# A new application area of electrochemical limiting diffusion current technique: measurement of mixing time and effect of some parameters

- 123 Downloads

**Part of the following topical collections:**

## Abstract

In this study, the electrochemical limiting diffusion current technique, which was employed first time for the mixing time measurement in the stirred tank system by the authors, was applied to determine the effect of some parameters on the global and local mixing times in the range of 3500 ≤ Re ≤ 10,000. The technique well sensed that increasing Reynolds number (Re), blade angle and blade number decreased the mixing time. The most effective parameter was observed to be Re, and the blade number and blade angle, in order. In addition, the mixing time was also investigated for solutions added in various volume fractions having different viscosity and density by using local mixing time measurement. It was determined that the mixing time decreased with the Reynolds number, the ratio of the initial solution volume to the total solution volume and the ratio of kinematic viscosity of the solution in the vessel initially to that of the added one viscosity. The ratio of initial solution volume to total solution volume was found to be a more effective parameter than the others in terms of mixing time.

## Keywords

Mixing time Stirred vessel Electrochemical methods Limiting current Mixing of miscible liquids## List of symbols

*A*Electrode area (m

^{2})*C*_{Ab}Bulk concentration of active species (mol/m

^{3})*D*_{a}Agitator diameter (m)

*D*_{t}Tank diameter (m)

*F*Faraday constant (= 96,485 C/mol or s A/mol)

*f*_{t}Dimensionless mixing factor

*g*Gravitational acceleration (= 9.81 m/s

^{2})*H*Solution height (m)

*i*_{L}Limiting current (A)

*k*_{c}Mass transfer coefficient (m/s)

*N*Rotating speed (rps)

*n*Blade number

*n*_{e}Number of transferred electron

*P*_{o}Power number

*RT*Rushton turbine

- Re
Reynolds number

*V*_{int}Initial volume of the solution in the vessel (L)

*V*_{tot}Total volume of the initial and added solutions (L)

*ϕ*Blade angle (˚)

*μ*Fluid viscosity (kg/m s)

*ν*Kinematic viscosity (m

^{2}/s)*ν*_{int}Kinematic viscosities of the initial solutions (m

^{2}/s)*ν*_{add}Kinematic viscosities of the added solutions (m

^{2}/s)*θ*_{m}Mixing time (s)

*θ*_{m,loc}Local mixing time (s)

*ρ*Fluid density (kg/m

^{3})

## 1 Introduction

Mixing as a discipline started to emerge with the fundamental and practical knowledge going back to 1950s. The studies about mixing has increased continuously and the investigations during the last 30 years made possible to design a mixing equipment for a process unit [1]. Mixing is one of the basic processes commonly used in chemical, pharmacy, petroleum, food, etc. processes, to get a homogenisation degree at any stage of the process, when necessary [2, 3]. Stirred vessels are the most commonly used process equipment for mixing purposes.

Mixing time is an important parameter in analysing the effectiveness and hydrodynamic of a vessel, and the time to reach at a certain homogenisation degree after an interruption to the system. The mixing time can generally be defined as the period necessary to achieve a certain degree of homogeneity after adding a tracer into the stirred vessel [4, 5]. The homogeneity of a system can be explained and determined by the gradients of concentration, temperature, pH, viscosity, colour and phase [1]. The important parameters affecting mixing time in a stirred vessel are the size and geometry of the vessel, the type, geometry, size and speed of impeller, the presence, size, shape and number of baffles, and the physical properties of fluid [6, 7, 8, 9].

The principle on which the mixing time is based can be the determination of the change of the properties of the fluid by imposing a tracer effect on the original system; that is, the measurement of the time to achieve a new stable or homogeneous mixture after imposing a tracer effect into the originally stable and homogeneous mixture. Many mixing time measurement techniques have been developed, and the most commonly used ones are conductivity [10], thermographic method [11], colorimetric technique [12], acid–base neutralisation reaction [13], planar laser-induced fluorescence (PLIF) [14], electrical resistance tomography [15], and particle image velocimetry (PIV) [16]. Every technique has its own advantages and disadvantages. A good review of these techniques and their comparisons are given by Ascanio [17].

However, for the measurement of mixing time, a novel and new measurement method in addition to the conventional ones was developed and proved by Aydin and Yapici [18], which is based on electrochemical limiting diffusion current technique (ELDCT). In their work they proved that this method is simple, cheap, flexible, straightforward and fast, and also able to measure mixing time both locally and globally in both intrusive and non-intrusive mode.

The purpose of the present work is to investigate the effect of fluid viscosity, blade number and blade angle on the mixing time in a four-baffled stirred vessel agitated by a Rushton turbine type impeller, and further prove that ELDCT is a good way of measuring mixing time, by employing this recently developed method it for this sort of measurements, which is a new application field for the technique. Another aim of the present work is to measure mixing time by adding a liquid with different viscosity in large amount into the main fluid present in the vessel; that is, mixing of two fluids having different viscosities. The reason to do this is the fact that this sort of mixing be encountered commonly in industrial operations, and therefore, to get an insight of mixing time for the systems of this kind.

## 2 Theoretical background

### 2.1 Mixing time in stirred tank

*K*is a constant depending upon the flow regime in the vessel, and the size, geometry of the vessel including impeller, and

*N*is the impeller speed in revolution per second [19].

*K*can be approached to be constant in the laminar and turbulent regimes, but not in the transition regime. Mixing time can be correlated in dimensionless form as a function of Reynolds number [20]

*α*and

*β*are the parameters depending on system parameters, and Re is the Reynolds number expressed as follow:

*ρ*is the density of fluid;

*D*

_{a}is the impeller diameter and

*μ*, the dynamic viscosity of fluid. For a stirred vessel with common type impellers, the flow regime is laminar for Re < 10, turbulent for Re > 1 × 10

^{4}, and transitional in the range of 10 ≤ Re ≤ 1 × 10

^{4}[21].

*f*

_{t}is dimensionless mixing factor,

*D*

_{t}; vessel diameter (m);

*g*, gravitation constant (m/s

^{2}); and

*H*, the height of liquid in the vessel (m). If one considers the change of

*f*

_{t}with Re, it is seen that it changes from approximately 700 to 7 as Re goes up from 1 to 10

^{3}while it changes, in a very narrow range, approximately from 7 to 4 when Re increases considerably from 10

^{3}to 10

^{6.}Therefore, for Re > 1000,

*f*

_{t}can approximately be taken as constant [21].

*P*

_{o}is dimensionless power number.

## 3 Experimental technique

### 3.1 Materials and method

*A*is electrode area (cathode), m

^{2};

*c*

_{Ab}, concentration of active ion in bulk solution, mol/m

^{3};

*F*, Faraday constant, (= 96,485 C/mol);

*k*

_{c}, convective mass transfer coefficient, m/s;

*i*

_{L}, limiting current, A.

If the system is operated under a fixed hydrodynamic condition, electrode surface area; consequently mass transfer coefficient is also become a constant. Under these conditions, changing bulk solution results in a change in the limiting current value. So by following the limiting current during the period for the system to get from one uniform condition to the after injecting tracer, the mixing time can be determined.

The chemicals used in the measurement were MERCK made, analytic grade with purity higher than % 99. The initial electrolyte consisted of a solution 0.0002 M of ferricyanide, 0.0008 M of ferrocyanide and 0.1 M of K_{2}CO_{3}, which is supporting electrolyte. The composition of tracer solution was 0.1 M K_{3}FCN_{6}, 0.4 M K_{4}FCN_{6} and 0.1 M K_{2}CO_{3}. The concentration of the solution was so arranged that the system becomes a cathodic controlled process; for this purpose the concentration of ferricyanide reduced at the cathode was kept one-fourth of that of ferrocyanide, and furthermore the area of the anode was arranged to be at least 3 times larger than that of the cathode. For global mixing time measurement, one baffle was used as cathode while the other three were used as anode.

The experiments for the determination of the over potential difference corresponding to about the plateau value of the limiting current were performed for two electrolyte solutions, namely the solutions with 0 and 50% glycerol at 6 different rotation speed (60–890 rpm) and for 5 different active ion concentrations (0.0008–0.0040 M), and was determined to be − 0.35 V.

### 3.2 Experimental

The experiments were carried out in the cylindrical vessel with an inner diameter of 150 mm which was made of Plexiglass material having a cooling/heating jacket around it. The 60 mm diameter Rushton turbine used as a mixer was placed concentrically, for which the ratio of the impeller diameter to the vessel’s is 0.4, and four baffles made of nickel were equally spaced into the tank. The height of the solution in the vessel was kept equal to the inner diameter of the vessel.

The experiments were performed in two series. In the first series, it was aimed to measure the effect of the blade number and blade angle of a Rushton type impeller on mixing time, by employing global mixing time measurements, for which one baffle was used as sensor while the other three as counter electrode [18]. In the second part of the experiments, the purpose was to measure the mixing time when a large amount of liquid having a different viscosity than that of the main liquid in the vessel was added in the vessel, which is a situation commonly encountered in industrial applications; that is mixing time of two different solutions in large proportion. For the second group, the local electrode was employed to measure mixing time since the total volume, therefore the active area of baffle used as cathode, was changed with the addition of second solution into the vessel.

For the electrochemical measurement, GAMRY made (Interface 1000) potentiostat/galvanostat was used and the readings were recorded by a computer to process data. After starting the experiment, a tracer of 5.2 mL was added into a solution of 2600 mL just at the 60th second, the change in limiting current was recorded, and the graph of the change was analysed by Boltzmann fitting method to determine the mixing time.

Viscosity and density values of solutions used in experiments

Glycerol ratio of solution (%) ( | Viscosity (cP) | Density (g/mL) |
---|---|---|

No glycerol | 1.0545 | 1.0115 |

10 | 1.3100 | 1.0220 |

20 | 1.8170 | 1.0550 |

30 | 2.5687 | 1.0820 |

40 | 3.7280 | 1.1050 |

50 | 6.1070 | 1.1393 |

The tip position of the local electrode in the vessel was 90 mm height from the bottom and 37.5 mm away from the side wall and it was so arranged that the tip point became on the radial line from the midst of two neighbouring baffles to axial centre of the vessel. Though these measurements the global mixing time values were not obtained, it gives an idea about the effects of the volumes and viscosities of the initial solution in the vessel and the added solution into it on mixing time. All the experiments were repeated at least twice, and their arithmetic averages were used in the calculations.

An example for proportions of original solution initially present in vessel and solution added onto it

Initial solution, glycerol (%) ( | Added solution, glycerol (%) ( | Volume ratio of added solution (%) \((v/v\)) | Volume of added solution (mL) | Volume of initial solution (mL) | Total volume (mL) |
---|---|---|---|---|---|

50 | 30 | 10 | 260 | 2340 | 2600 |

50 | 30 | 20 | 520 | 2080 | 2600 |

50 | 30 | 30 | 780 | 1820 | 2600 |

Reynolds numbers corresponding to stirring speeds for each solution

Glycerol (%) ( | Stirring speed (rpm) | |||||
---|---|---|---|---|---|---|

Re | 0 | 10 | 20 | 30 | 40 | 50 |

3500 | 60 | 70 | 100 | 140 | 200 | 310 |

5200 | 90 | 110 | 150 | 210 | 290 | 460 |

6900 | 120 | 150 | 190 | 270 | 390 | 620 |

8600 | 150 | 180 | 240 | 340 | 480 | 770 |

10,000 | 180 | 210 | 280 | 400 | 560 | 890 |

## 4 Results and discussion

### 4.1 Effect of blade angle

Figure 4 shows that increasing Reynolds number has an effect of reducing mixing time. For the same Reynolds number the mixing time values are quite close each other for the solutions having different viscosities at the same blade angle. The differences are due to the fact that the constant *K* in the equation of \(\theta_{m} N = K\) is not exactly constant for the transient flow regime.

### 4.2 Effect of blade number

*R*= 0.9410,

*k*

_{1}= 7.36 × 10

^{6}s for the present vessel and impeller type, a coefficient which might depend on type of vessel and impeller, and flow regime; \(\phi\), blade angle (°);

*n*, blade number.

*R*= 0.9201.

### 4.3 Mixing factor

*f*

_{t}with Re is given in Fig. 10 as the averaged values of

*f*

_{t}for each Reynolds versus Re. As seen from this graph, the value of

*f*

_{t}goes from approximately 6 to 4 as Re goes from 3 × 10

^{3}to 10

^{4}, showing very good agreement with the values of the correlation given in Geankoplis [21]. This result also proves the validity of ELDCT for the mixing time measurement.

### 4.4 Local mixing time of bulk mixing of solutions with different viscosities

Mixing time values for solutions having different viscosity values for mixing in bulk volumes

Exp. no | Glycerol ratio of the initial solution (w/w) | Glycerol ratio of added solution (w/w) | Volume ratio of added solution (%) | Mixing time from local cathode (s) |
---|---|---|---|---|

1 | 50 | 50 | 10 | 12.05 |

2 | 50 | 50 | 20 | 15.45 |

3 | 50 | 50 | 30 | 17.2 |

4 | 50 | 30 | 10 | 6.75 |

5 | 50 | 30 | 20 | 7.80 |

6 | 50 | 30 | 30 | 14.10 |

7 | 50 | 10 | 10 | 5.90 |

8 | 50 | 10 | 20 | 7.68 |

9 | 50 | 10 | 30 | 8.00 |

10 | 30 | 50 | 10 | 5.69 |

11 | 30 | 50 | 20 | 7.43 |

12 | 30 | 50 | 30 | 10.23 |

13 | 30 | 30 | 10 | 5.42 |

14 | 30 | 30 | 20 | 6.90 |

15 | 30 | 30 | 30 | 10.61 |

16 | 30 | 10 | 10 | 6.24 |

17 | 30 | 10 | 20 | 7.60 |

18 | 30 | 10 | 30 | 8.57 |

19 | 10 | 50 | 10 | 6.95 |

20 | 10 | 50 | 20 | 7.30 |

21 | 10 | 50 | 30 | 15.92 |

22 | 10 | 30 | 10 | 4.03 |

23 | 10 | 30 | 20 | 6.46 |

24 | 10 | 30 | 30 | 7.95 |

25 | 10 | 10 | 10 | 5.45 |

26 | 10 | 10 | 20 | 6.65 |

27 | 10 | 10 | 30 | 7.30 |

Although these measurements give mixing time values at a certain location, not the global ones, it gives an idea about the effects of the volumes and viscosities of the initial solution in the vessel and the added solution into it.

*θ*

_{m,loc}is the local mixing time, s;

*k*

_{3}= 612.3 s, a coefficient dependent upon vessel and impeller type, and flow regime;

*V*

_{int}is the initial volume of the solution in the vessel, L;

*V*

_{tot}is the total volume of the initial and added solutions, L;

*ν*

_{int}and

*ν*

_{add}are the kinematic viscosities of the initial and the added solutions, m

^{2}/s, respectively. When the analysis repeated with the Reynolds number based on the properties of the end solution, the correlation takes the following form with a little higher regression coefficient of 0.8701.

**k**_{4}= 514.2 s. These empirical equations show that increasing Reynolds number reduces the mixing time; that is speed up mixing. However the power of the Reynolds number shows deviation from the expected value of − 1.0; this can be attributed to the fact that correlation was developed on the basis of the properties of the initial and end solutions, separately, in the vessel, and that the Reynolds number changes after the addition of second solution depending upon the differences in their glycerol percentage. Furthermore, the measurements are local and they may not represent exactly the same behaviour as the global mixing time of the vessel. The most effective parameter is the volume ratio of the initial solution in the vessel. The correlation shows that when the initial amount of the solution is much more than the added one, that is as the amount of the added solution is smaller, the mixing becomes faster. Moreover, when the difference between the kinematic viscosities of the initial and the added solutions is higher, the mixing gets faster as well, reducing mixing time. This behaviour reflects the effect of the density as well. This can be explained by that as the difference between the densities of the solution increases, the centripetal and centrifugal forces become more effective, thus speeding up the mixing.

The mixing time is effected by hydrodynamics of the system governed by a lot of parameters including the geometry of the system, the type of the agitators, type of blades, orientation of the agitators which can be concentrically or eccentrically oriented, shape of the vessel, flow regime of the mixed fluid. Therefore the values of the coefficient *k* in Eqs. 9–13 will be depend upon the effect of these parameters, which is fairly complicated; it might differ depending upon the system characteristics similar to the coefficients in the empirical equations of heat/mass transfer. When these equations were developed as a function of the same variables with a slight change in system geometry, their coefficients differ greatly even though the flow regime and fluid characteristics were kept constant; in some instances, the coefficient has different values at different flow rate intervals without any change in flow regime and system geometry. Similarly, for example, for a system having a the same stirring speed range and vessel type as above, but with e different impeller type, the coefficient *k* will probably be different when mixing time equation is developed as a function of the same variables.

## 5 Conclusions

Increasing blade number and blade angle had a reducing effect of the mixing time.

Increasing Reynolds number reduced the mixing time.

The developed empirical correlation showed that the most effective parameter for a faster mixing is Reynolds number with a power of − 1.13, which is very close to the values reported in the literature.

The blade number was found to be more effective for reducing mixing time than blade angle in the investigated range.

Dimensionless mixing factor values were found to be in very good agreement with those given in the literature.

The local mixing time values for the solutions mixed in considerably higher proportions instead of adding trace amounts into the main solution showed that mixing time reduced with increasing Reynolds number, initial solution volume proportion and the ratio of the kinematic viscosity of the initial solution to that of the added one.

These results show that the electrochemical limiting current technique has a promising potential for mixing time measurements as a good, easy, simple, straightforward, fast and effective method.

## Notes

### Acknowledgements

Thanks to Ataturk University for the support with the BAP Project No: 2013/343.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## References

- 1.Paul EL, Atiemo-Obeng VA, Kresta SM (2004) Handbook of industrial mixing: science and practice. Wiley, HobokenGoogle Scholar
- 2.Cullen PJ (2009) Food mixing: principles and applications. Wiley, HobokenCrossRefGoogle Scholar
- 3.Ghotli RA, Raman AAA, Ibrahim S, Baroutian S (2013) Liquid–liquid mixing in stirred vessels: a review. Chem Eng Commun 200(5):595–627CrossRefGoogle Scholar
- 4.Harnby N, Edwards M, Nienow AW (1997) Mixing in the process industries. Butterworth-Heinemann, OxfordGoogle Scholar
- 5.Zhang GA, Cheng YF (2009) Electrochemical corrosion of X65 pipe steel in oil/water emulsion. Corros Sci 51(4):901–907CrossRefGoogle Scholar
- 6.Montante G, Mostek M, Jahoda M, Magelli F (2005) CFD simulations and experimental validation of homogenisation curves and mixing time in stirred Newtonian and pseudoplastic liquids. Chem Eng Sci 60(8–9):2427–2437CrossRefGoogle Scholar
- 7.Jakobsen H (2008) Chemical reactor modeling. Springer, Berlin, pp 679–755CrossRefGoogle Scholar
- 8.Doran PM (1995) Bioprocess engineering principles. Academic Press, CambridgeGoogle Scholar
- 9.Varzakas T, Polychniatou V, Tzia C (2014) Food engineering handbook: food process engineering. CRC Press, Boca Raton, pp 181–252CrossRefGoogle Scholar
- 10.Bouwmans I, Bakker A, Van Den Akker HEA (1997) Blending liquids of differing viscosities and densities in stirred vessels. Chem Eng Res Des 75(8):777–783CrossRefGoogle Scholar
- 11.Lee KC, Yianneskis M (1997) A liquid crystal thermographic technique for the measurement of mixing characteristics in stirred vessels. Chem Eng Res Des 75(8):746–754CrossRefGoogle Scholar
- 12.Takahashi K, Motoda M (2009) Chaotic mixing created by object inserted in a vessel agitated by an impeller. Chem Eng Res Des 87(4):386–390CrossRefGoogle Scholar
- 13.Delaplace G, Bouvier L, Moreau A, Guérin R, Leuliet J-C (2004) Determination of mixing time by colourimetric diagnosis—application to a new mixing system. Exp Fluids 36(3):437–443CrossRefGoogle Scholar
- 14.Zadghaffari R, Moghaddas J, Revstedt J (2009) A mixing study in a double-Rushton stirred tank. Comput Chem Eng 33(7):1240–1246CrossRefGoogle Scholar
- 15.Holden P, Wang M, Mann R, Dickin F, Edwards R (1998) Imaging stirred-vessel macromixing using electrical resistance tomography. AIChE J 44(4):780–790CrossRefGoogle Scholar
- 16.Alvarez MM, Guzmán A, Elías M (2005) Experimental visualization of mixing pathologies in laminar stirred tank bioreactors. Chem Eng Sci 60(8):2449–2457CrossRefGoogle Scholar
- 17.Ascanio G (2015) Mixing time in stirred vessels: a review of experimental techniques. Chin J Chem Eng 23(7):1065–1076CrossRefGoogle Scholar
- 18.Aydin Ö, Yapici S (2018) A novel method for the measurement of mixing time: a new application of electrochemical limiting diffusion current technique. Exp Therm Fluid Sci 99:242–250CrossRefGoogle Scholar
- 19.Tatterson GB (1991) Fluid mixing and gas dispersion in agitated tanks. McGraw-Hill Companies, New YorkGoogle Scholar
- 20.Moo-Young M, Tichar K, Dullien FAL (1972) The blending efficiencies of some impellers in batch mixing. AIChE J 18(1):178–182CrossRefGoogle Scholar
- 21.Geankoplis CJ (2003) Transport processes and separation process principles: includes unit operations. Prentice Hall Professional Technical Reference, Upper Saddle RiverGoogle Scholar
- 22.Wragg A (1977) Applications of the limiting-diffusion-current technique in chemical engineering. Chem Eng (Lond) 316:39–44Google Scholar
- 23.Selman JR, Tobias CW (1978) Mass-Transfer measurements by the limiting-current technique. In: Drew TB, Cokelet GR, Hoopes JW, Vermeulen T (eds) Advances in chemical engineering, vol 10, Chap 4. Academic Press, pp 211–318Google Scholar
- 24.Landau U (1981) In: AIChE symposium, series, vol 204, pp 75–87Google Scholar
- 25.Selman J (1981) In: AIChE symposium series (United States), vol 77. Illinois Institute of Technology, Chicago, pp 88–102Google Scholar
- 26.Böhm L, Jankhah S, Tihon J, Bérubé PR, Kraume M (2014) Application of the electrodiffusion method to measure wall shear stress: integrating theory and practice. Chem Eng Technol 37(6):938–950CrossRefGoogle Scholar
- 27.Berger FP, Ziai A (1983) Optimisation of experimental conditions for electrochemical mass transfer measurements. Chem Eng Res Des 61(6):377–382Google Scholar