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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Optimal location of axial impellers in a stirred tank applying evolutionary programing and CFD J. Arturo Alfaro-Ayala a,∗ , Víctor Ayala-Ramírez b , ˜ c , Agustín R. Uribe-Ramírez a Armando Gallegos-Munoz a

Department of Chemical Engineering, University of Guanajuato, DCNE, Col. Noria Alta s/n, C.P. 36050 Guanajuato, GTO, Mexico b Department of Electronics Engineering, University of Guanajuato, DICIS, Carretera Salamanca-Valle de Santiago Km. 3.5+1.8, Comunidad de Palo Blanco, C.P. 36885 Salamanca, GTO, Mexico c Department of Mechanical Engineering, University of Guanajuato, DICIS, Carretera Salamanca-Valle de Santiago Km. 3.5+1.8, Comunidad de Palo Blanco, C.P. 36885 Salamanca, GTO, Mexico

a r t i c l e

i n f o

a b s t r a c t

Article history:

The optimal location of one and two impellers on the central shaft of a tall stirred tank was

Received 9 December 2014

obtained using a combination of Evolutionary Programing (EP) and Computational Fluid

Received in revised form 28 March

Dynamics (CFD) techniques; this location was evaluated through the determination of the

2015

normalized mixing time (TN ) and power consumption (PC ). The systems were simulated

Accepted 23 May 2015

using 45◦ pitched, down-pumping, 4-blade turbine impellers (PBT). The EP method consid-

Available online 2 June 2015

ered a small population of 6 individuals (systems) where the impeller(s) were located on the shaft in order to cover most of the search space through the ﬁtness function. Popula-

Keywords:

tions were generated by applying the selection and mutation of genetic operators to the

Optimal system

best individuals for the next generation. Four generations of population were needed to ﬁnd

Evolutionary programing

the optimal location of the impeller(s) in the shaft. The results show the best individual

CFD

performance for the systems with one and two impellers.

PBT impeller

© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Mixing time Power consumption

1.

Introduction

Stirred tanks are widely used in most industries, such as: chemical, petrochemical, biotechnological, pharmaceutical, metallurgical and so on (Yapici et al., 2008). They are used in unit operations, such as: polymerization, emulsiﬁcation and solvent extraction, among others (Cheng et al., 2013). The stirred tanks are also widely used for blending two miscible ﬂuids in the food, pharmaceutical and chemical industries (Zadghaffari et al., 2009). An appropriate combination of the number and the location of impellers in a stirred tank allows the reduction of the power consumption and the mixing time. Supplying mechanical rotary motion with minimal power consumption

∗

is important in a large variety of processes in the chemical and process industry, because the cost associated with the power input contributes signiﬁcantly to the overall operation cost of the plant. Various works have been reported on the effect of the impellers location on mixing time and power consumption. Hiraoka et al. (Hiraoka et al., 2001) studied the best set-up positions of impellers and determined the power consumption and the mixing time for double impellers in stirred tanks, deﬁning the relationship of mixing time to power consumption for all system studied. Montante et al. (Montante et al., 2005) applied CFD simulation strategy for the calculation of mixing time in stirred tanks for ﬂuids characterized by either Newtonian or non-Newtonian rheological behavior. They stated that the turbulent Schmidt number was critically

Corresponding author. Tel.:+52 473 7320006x8136. E-mail address: [email protected] (J.A. Alfaro-Ayala). http://dx.doi.org/10.1016/j.cherd.2015.05.036 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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Nomenclature B CL D Ds DA H p PC Re S1 Sct T TN ui , uj W wA

bafﬂe width (m) clearance (dimensionless) impeller diameter (m) shaft diameter (m) diffusivity of species A in the mixture (m2 s−1 ) liquid level (m) total pressure (Pa) power consumption (dimensionless) Reynolds number distance between the lower and upper impeller (dimensionless) turbulent Schmidt number tank diameter (m) Normalized mixing time (dimensionless) velocity component (m s−1 ) blade width (m) mass fraction of species A

Greek letters production rate of kinetic energy (m2 s−3 ) ε turbulent kinetic energy dissipation rate (m2 s−3 ) turbulent kinetic energy (m2 s−2 ) dynamic viscosity (Pa s) turbulent dynamic viscosity (Pa s) t kinematic viscosity (m2 s−1 ) turbulent kinematic viscosity (m2 s−1 ) t density (kg m−3 ) Prandtl number for (1.0) ε Prandtl number for ε (1.2) angular velocity (rpm) ω

important for obtaining a good forecast of both homogenization curves and mixing time. Taghavi et al. (Taghavi et al., 2011) studied experimentally the power consumption and ﬂow regimes in dual Rushton impeller stirred tank in both single and two phase conditions, proposing some correlations to predict the ﬂow regime transitions as well as local and total power consumption. Woziwodzki and Jedrzejczak (Woziwodzki and Jedrzejczak, 2011) examined the size, positions and structure of the isolated mixing regions (IMR) as a function of the Reynolds number and the eccentricity ratio in vessel equipped with double turbine impellers. They found that the eccentricity causes a strong compartmentalization effect, which affects the mixing pattern and the occurrence of ribbon-like IMR (RIMR), where for a small eccentricity; the structure was similar to structures in concentric systems. Somnath and Sumanta (Somnath and Sumanta, 2012) studied the mixing time in a stirred tank to identify the effects of low frequency macroinstability oscillations in the impeller speed. The mixing time is signiﬁcantly reduced when the ﬂow is perturbed using a steepchange in the impeller speed with a speciﬁc macroinstability frequency, also leading to an energy improvement strategy for faster mixing. However, the sudden changes in rotational speed during a perturbation cycle can result in higher wear of rotating components in mechanical seal assemblies, and hence can reduce the seal life. Machado et al. (Machado et al., 2013) analyzed the transitional ﬂow in stirred tanks considering impeller Reynolds number, the scaling of mean velocity proﬁles and energy dissipation to keep the ﬂow in the fully

turbulent regime, concluding that the transitional regime for the ﬂow in a stirred tank is at Re = 20,000 in a substantial part of the bench scale tank, and a Reynolds number higher than 20,000 is enough to sustain fully turbulent ﬂow; however, it is clear that this general rule is only valid for regions close to the impeller; therefore, having regions with transitional ﬂow inside the stirred tank can produce uncertainties in the design. Machado and Kresta (Machado and Kresta, 2013) presented a new design of the conﬁned impeller stirred tank (CIST), where active circulation and fully turbulent ﬂow are sustained in the entire tank. This design considers the use of multi-impellers to obtain a more uniform turbulence ﬁeld than the conventional stirred tank; however, there are still several gaps related to the use of the impeller locations and number of impellers in a stirred tank. Surrogate modeling consists of building an approximate model for the objective function to be used in the solution process of the given optimization problem. When used in conjunction with evolutionary algorithms, surrogate models can help to reduce complexity, to smooth the ﬁtness landscape and to deal effectively with noisy environments (Fonseca et al., 2012). Jin et al. (Jin, 2011) highlights the importance of surrogate modeling for CFD problems. The approximation of CFD processes is performed by stopping simulations before convergence and only computing exact simulation for a small number of candidate solutions (Giannakoglou et al., 2006). Asouti et al. (Asouti et al., 2009) combined the use of evolutionary algorithms and surrogate models for the design of aerodynamic shapes. The surrogate metamodels used in this work are Multi Layer Perceptrons and Radial basis Function Networks. Both of them are Artiﬁcial Neural Network Models. The optimization of a mixing process is quite complex and just few papers can be found in the literature about this topic; e.g. Mohammadpour et al. (Mohammadpour et al., 2015) developed experimentally an statistical design which included the response surface methodology (RSM) in a surface aeration tank. On the other hand, evolutionary algorithms (EA) have been used as an alternative tool to classical optimization algorithms in Computational Fluid Dynamics (CFD) related problems, e.g. the optimization of micro heat exchangers, proposed by Foli et al. (Foli et al., 2006). Their approach uses Multi-Objective Evolutionary Algorithms (MOEA) and they compared their performance top the analytical approach. The main advantage of the evolutionary approach has been found to be its capability to simultaneously compute the parameters of the optimal shape and the optimal dimensions of the heat exchanger. Hilbert et al. (Hilbert et al., 2006) have also addressed a heat exchanger design problem using parallel genetic algorithms, where the goal is to optimize the shapes of the blades of the heat exchanger. On the other hand, Li et al. (Li et al., 2013) carried out the optimization of the operation of the ventilation system of ofﬁces. The authors found an increased resolution in the positioning of the thermal comfort zones, an optimization of the energy cost and an improvement of the indoor air quality after using a genetic algorithm (GA). Applications of evolutionary and CFD techniques to reach optimal design have been reported mainly to airfoils. Kumar et al. (Kumar et al., 2011) used an ant colony optimization (ACO) method to develop a CFD solver to optimize the shape of airfoils. Jahangirian and Shahrokhi (Jahangirian and Shahrokhi, 2011) also propose the use of genetic algorithms for aerodynamic shape optimization problems. Andrés et al. (Andrés et al., 2012) proposed to integrate EP and support vector regression algorithms for optimal airfoil design. In the case

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of mixing process, Batres (Batres, 2013) used a micro genetic algorithm to generate operating procedures. His work focuses more on a planning task rather than a design task; however it illustrates the interest of using evolutionary techniques in chemical engineering problems, being the only works reported to this date on the application of genetic algorithms to the optimization of mixing tanks that the authors are aware. In this work, it is proposed the use of Evolutionary Programing (EP) to ﬁnd the optimal location of one and two impellers on the central shaft of a stirred tank. Given the high computational cost of the CFD simulation, it is considered a small population of individuals as in the Micro Genetic Algorithm proposed by Krishnakumar (Krishnakumar, 1989), and as in the one used for the planning of the operation of a mixing tank by Batres (Batres, 2013). The rationale of using a small number of individuals is to reduce the computational cost of the entire optimization process and to show that a small number of individuals can improve the performance of the traditional experimental design methodology. Even if a larger number of individuals could result in a better performance, which could be obtained only at a very high computational cost. This is the reason to look for an optimum positioning in a surrogated search space. This approach is applied in the search space of several parameters, such as: the clearance CL , the separation between impeller S1 (for systems with two impellers) and the number of impellers (one impeller and two impellers). The optimal systems are searched in a tall stirred tank with an aspect ratio of 1.5 and a PBT impeller aspect ratio (impeller diameter/tank diameter) of 0.523, to reach homogenization through the prediction of the normalized mixing time (TN ) and power consumption (PC ).

2.

Evolutionary programing

Evolutionary Programing (EP) is an evolutionary computing technique that is characterized by only using mutation operators for the evolution of the population. EP has been originally proposed by Kennedy and Eberhart (Kennedy and Eberhart, 2001) and it has been used as variant of Genetic Algorithms (GA) in order to avoid the degradation of performance when crossover is used in order-based problems (Chiong and Beng, 2007). Another advantage of EP vs GA is the ﬂexibility of representation that can be encoded in the individual representation.

2.1.

Individual representation

In this approach, the individuals are encoded as a sequence of real numbers representing the ordered position of the blades of a mixing axle. That is, we work in the phenotypic space for the application of mutation operators.

2.2.

Mutation operators

The mutation operators are described in Fig. 1. At each iteration, one of these operators is applied to each individual in the EP population to generate an offspring. The mutation operators perturb each individual in the current population to increase the exploration of the design search space. A one impeller individual can mutate into an offspring with the impeller in a different position or into a two impeller individual. A two impeller solution can evolve into a two impeller

Table 1 – Evolutionary programing algorithm. Algorithm Initialize the population Do { Expose the population to the environment; Calculate the ﬁtness for each member; Randomly mutate each parent; Evaluate parents and children; Select members of new population; } until (the ﬁnal condition is satisﬁed).

individual but with different positions for one of the impellers or into an individual with only one impeller.

2.3.

Selection operator

At each iteration, a duplicated size pool of parents and their offsprings are obtained. From the entire pool of parent and children, the best half to pass into the next generation was selected. In this problem, a small population size (6 individuals) was used because of the time needed to perform the CFD simulation associated to each individual. The selection of the best individual is done by using a deterministic rule, the best half in terms of their ﬁtness score is kept. That guarantees that the algorithm follows a steady-state policy.A general sequence of an EP algorithm is described in Table 1. The interaction between the EP and the CFD techniques for the optimization process is shown in the activity diagram depicted in Fig. 2. The ﬁtness score of the individuals is the mixing time resulting from the mixing simulations and power consumption. Lower times and power consumption are associated to better individuals. The rate of convergence of the ﬁtness function is shown in Table 2. The evolution was stopped at the fourth generation because a minimum was reached (probably a local minimum). However, this ﬁtness value is considered sufﬁcient, since looking for a global minimum will involve running more costly CFD simulations.

3.

CFD simulation

3.1.

Stirred tank conﬁguration

The tank conﬁguration is shown in Fig. 3(a) and the location of the probes is shown in Fig. 3(b). The inner diameter is T = 0.287 m and the liquid level height is H = 0.45 m (1.56 T) with a torispherical ASME 10% bottom and an open top. Four bafﬂes with a height BH = 0.39 m and a width B = 0.025 m are mounted symmetrically near the wall with a clearance B = 0.008 m. The tank has a concentric shaft with a diameter Ds = 0.01 m, an impeller diameter D = 0.15 m (0.523 T) and the width of each impeller blade is W = 0.038 m. The ﬁrst four generations of the population created by EP along with the six individuals of the population per Table 2 – Convergence of the ﬁtness function.

Generation 1 Generation 2 Generation 3 Generation 4

Average ﬁtness

Optimal ﬁtness

10.51 9.40 8.52 8.52

8.2 8.0 7.0 7.0

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Fig. 1 – A ﬂowchart description of the EP mutation operators. generation named from A to X, the number of impellers for each system and the location of the impeller on the shaft are shown in Table 3. It includes the dimensionless clearance between the bottom of the tank and the lower impeller (CL ) and the dimensionless separation between two impellers (S1 ) taking the liquid height (H) as the reference distance, see Fig. 3(a). These geometrical parameters were used in

the EP method to ﬁnd the optimal individual, as described above.

3.2.

Boundary conditions and stirred tank mesh

A constant rotary speed of 100 rpm was used as angular velocity (ω) during the mixing process for all the systems. The

Table 3 – Generations and individuals (systems) of the population. Generations

Individuals(systems)

Number of impeller

CL

A B C D E F

Two One One One Two One

0.74892 0.57621 0.42462 0.44569 0.66262 0.83571

02

G H I J K L

One One One One One One

0.74892 0.13418 0.30036 0.15222 0.87875 0.39693

03

M N O P Q R

One One One Two Two One

0.72834 0.66262 0.55471 0.26441 0.44569 0.88429

S T U V W X

One Two Two One Two One

0.26441 0.65368 0.64203 0.78104 0.39693 0.5510

01

04

S1 0.1354

0.2161

0.1601 0.0461

0.1200 0.1200 0.4534

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Fig. 2 – Interactions between the EP and the CFD simulations. Reynolds number Re = 37,329.6 shows that the system operates in the turbulent regime. The free surface was set as a symmetry boundary condition and all the solid walls were set as no slip. For simplicity, the impellers were modeled as thin surfaces, since this does not have a signiﬁcant impact on the performance (Chiu et al., 2009). The computational

geometry and the mesh are shown in Fig. 4. A full analysis of the grid size was taken in a count, the numbers of unstructured cells were considered between 250,000 cells and 2700,000 cells (tetrahedrons). Avoiding any unnecessary computational effort required for the calculations with a large number of cells. The higher normalized mixing time found between the

Fig. 3 – (a) Stirred tank conﬁguration; (b) location of the probes.

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Fig. 4 – Computational geometry and mesh.

coarse and ﬁne meshes were around 9.5%. The time required to reach a CFD solution of an individual with 1.5 million of cells was around 8 h in a computer with a quad-core CPU (4th Generation i7 Processor) and 8 GB in RAM.

3.3.

Mathematical model

The Reynolds averaged mass and momentum balance equations are solved for the ﬂuid. The governing equations are given below: Continuity equation: ∂ui =0 ∂xi

(1)

Momentum equation: ∂ui uj ∂xi

=−

1 ∂p ∂ + ∂xi ∂xi

( + t )

∂uj ∂xi

+

∂ui ∂xj

(2)

In order to simulate the turbulence on the ﬂuid ﬂow, the 3D – realizable turbulent model, proposed by Shih (Shih et al., 1995), was applied as follows: Turbulent kinetic energy, , equation: ∂ ∂ui = ∂xi ∂xi

+

t k

∂ ∂xi

+ −ε

(3)

Rate of energy dissipation, ε, equation: ∂ui ε ∂ = ∂xi ∂xi

+

t ∂e

∂ε ∂xi

+ C1 ε − C2

ε2 √ + ε

(4)

Mixing time calculations were performed by solving the transport equation for a non-reacting species (i.e. the tracer), as follows:

∂(uj wA ) ∂wA ∂ = + ∂t ∂xj ∂xj

DA +

t Sct

∂w A

∂xj

second order upwind discretization scheme for the Momentum, Turbulent Kinetic Energy and Dissipation Rate, while the Pressure–Velocity coupling was computed with the SIMPLE scheme. After that, the ﬂow equations were uncoupled and a tracer material was added at the top of the stirred tank. Then, the unsteady solution for the transport of the tracer was performed with a time step equal to 0.01 s. The concentration of the tracer was recorded until no longer varied with time for all the systems.

4.

Results and discussion

4.1.

Mixing time

To ﬁnd the optimal system for the homogenization process in the stirred tank, the location of the impellers in the tank for the populations which contain one or two impellers were studied. The optimal location can reduce the mixing time and the power consumption to achieve homogeneity in the tank. Levels of homogeneity such as 90%, 95%, 98% and 99% of the ﬁnal concentration are common in the literature (Zhao et al., 2001; Ochieng and Onyango, 2008), but according to the chosen degree of homogenization, a difference in the mixing time can be obtain; thus, a normalized mixing time, TN , is presented in this work, it was normalized dividing the time obtained for all systems (individuals A to X) and the system with the highest time reached (individual W). The normalized mixing time for systems with one impeller, sorted according to the dimensionless clearance CL , is shown in Fig. 5. The geometry of the stirred tank with the location of the impeller for its corresponding system can be observed at the bottom of Fig. 5. The minimum value TN found in one of the probes, correspond to the systems G and H, with values of TN =0.2759 and 0.2966, respectively.

(5)

For a more detailed explanation of the above equations and the description of the different parameters used, the reader is referred to any reference on the theory of CFD, e.g. ANSYS Fluent user manual (ANSYS Inc., 2011; Versteeg and Malalasekera, 2007). The numerical solution of the ﬂow was obtained solving the Reynolds Average Navier–Stokes (RANS), the Realizable Turbulence – model and the Multiple Frames of Reference (MFR) model through the use of the commercial software ANSYS Fluent® v14.0. This solution was obtained using the

Fig. 5 – Minimum normalized mixing time for the systems with one impeller.

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Fig. 6 – Minimum, average and maximum normalized mixing time vs clearance CL . The set of probes considered gives a better understanding of the state of the homogenization inside the entire tank. The minimum and maximum values of TN found in one of the probes and the average TN of all probes for the systems H, S, L, O, G and R are shown in Fig. 6. It can be observed that the systems have a small or large difference of normalized mixing time (between minimum and maximum TN ); a small difference leads to a better homogenization process in the entire tank while a large difference of normalized mixing time leads to a worse homogenization process in the entire tank. It can be observed that system H shows a small difference between the minimum and maximum normalized mixing time, while System G presents a large difference between these values. Thus, the best homogenization inside the entire tank is for the system H, since it presents the smallest difference between the minimum and maximum normalized mixing time and the average TN = 0.3241 is the lowest among all systems with one impeller. The minimum, average and maximum normalized mixing time TN , for systems with two impellers, are shown in Fig. 7. The best homogenization process inside the entire tank is for the system P, because it presents a small difference between the minimum and maximum normalized mixing time and the average TN = 0.2672 is the lowest among all systems with two impellers.

209

Fig. 7 – Minimum, average and maximum TN for the systems with two impellers.

4.2.

Velocity vectors

The velocity vectors for the best systems H and P and the worst system W are shown in Fig. 8. These vectors are plotted on a vertical plane passing through the axis of the impeller at a selected angular position, so that it passes through the middle point between two adjacent bafﬂes. For the system H with one impeller (CL = 0.13418) and the system P with two impellers (CL = 0.26441 and S1 = 0.1601) one circulation loop can be observed that suctions the liquid form the top of the tank and mixes the tracer inside the entire tank. In the case of the system W (clearance CL = 0.39693 and S1 = 0.4534), it reaches the higher normalized mixing time of all individuals in the generations studied. This is due to the presence of two circulation loops formed around the impellers; provoking a decrement in the interaction between the ﬂuid regions located at the bottom and top of the tank.

4.3.

Power consumption

A comparison of the power consumption (PC ) for all systems with one and two PBT impellers is shown in Fig. 9. It was normalized similar to the TN , dividing the power consumption obtained for all systems (individuals A to X) and the system with the highest power consumption reached (individual W). A

Fig. 8 – Velocity vectors for systems.

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Fig. 9 – Power consumption. comparison for all individuals with only one impeller showed a slightly different normalized power consumption according to the location of the impeller on the shaft; the highest difference is around 18.85% between systems L (PC = 0.59168) and S (PC = 0.48013). On the other hand, for the individuals with two impellers the highest difference is around 21.17% between systems W (PC = 1) and A (PC = 0.78835). An important increment of the normalized power consumption was observed when an extra PBT impeller is added to the system, caused by the extra mass and area added which is used to move the ﬂuid in the stirred tank, thus, a higher torque is induced and subsequently the normalized power consumption is increased. The largest difference between the systems with one and two impellers is about 51.98% between system W (PC = 1) and system S (PC = 0.48013). Thus, the systems with one PBT impeller consume less power than the systems with two impellers. Among all individuals created by the evolutionary programing, the systems that show the best performance are systems H and P with one and two impellers, respectively, these systems differ by around 35.5% for the power consumption and 17.6% for the normalized mixing time. This represents around 49.6% decrement in the normalized mixing time per 100% increment in the normalized power consumption. This leads to the conclusion that two impellers do not work in a tank with the geometric parameters considered in this study. As can be observed in Fig. 9, the power consumption for the systems with one impeller does not vary signiﬁcantly, regardless of the position of the impeller (between 0.5 and 0.6). The same behavior is observed in the systems with two impellers (between 0.8 and 0.9). The highest power consumption is observed in system W, which corresponds to a normalized value of 1. Therefore, it can be concluded that the power consumption is only important when choosing whether to use a system with one or two impellers.

5.

Conclusions

The optimization of a mixing process was achieved applying by the Evolutionary Programing (EP) method and CFD techniques. The evolutionary programing generated several individuals in different generations of population. These individuals were represented in terms of different conﬁguration systems for one or two PBT impellers. The optimization allowed the exploration of large geometric search spaces, such

as the search in the group of systems of one or two impellers (number of impellers) and location of impeller(s) along the central shaft that are not intuitive for a human designer. The optimal one-impeller system is the system H, while the optimal two-impeller system is the system P. The evolution process ﬁlters the best individuals according to the operators: selection and mutation. This approach represents an efﬁcient method for optimizing the mixing process in stirred tanks which has not been reported before. Also, the approach allows the solution of an expensive engineering problem through the use of a reduced population of individuals, minimizing the CFD computational cost through the ﬁtness evaluation.

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