configurations, with different liquids, and under different hydrodynamic regimes by an RBI in a 61-L vessel that was built as an actual scale-down replica of the glass-lined vessels typically used for API manufacturing in the pharmaceutical industry. A secondary objective of this work was to obtain semiempirical Po-vs-Re correlations that could be used to predict the power dissipated in those industrial systems.

Po =

(2)

where D is the impeller diameter, and ρ is the fluid density. Using the Buckingham Pi Theorem, Po can be related to other dimensionless groups of relevance in a stirred tank such as the impeller Reynolds number (Re = ρND2/μ), the Froude number (Fr = N2D/g), the impeller type and geometry, baffling configuration, and geometrical ratios such as D/T, C/T, and H/ T, as follows12,13 ⎛ D C H ⎞ Po = f ⎜Re , Fr , baffling type, impeller type, , , , ...⎟ ⎝ T T T ⎠ (3)

where Fr is relevant only if a vortex forms in the liquid, and hence the gravity effects become important. In a fully baffled system and for most partially baffled systems, the vortex is either not formed at all or is very small, which implies that Fr can often be neglected.8,12 In addition, in recent work in unbaffled tanks, Scargiali et al.9 have shown that Fr is unimportant in the quantification of Po even in unbaffled vessels as long as the vortex generated by the impeller is not deep enough to reach the impeller itself. Since operating industrial tanks at the high agitation speeds required to generate such deep vortexes is highly unlikely, Fr can be reasonably ignored even in most unbaffled vessels, which means that the only dynamic variable of relevance for Po is Re. Finally, for geometrically similar systems in which all geometrical ratios are kept constant during scaling-up, the geometric ratios can also be neglected, and Po becomes only a function of Re and baffling type. Under these conditions, experimental power dissipation data can be reported as Po-vs-Re plots, using the baffling type as a parameter. Therefore, eq 3 becomes Po = f (Re , baffling type)

(4)

Clearly, by careful equipment manufacturing and positioning, the most important geometric ratios such as D/T, C/T, and H/ T can relatively easily be made identical in the smaller scale system as in the typical full-scale industrial system. However, in practice it is more difficult to manufacture smaller-scale components with exact geometric ratios as in the industrial scale equipment as far as some less critical ratios are concerned, such as the impeller blade thickness-to-the impeller diameter ratio. Therefore, in smaller-scale systems this ratio can be expected to be very similar, but not necessarily identical, to that in full-scale systems (as it was also the case in this work). Actually, the cross section of most industrial glass-coated RBI blades is not rectangular but somewhat ellipsoid and rounded at the edges because of the glass-coating process, which would make the manufacturing of exact smaller-scale replica of industrial RBIs even more challenging to achieve. In addition, different equipment manufacturing companies may manufacture slightly different RBIs, depending on their manufacturing process. On the other hand, such small differences in geometry between the smaller-scale equipment and full-scale apparatuses can be expected to have only a minor impact on power dissipation and Power number. Therefore, their effects can be neglected, and eq 4 can still be used.

2. THEORETICAL BACKGROUND The power, P, dissipated by an impeller rotating in a liquid depends on the following: (1) the system’s geometry, including impeller type, diameter, and location, the number and location of baffles, and tank dimensions; (2) the physical properties of the liquid; and (3) the dynamic operating conditions of the system, mainly related to the impeller agitation speed.2,12 Power dissipation can be obtained experimentally from measurements of the torque applied to the impeller, Γ, and the impeller rotational speed, N (in rps):2 P = 2πN Γ

P ρN3D5

(1)

3. EQUIPMENT, MATERIALS, AND METHODS 3.1. Equipment. Vessel. An open cylindrical vessel with a torispherical (dish) bottom, similar to the dish bottom commonly found in industrial stirred tanks, was utilized as

The Power number, Po (also referred to as NP or Ne in the literature), is the dimensionless group used to quantify the power dissipation using the nondimensional expression12,13 B

DOI: 10.1021/acs.iecr.7b02634 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

system. The beavertail baffle, similar to that commonly found in pharmaceutical reactors, had the following dimensions: diameter of the top section: 15.24 mm; length of the top section: 142.9 mm; diameter of the middle section: 22.2 mm; length of the middle section: 199.7 mm; diameter of the bottom section: 20.1 mm; length of the bottom section: 70.6 mm. The beavertail baffle was placed midway between the center of the vessel and the vessel wall, and its off-bottom clearance was 182 mm, measured from the bottom of the baffle to the bottom of the stirred vessel. The fully baffled system included four baffles with a width of 44 mm, a thickness of 3 mm, and a length equal to 430 mm, mounted from the top of the vessel. The lower edge of the baffles was 110 mm from the vessel bottom. Impeller. A single, three-blade RBI, geometrically similar to those typically used in commercial glass-lined vessels in the pharmaceutical industry, was used in all experiments (Figure 1e) except in some preliminary calibration experiments, as explained below. The RBI was manufactured based on the large-scale model from a commercial equipment manufacturer for the pharmaceutical and chemical industries (De Dietrich Company, Union, NJ). The dimensions of the impeller, measured with a caliper, were as follows: impeller diameter, D = 202.5 mm; radius of curvature of blades = 92.08 mm; height of blade = 25.4 mm; thickness of blade = 12.7 mm; and an impeller diameter-to-vessel diameter ratio, D/T = 0.487. The impeller was attached to the end of a 12.5 mm shaft and was centrally located inside the stirred vessel. The impeller clearance off the vessel bottom, measured from the bottom of the impeller, Cb, was always 40 mm in all experiments, and the corresponding impeller clearance-to-vessel diameter ratio, Cb/ T, was 0.089, similar to the large-scale configuration. Agitation Systems and Torque Measurement Apparatuses. Two different systems were used to rotate the impeller and to measure the torque and hence the impeller power dissipation. In the first one (Figure 2a), the impeller was driven by a variable-speed, 1/3 HP Lightnin motor (Model XJ-33 VM, Serial No. 88/365321, Lightning, Rochester, NY, USA), with a maximum rotational speed of 5500 rpm. The torque required to rotate the impeller and determine the power dissipated by the impeller in the liquid was experimentally obtained using an external strain gauge-based rotary torque transducer (Model, T6-5-Dual Range, Interface, Inc. Scottsdale, AZ) mounted in line between the motor and the impeller shaft. The transducer

the stirred vessel for the entirety of this work (Figure 1a). The vessel was made of a thin (0.5 mm) fluorinated ethylene

Figure 1. Experimental mixing system: (a) unbaffled vessel; (b) partially baffled vessel with beavertail baffle; (c) fully baffled vessel; (d) single beavertail baffle used for partial baffling; (e) retreat-blade impeller.

propylene (FEP) copolymer and had an internal diameter, T, of 450 mm and an overall height of 540 mm. The height included the cylindrical and torispherical bottom sections, measuring 430 mm and 110 mm, respectively. A rigid collar and lip at the top of the vessel allowed it to be suspended in a larger “host” Plexiglas square tank, with a side equal to 0.61 m. The square tank was filled with water and connected in a closed-loop recirculation mode to a heating/refrigeration circulating bath with a digital controller (Model 12108, Cole-Parmer, USA) to provide precise temperature control of the water circulating in the square tank and thus in the liquid inside the FEP vessel. Baffling. The FEP vessel was operated under three baffling configurations, i.e., unbaffled (Figure 1a), partially baffled (Figure 1b), and fully baffled (Figure 1c). A single beavertail baffle, shown in Figure 1d, was used for the partially baffled

Figure 2. Experimental system: (a) Lightnin motor and Interface transducer and (b) Heidolph motor with a torque meter and tachometer. C


Article


transducer, mounting a 6-blade disk turbine of known diameter on the shaft below the Interface transducer, applying a known force, F, to the tip of the impeller perpendicularly to the impeller diameter D, measuring the applied force with dynamometer (Shimpo FGV-0.5XY Force Gauge, 8 OZ Capacity, with a precision of ±2 N (2% high accuracy), calculating the torque from Γ = F·D/2, while at the same time measuring the torque reading from the Interface torque meter indicator. The same procedure was repeated by applying forces of different intensities. A comparison of the torque measurements from the Interface unit with those obtained using the dynamometer resulted in close agreement (standard deviation = ±0.44%) over a wide range of torques (0−0.5 N m). Dynamic testing of the Interface torque/power meter was additionally conducted to confirm the results of the static calibration test and to ensure that no additional external dynamic friction forces were introduced when power measurements were taken with the system in motion. Dynamic testing consisted of measuring the power dissipation and hence determining the Power number for a standard 6-blade disk turbine (D = 0.128 m) in water under turbulent conditions (Re > 10,000) in a flat-bottomed, baffled, Plexiglas tank (T = 0.29 m) for a typical configuration (C/T = 1/3; D/T = 0.44) and comparing the Po results with the commonly reported value for such an impeller, i.e., Po ≅ 5.2,11 The results, reported in Table 2, show that the experimental Po value for disk turbines was

was connected to an Interface series 9850 Multi-Channel Load Cell Indicator transferring data to a computer with M700 software (Interface) for data acquisition and processing. The transducer could measure the torque, Γ, in two different scales, i.e., 0−0.5 N m and 0−5 N m although only the first scale was used in this work. The same instrument could also measure the agitation speed, N, and internally calculate the instantaneous power, P, delivered through the shaft to the impeller and the liquid, according to eq 1. This system could measure torque with high accuracy (±0.1% FS) and precision (nonrepeatability = ±0.02%), as specified by the manufacturer and as also tested in this work (as explained below). However, this system could not be used effectively when the agitation speed was low, typically below 50 rpm. Therefore, a second agitation system was additionally used (Figure 2b), consisting of a 100 W motor (Heidolph RZR 2102 Control, Heidolph, Schwabach, Germany) operating in the range 12−400 rpm, and directly connected to the impeller shaft. This unit included its own internal torque meter with a torque measurement resolution of ±0.001 N m, thus resulting in good precision. However, the accuracy of this instrument was undetermined, especially considering that the torque in this unit was internally obtained by measuring the electric power consumed by the stirrer motor. Therefore, using this device to determine power dissipation required first calibrating this unit with the Interface unit, as described below, and then using the resulting calibration function to determine the power dissipation from experimental measurements. 3.2. Materials. The liquids undergoing agitation were either water or molasses of different sugar concentrations, viscosities (range: 0.6−42,000 cP), and densities (range: 990−1589 kg/ m3), so as to determine power dissipation over a large range of Reynolds numbers. A high-viscosity commercial molasses (Grade 42 DE Corn Syrup, Golden Barrel, www.goldenbarrel. com) was used as the initial liquid in experiments. Molasses with different viscosities and densities (labeled Molasses A through Molasses L) were obtained by successive dilutions of the initial molasses with distilled water, as well as by varying the liquid temperature. The properties of the different molasses used here are reported in Table 1. The viscosities and densities of the molasses were experimentally obtained as described below. 3.3. Methods. 3.3.1. Calibration of Torque/Power Meters. The Interface torque/power meter was calibrated in two separate ways, i.e., statically and dynamically. The static method consisted of blocking the shaft above the inline torque

Table 2. Experimental Po Values for Disk Turbines Used To Calibrate Dynamically the Interface Torque/Power Meter

Molasses Molasses Molasses Molasses Molasses Molasses Molasses Molasses Molasses Molasses water

A B C D E G−H I−J K L

temp (°C)

density, ρ (kg/m3)

viscosity, μ (cP)

30−48 50 50 45 40−45 45.1 27.9−39.9 31.5−39.1 23.6-31.9 20.5−44.3 16.5−43.9

1589 1416 1400 1393 1375 1364 1290−1308 1241−1256 1197−1202 1142 990−998

6475−42384 2,807 1,606 1,050 307−443 296 53−96 19.5−22.4 9.7−15.9 3.4−6.9 0.6−1.1

D/T

N (rpm)

Re

Po (this work)

Po (refs 2 and 11)

water

0.441

119.8 151.1 180.5 203.1

26383 33287 39751 44739

4.83 4.91 4.94 4.98

5.0 5.0 5.0 5.0

4.917 ± 0.016, i.e., in close agreement with the literature value, and with an experimental standard deviation similar to that of the static test (= ±0.33%). As for the Heidolph motor and torque meter, this system could not be calibrated statically since the torque was internally obtained by measuring the electric power consumption. However, dynamic calibration experiments for this system, similar to those described above, indicated that values of the Power number obtained with the Heidolph system were typically 8.32% lower than those reported in the literature.2,11 Similarly, initial experiments with the Heidolph system showed that the experimental power and Power number for the RBI were in substantial agreement with those from the Interface system, but they were typically smaller by a few percentage points (∼6%). Therefore, some 200 separate calibration experiments were conducted with Molasses A, B, C, and F in a small, flat-bottomed, baffled, cylindrical tank (T = 0.242 m), in which disk turbines of different sizes (D = 0.0768, 0.1030, 0.1279, 0.1529, and 0.1780 m) rotating at different agitation speeds were used, so as to vary N and P extensively and approximately independently of each other by changing the size of the impeller at the same agitation speed. These calibration experiments were repeated using the Heidolph system as well as the Interface system, and the percentage deviation of the power values in the two systems, β, defined as

Table 1. Densities and Viscosities of Liquids Used in This Work liquid

liquid

D


Article

Industrial & Engineering Chemistry Research β=

precisely measured diameters and known densities in the unstirred molasses in the vessel. The time, t, for the sphere to drop by a known vertical distance, L, was measured (an “acceleration zone” was allowed for the sphere to reach its terminal velocity), and the viscosity was obtained from Stokes’ law for the terminal velocity of a sphere falling in a fluid:

(PInterface − PHeidolph) PHeidolph

(5)

was obtained as a function of N and P. The average value of β was found to be equal to 6.19%. The experimental β values were then linearly regressed as a function of N and P (i.e., β = a + bN + cP, with N in rpm and P in W, respectively), and the coefficients were found to be as follows: a = 4.2245, b = 0.0204, and c = −0.0944 (in the appropriate units). This function was found to predict adequately the experimental values of β, with an average prediction error for β of 6.5%. Therefore, when collecting experimental power dissipation data, the raw power data obtained with the Interface system were used as such, and the Heidolph raw power dissipation data were converted to the corrected power values using eq 5, i.e. ⎧ P ⎪ Interface P=⎨ ⎪ ⎩ PHeidolph·(1 + β)

μ=

2 g (ρsphere − ρ)dsphere

18L

(7)

At a minimum, triplicate viscosity measurements were conducted. The recorded drop times were on the order of tens or, more often, hundreds of seconds, thus ensuring easy reproducibility of the test (relative standard deviation = ±1.99%). Viscosity measurements for lower-viscosity molasses (0.6 × 10−3 Pa·s < μ < 2 Pa·s) were obtained with different Cannon-Ubbelhode viscometers (Viscometer size 200, 350, and 450) immersed directly in the unstirred molasses inside the FEB vessel, filled with the same molasses, and after allowing sufficient time for temperature equalization with the surrounding molasses to occur. Triplicate efflux time measurements in the viscometer were typically taken (standard deviation = ± 1.27%). The properties of water at any temperature were obtained from standard references (https://www.thermexcel. com/english/tables/eau_atm.htm). The densities of all molasses were obtained with a calibrated pycnometer.

(6)

3.3.2. Determination of Power Dissipation and Power Number. The molasses or water was placed in the FEP vessel, the impeller and the desired baffling system were mounted, the temperature in the circulation bath was set, and the system was allowed to reach the desired temperature, typically overnight. Each experiment consisted of measuring the torque and/or power dissipation at different agitation speeds. When using the Heidolph unit, a single power data point measurement consisted of taking ten recordings within a 3 min period and then averaging them (with typical fluctuations within ±1.50%). Similarly, when the Interface unit was used, the system was allowed to stabilize for 3 min, and after steady state was reached, continuous power data were recorded for 3 min and averaged (with typical fluctuations within ±2.65%). Key experiments were conducted in triplicate to determine the experimental reproducibility, which was found to be ±1.26%. Before and/or after an experiment the viscosity of the liquid was measured directly in the vessel, as described below, and the fluid density was measured pycnometrically by removing a sample. After the experiments with a given molasses concentration were concluded, a portion of the molasses was removed, distilled water was added, and the system was allowed to become homogenized by stirring the vessel overnight. By varying the temperature of different molasses, different fluids with similar viscosities but different water concentrations could be obtained, thus ensuring that overlapping N−P regions could be explored with different fluids. Some 1,200 P values were experimentally obtained over a wide range of Re, i.e., in both the laminar and turbulent flow regions, and Po and Re were calculated. 3.3.3. Determination of Fluid Viscosities and Densities. Molasses and sucrose solutions in general are Newtonian fluids, and their viscosity can be easily measured.24 Since the viscosity of molasses is extremely sensitive to temperature, all viscosity measurements were conducted directly in the molasses inside the FEP vessel and at the same temperature of the fluid in the actual experiment, i.e., in the range 16.50−50 °C. The viscosity of highly viscous molasses, having the consistency of honey, could not be easily measured by transferring the fluid to an external viscometer without possibly altering its temperature. Therefore, for higher-viscosity molasses (2 Pa·s < μ < 43 Pa·s), the viscosity was measured by dropping small steel spheres of

4. RESULTS Figure 3a shows the experimentally obtained Po values as a function Re for the RBI in the fully baffled vessel for 1 < Re < 500,000. For Re < 10, i.e., in the laminar region, the values of log10(Po) varied linearly with log10(Re), implying that Po is inversely proportional to Re, as expected.13 For Re > 10,000, i.e., in the fully turbulent flow regime, Po remained approximately constant and nearly independent of the Re (Figure 3a), as reported previously for different types of impellers at high Re when the system was fully baffled.2,11 The liquid free surface in the fully baffled systems was observed to be nearly perfectly horizontal because of the presence of baffles. It should be remarked that the Po data for Re > 1,000 did not align themselves on a perfect horizontal line, but they tended to “bulge up” for Re ≈ 10,000. There is no fundamental reason to expect that the Po value in this region should remain absolutely constant, and in fact previous investigations using other type of radial impellers have reported a similar phenomenon.2,11 However, for those radial impellers, such as disk turbines, the Po value was reported to decrease at intermediate Reynolds number before flattening out at very high Re values (> ∼30,000).2,11 In the RBI case, the Po value instead increased slightly with Re before becoming approximately constant at very high Re. In order to make sure that our data were not artifacts, the same Re range where this phenomenon occurred was investigated using molasses of different concentrations as well as by using water, i.e., with solutions of very different viscosities. This in turn required using different agitation speeds to achieve similar Reynolds numbers. In all cases, the Po values for Re ≈ 10,000 presented a small maximum (with Po ≈ 0.85−0.9), thus confirming that the results were correct. The results for the partially baffled system with a single beavertail-style baffle, presented in Figure 3b, show that Po decreased inversely with Re in the laminar regime (Re < 10). For higher values of Re, Po decreased approximately linearly on a logarithmic scale and kept slightly decreasing with Re even for E


Article


experiments in the unbaffled vessel, even those at very high Re, the vortex depth never reached the impeller, i.e., the system always operated in the “subcritical” regime defined by Scargiali et al. as the regime in which the vortex did not reach the impeller.9 In fact, the data reported in Figure 3c show that Po could be expressed as a function of Re only, which is consistent with the results of those investigators, who also reported that Po became a function of Fr only when the vortex depth was equal to, or larger than, the impeller depth (“supercritical” region).9 Therefore, even for the unbaffled case, Po was plotted here as a function of the Reynolds number alone. Figure 4 presents plots of the same data as in Figures 3a, 3b, and 3c but on the same chart. For Re < ∼100, all data overlapped irrespective of baffling conditions, which is consistent with the concept that the flow in region around the impeller, i.e., the region where most of the power dissipation occurs, is not affected by the presence of baffles away from the impeller. As a result, baffling has little impact on Po in the laminar regime.2,11 Only when Re > 100 did the data start diverging. For Re < 10, Po was not only unaffected by baffling, but it was highly laminar and inversely proportional to Re. At higher Re values (Re > ∼1000), the effect of turbulence became obvious, the system operated under increasingly turbulent conditions, and baffling became relevant. The power consumption in the presence of four wall baffles was the largest and essentially constant for Re > 10,000. As the number of baffles increased, Po in the turbulent regime also increased (Figure 4).

5. DATA ANALYSIS 5.1. Determination of Correlating Equations. The data presented in the previous section can be used to obtain Po from Re and thus predict power dissipation in any industrial system geometrically similar to the RBI system used here. However, this requires manually entering the value of Re in Figure 4 to read the corresponding value of Po. It is advantageous to use instead equations that correlate the data so as to make the calculation of power dissipation possible using an algorithm. Therefore, here the Po−Re data were regressed to obtain such correlations. This in turn required the selection of appropriate functions incorporating the smallest number of adjustable coefficients. Two approaches were used here to derive correlations fitting the data. Type 1 Correlation. The first approach to regress the Po-vsRe data was by using a simple power-law function A Po = + FReq (8) Re where the coefficients A, F, and q must be determined from data regression. This correlating function was selected here not only because of its simplicity (power-law functions are widely used in the scientific literature) but also because the coefficients appearing in eq 8 can be readily associated with the slopes and intercepts of different linear sections (on a log−log scale) of the curves that can be obtained from the regressions of the data in Figure 4 for different flow regimes (such as laminar vs turbulent). Hence, this equation can provide an immediate insight into the dependence of Po on Re under different operating conditions. The first term in eq 8 represents the laminar regime contribution, while the second term represents the turbulent regime contribution. The coefficient A was calculated first by

Figure 3. Experimental RBI Power number vs Reynolds number: (a) fully baffled system; (b) partially baffled system; and (c) unbaffled system.

Re > 100,000. The liquid free surface in this system was nearly always horizontal, as in the fully baffled systems, although a very small vortex formation could be observed at higher agitation speeds, thus justifying that effect of the Froude number on Po could be ignored. As for the unbaffled system, the results presented in Figure 3c show that the Power number was also inversely proportional to Re for Re < 10, as in the other baffling configurations, and that Po decreased approximately linearly on a log−log scale at high Re values (Figure 3c). At higher Re values the air surface began to deform and a central vortex appeared. In general, vortex formation and vortex depth in unbaffled systems depend on agitation speed. Very deep vortices are typically undesirable as axial recirculation decreases and vibrations and instabilities appear. However, in all our F


Article


Figure 4. Comparison of experimental RBI Power numbers as a function of Reynolds numbers for different baffling configurations.

baffling system configuration were obtained as shown in Figures 6a, 6b, and 6c, respectively. The coefficients A, F, and q for each baffling type are summarized in Table 3. Type 2 Correlation. The second correlation approach consisted of using a modified Nagata’s equation.8 Nagata extensively studied unbaffled systems and, to a more limited extent, baffled systems and generated several semiempirical correlations for the Power number. Many of such correlations relied on the empirical determination of a large number of adjustable coefficients to account for the effects of different geometric variables. While the use of many adjustable coefficients can result in a better fit, this approach is less than optimal conceptually, as well as practically. Nevertheless, some of Nagata’s more fundamental insights can be very useful in data analysis, and this is why a method based on this approach was adopted here. The general form of Nagata’s correlation that was used here as the starting point to correlate the data was as follows:8

using only the combined laminar regime data (Re < 10) for all baffling conditions. Accordingly, the second term on the righthand side of the above equation could be neglected, and the equation reduced to log10(Po) = log10 A − log10(Re)

for Re < ∼10

(9)

The value of A was obtained as the intercept of the Po-vs-Re data using the logarithmic plot in Figure 5 where the slope of

Po =

(11)

As before, the first term on the right-hand side in this equation accounts for the laminar regime contribution, while the second term represents the turbulent contribution. Nagata provided some rationale for the use of the numerical values of the constants (1000, 1.2, 3.2, 0.66) in this equation. Nevertheless, the coefficients A, B, and p must be determined from data regression. In this work, we attempted not only to minimize the number of adjustable coefficients (limited here to A, B, and p) but also to use common values for some of these coefficients irrespective of baffling, whenever possible. Specifically, coefficients A and B in eq 11 were forced to be independent of baffling, while only p was allowed to vary with baffling type. The determination of A, B, and p was accomplished in three steps. The first was to calculate the value of A, as already done for Type 1 Correlation, resulting in the same value reported in Table 3. The second step consisted of obtaining B. By taking the limit in eq 11 for Re → ∞ one obtains20,21

Figure 5. Regression of experimental Po−Re data using all data in the laminar regime (Re < 10) irrespective of baffling type, from which A was obtained.

the straight line was forced to be equal to −1. From this regression the value of A = 39.724 ± 0.256 was found, which could be more conveniently approximated to A = 40 with minimal loss of accuracy. Whereas A was independent of baffling in eq 8, F and q were allowed to vary with baffling type. To obtain them, eq 8 was rewritten as ⎛ A ⎞⎟ log10⎜Po − = log10 F + q·log10(Re) ⎝ Re ⎠

⎛ 1000 + 1.2Re 0.66 ⎞ p A + B⎜ ⎟ Re ⎝ 1000 + 3.2Re 0.66 ⎠

(10)

where the data for the term on the left-hand side of this equation were plotted vs Re for each baffling type using a log− log scale, and F and q were obtained as the intercept and slope, respectively. Only data for Re > 40,000 were considered, to ensure good fitting in the turbulent region rather than in the transition region. The best-fit coefficients for each individual

⎛ 1.2 ⎞ p lim Po = B⎜ ⎟ ⎝ 3.2 ⎠ Re →∞ G

(12) DOI: 10.1021/acs.iecr.7b02634 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


0, and B can be obtained by regressing the data for the fully baffled case for Re > 40,000 using the equation A Po = + B (fully baffled system; Re > 40,000) (13) Re B was found to be equal to 0.749, as shown in Figure 7a, which can be approximated to B = 0.75 with minimal loss accuracy.

Figure 6. Regression of experimental Po−Re data (only data with Re > 40,000 were used in the regression) using Type 1 Correlation to obtain F and q for each baffling configuration: (a) fully baffled system; (b) partially baffled system; and (c) unbaffled system.

Table 3. Summary of Coefficients A, F, and q and Their Standard Errors for Each Baffling Type Using Type 1 Correlation baffling system

A

F

q

fully baffled system partially baffled system unbaffled system

39.724 ± 0.256

0.750 ± 0.002

0

39.724 ± 0.256

1.268 ± 0.013

−0.081 ± 0.003

39.724 ± 0.256

3.143 ± 0.019

−0.211 ± 0.004

Figure 7. Regression of experimental Po−Re data (only data with Re > 40,000 were used in the regression) using Type 2 Correlation to obtain p for each baffling configuration: (a) fully baffled system; (b) partially baffled system; and (c) unbaffled system.

This value is obviously nearly identical to the value of F for the same fully baffled case using Type 1 Correlation (Table 3). The third step consisted in determining the values of p for each baffling configuration while retaining the same values A and B irrespective of baffling. Accordingly, p was taken to be zero for the fully baffled case and was obtained from the regression of the corresponding data for the other baffling configurations. In all cases, only data for Re > 40,000 were considered, to ensure good fitting in the turbulent region rather than in the transition region. The results are shown in Figures 8a, 8b, and 8c,

which cannot be used to obtain B and p independently. However, for the fully baffled case Po becomes independent of Re in the highly turbulent regime. Therefore, for such a case p = H


Article

Industrial & Engineering Chemistry Research respectively, and values of the coefficients for each baffling type are summarized in Table 4.

Po =

40 + 3.14Re−0.21 Re

(unbaffled system; Type 1 Correlation)

(16)

Figure 8a shows a comparison of the experimental Po−Re data with the predictions from these equations. The average deviations between the predicted values and the experimental data for the fully baffled, partially baffled, and unbaffled systems were 7.52%, 5.09%, and 14.13%, respectively, indicating substantial agreement with the data. This was especially so for the partially baffled system, the most relevant for industrial applications, and the baffled systems but less so for the unbaffled system. The reason for the larger error for the unbaffled cases can be attributed to the larger deviation between data and correlation equation in the region for 10 < Re < 100. Similarly, the correlating equations based on Type 2 Correlation with the rounded-off coefficients from Table 4 are as follows: 40 Po = + 0.75 Re (fully baffled system; Type 2 Correlation)

(17)

⎛ 1000 + 1.2Re 0.66 ⎞0.52 40 Po = + 0.75⎜ ⎟ Re ⎝ 1000 + 3.2Re 0.66 ⎠ (partially baffled system; Type 2 Correlation) Po =

⎛ 1000 + 1.2Re 0.66 ⎞1.26 40 + 0.75⎜ ⎟ Re ⎝ 1000 + 3.2Re 0.66 ⎠

(unbaffled system; Type 2 Correlation)

Table 4. Summary of Coefficients A, B, and p and Their Standard Errors for Each Baffling Type Using Type 2 Correlation A

B

p

fully baffled system partially baffled system unbaffled system

39.724 ± 0.256 39.724 ± 0.256

0.749 ± 0.002 0.749 ± 0.002

0 0.518 ± 0.005

39.724 ± 0.256

0.749 ± 0.002

1.262 ± 0.010

6. DISCUSSION The large number of experimental Po results presented in this work (some 1200) was obtained for a system of significant industrial relevance for which only limited or incomplete information was available, over a wide range of Reynolds numbers (about 6 orders of magnitude), for different flow regimes, from very laminar to highly turbulent, in a system of appropriately large size to generate reliable results for industrial-scale processes, geometrically similar to the commercial large-scale tanks, and for the different baffling systems commonly found in pharmaceutical operations. Therefore, the results obtained here, such as those shown in Figure 4, should be a significant relevance and direct applicability to engineers and scientists working in industry. In addition, the two approaches used here to correlate the data resulted in equations (eqs 14−19), which, despite their simplicity, produced predictions that were overall in substantial agreement with the experimental results. This was especially so for the partially baffled system (∼5% deviation), the most relevant for industrial applications. The agreement was satisfactory for the fully baffled system (∼7.5% deviation) and

5.2. Comparison between Experimental Data and Prediction Bases on Correlating Equations. The Type 1 Correlation equations with the rounded-off coefficients from Table 3 are as follows: Po =

40 + 0.75 Re

(fully baffled system; Type 1 Correlation) Po =

(14)

40 + 1.27Re−0.08 Re

(partially baffled system; Type 1 Correlation )

(19)

Figure 8b presents a comparison of the experimental Po−Re data with the predictions from these equations. The average deviations between the predicted values and the experimental data were found to be 7.52%, 5.33% and 8.24%, for the fully baffled, partially baffled, and unbaffled systems, respectively, indicating substantial agreement with the data for all three baffling cases.

Figure 8. Comparison between experimental Po−Re data and correlating equations: (a) using Type 1 Correlation and (b) using Type 2 Correlation.

baffling system

(18)

(15) I


Article


• The data in the laminar region (Re < 10) for different types of baffling systems were superimposable, indicating that the presence or absence of baffles in this region is unimportant as far as Po is concerned; • The data for different baffling types started to diverge when Re > 100. The data for the fully baffled system at high Re showed a horizontal asymptote. The Po data for the partially baffled system and the unbaffled system showed instead a decreasing trend with increasing Re values; • These results followed the typical relationships expected for a radial impeller; • Correlation equations were obtained to predict Po as a function of Re using two approaches and requiring a minimum number of independently obtained experimental coefficients. The correlating equations were typically in good agreement with the experimental data over a large range of Re (6 orders of magnitude). The results obtained here can be expected to be of significant applicability to industrial systems, such as glass-lined vessels, typically used in the pharmaceutical industry, as long as geometric similarity is maintained.

less so, although still acceptable, for the unbaffled system. In this case, Type 2 Correlation (eq 19) was more satisfactory (∼8% deviation). In addition, the equations based on Type 2 Correlation had the additional conceptual advantage of predicting the asymptotic limits of the Power numbers for Re → ∞, which is equal to 0.75, 0.45, and 0.22 for the fully baffled, partially baffled, and unbaffled systems, respectively. The correlating equations presented here can also be of direct applicability to industrial systems, including their incorporation in numerical programs for routine use without resorting to the use of figures or tables. The results of this work can be compared with the results available in the literature. The data obtained here for the partially baffled case are in close qualitative agreement with the curve reported in the book chapter by Hemrajani and Tatterson2 for an RBI in a vessel equipped with a single finger baffle, although, as already pointed out, no experimental points were reported and the curve was of unknown origin. Campolo et al.20 and Campolo and Soldati21 generated experimental results as well as numerical results for an RBI in a system equipped with two beavertail baffles and reported that Po at high Re was between 0.819 and 0.830 in a large-scale system and about 0.7 in a lab-scale system. These values are in overall agreement with the results found in our work and especially the latter result, which is between the 0.75 value found here for the fully baffled system and the 0.51 value for the partially baffled system (with one baffle). Those investigators also reported that their values were in agreement with those calculable by using one of Nagata’s equations.8 Also, the value of “Ne∞” that Campolo et al.20 found using one of Nagata’s equations (0.228) matches well the corresponding asymptotic value of Po for Re → ∞ found here for the unbaffled vessel case (0.22). Kato et al.22 and, more recently, Furukawa et al.23 of the same group obtained results in a small (200 mm) flat-bottom tank that were similar (as discernible from their figures) to those reported here for the high-Re value of Po in the fully baffled and unbaffled systems. More recently, Scargiali et al.9 reported the dependence of Po for a disk turbine in an unbaffled system but only in the turbulent regime. They found that as long as the vortex did not reach the impeller, Po depended only on Re, and this functionality could be expressed as Po ∝ Re−0.3. In our work with an RBI, Po was also found to depend only on Re, but the functionality was found to be Po ∝ Re−0.21 (eq 16). This difference could possibly be attributed to the different types of impellers. Finally, the Po-vs-Re trends observed here for different baffling systems are consistent with the trend observed in previous studies for other types of radial impellers.2,11 In summary, the results obtained in this work for an RBI are similar to those reported by the limited number of investigators who worked with similar systems, although such a comparison can only be made for the limited number of cases examined in previous studies.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: (973) 596-3548. Fax: (973) 596-8436. E-mail: piero. [email protected]. ORCID

Piero M. Armenante: 0000-0003-0753-1384 Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The partial financial support of Merck & Co. is gratefully acknowledged. NOTATION coefficient in eq 8 and eq 11 coefficient in eq 11 impeller clearance off the vessel bottom, m impeller clearance off the vessel bottom, measured from the bottom of the impeller, m dsphere diameter of falling sphere for viscosity measurements, m D impeller diameter, m F coefficient in eq 8 H liquid height, m L distance traveled by falling sphere during viscosity measurements, m N rotational speed, revolutions per seconds (rps) or revolutions per minutes (rpm) p coefficient in eq 11 PHeidolph power dissipation from Heidolph torque measurement, W PInterface power dissipation from Interface torque measurement, W P power dissipation, W q coefficient in eq 8 T tank diameter, m β percentage deviation between power measurements in eq 5

A B C Cb

7. CONCLUSIONS In this work the power dissipation, P, and the impeller Power number, Po, were obtained in a torispherical-bottomed vessel equipped with a retreat blade impeller under different baffling conditions. The experimental data, some 1,200 in total, were used to generate Po-vs-Re plots for different baffling conditions and flow regimes (0.5 < Re < 400,000). A number of conclusions can be drawn from these results:

Greek Symbols

Γ J

torque applied to impeller, N·m DOI: 10.1021/acs.iecr.7b02634 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research ρ ρsphere μ ω

liquid density, kg/m3 density of falling sphere for viscosity measurements, kg/ m3 liquid viscosity, kg/(m s) or centipoise (cp) angular velocity, rad/s

(19) Motamedvaziri, S.; Armenante, P. M. Flow Regimes and Surface Air Entrainment in Partially Filled Stirred Vessels for Different Fill Ratios. Chem. Eng. Sci. 2012, 81, 231−250. (20) Campolo, M.; Paglianti, A.; Soldati, A. Fluid Dynamic Efficiency and Scale-up of a Retreated Blade Impeller CSTR. Ind. Eng. Chem. Res. 2002, 41, 164−172. (21) Campolo, M.; Soldati, A. Appraisal of Fluid Dynamic Efficiency of Retreated-Blade and Turbofoil Impellers in Industrial-Size CSTRs. Ind. Eng. Chem. Res. 2002, 41, 1370−1377. (22) Kato, Y.; Tada, Y.; Takeda, Y.; Hirai, Y.; Nagatsu, Y. Correlation of power consumption for propeller and Pfaudler type impellers. J. Chem. Eng. Jpn. 2009, 42, 6−9. (23) Furukawa, H.; Kato, Y.; Inoue, Y.; Kato, T.; Tada, Y.; Hashimoto, S. Correlation of Power Consumption for Several Kinds of Mixing Impellers. Int. J. Chem. Eng. 2012, 2012, 106496. (24) Sucrose: Properties and Applications; Mathlouthi, M., Reiser, P., Eds.; Springer: New York, 1995; DOI: 10.1007/978-1-4615-2676-6.

Nondimensional Groups

Fr Froude number = N2D/g Re Reynolds number = ρND2/μ Po Power number = P/(ρN3D5)

■

REFERENCES

(1) Paul, E. L.; Midler, M.; Sun, Y. Mixing in the Fine Chemicals and Pharmaceutical Industries. In Handbook of Industrial Mixing: Science and Practice; Paul, E. L., Atiemo-Obeng, V. A., Kresta, S. M., Eds.; John Wiley & Sons, Inc.: Hoboken, 2004; pp 1027−1069, DOI: 10.1002/ 0471451452.ch17. (2) Hemrajani, R. R.; Tatterson, G. B. Mechanically Stirred Vessels. In Handbook of Industrial Mixing: Science and Practice; Paul, E. L., Atiemo-Obeng, V. A., Kresta, S. M., Eds.; John Wiley & Sons, Inc.: Hoboken, 2004; pp 345−364, DOI: 10.1002/0471451452.ch6. (3) Oldshue, J. Y. Fluid Mixing Technology; Chemical Engineering McGraw-Hill Publications Co.: New York, 1983. (4) Harnby, N.; Edwards, M. F.; Nienow, A. W. Mixing in the Process Industries, 2nd ed.; Reed Educational and Professional Publishing Ltd.: Oxford, UK, 1985. (5) Bates, R. L.; Fondy, P. L.; Fenic, J. G. Impeller Characteristics and Power. In Mixing: Theory and Practice; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1966; pp 112−178. (6) Rushton, J. H.; Costich, E. W.; Everett, H. J. Power characteristics of mixing impellers-Part I. Chem. Eng. Prog. 1950, 48, 395−404. (7) Rushton, J. H.; Costich, E. W.; Everett, H. J. Power characteristics of mixing impellers-Part II. Chem. Eng. Prog. 1950, 46, 467−476. (8) Nagata, S. Mixing Principles and Applications; Halsted Press: New York, 1975. (9) Scargiali, F.; Busciglio, A.; Grisafi, F.; Tamburini, A.; Micale, G.; Brucato, A. Power Consumption in Uncovered Unbaffled Stirred Tanks: Influence of the Viscosity and Flow Regime. Ind. Eng. Chem. Res. 2013, 52, 14998−15005. (10) Dickey, D. S.; Bittorf, K. J.; Ramsey, C. J.; Johnson, K. E. Understand Flow Patterns in Glass-lined Reactors. Chem. Eng. Prog. 2004, 100, 21−25. (11) Bates, R. L.; Fondy, P. L.; Corpstein, R. R. An Examination of Some Geometric Parameters of Impeller Power. Ind. Eng. Chem. Process Des. Dev. 1963, 2, 310−314. (12) Bates, R. L.; Fondy, P. L.; Fenic, J. C. Impeller Characteristics and Power. In Mixing-Theory and Practice; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1966; Vol. I, pp 111−178. (13) Tatterson, G. B. Fluid Mixing and Gas Dispersion in Agitated Tanks; McGraw-Hill: New York, 1991. (14) Armenante, P. M.; Huang, Y.-T.; Li, T. Determination of the Minimum Agitation Speed to Attain the Just Dispersed State in SolidLiquid and Liquid-Liquid Reactors Provided with Multiple Impellers. Chem. Eng. Sci. 1992, 47, 2865−2870. (15) Armenante, P. M.; Huang, Y.-T. Experimental Determination of the Minimum Agitation Speed for Complete Liquid-Liquid Dispersion in Mechanically Agitated Vessels. Ind. Eng. Chem. Res. 1992, 31, 1398− 1406. (16) Armenante, P. M.; Chang, G. M. Power Consumption in Agitated Vessels Provided with Multiple-Disk Turbines. Ind. Eng. Chem. Res. 1998, 37, 284−291. (17) Armenante, P. M.; Mazzarotta, B.; Chang, G.-M. Power Consumption in Stirred Tanks Provided with Multiple Pitched-Blade Turbines. Ind. Eng. Chem. Res. 1999, 38, 2809−2816. (18) Armenante, P. M.; Uehara Nagamine, E. Solid Suspension and Power Dissipation in Stirred Tanks Agitated by One or Two Impellers at Low Off-Bottom Impeller Clearances, Proc. 9th European Conference on Mixing, Paris, France, March 18−21, 1997; pp 73−80.

■

NOTE ADDED AFTER ASAP PUBLICATION After this paper was published ASAP August 28, 2017, a correction was made to the title. The corrected version was reposted August 29, 2017.

K


Power Dissipation and Power Number Correlations for a Retreat

Recommend Documents