Optimization of the Efficiency of a Michell-Banki Turbine through the
Variation of its Geometrical Parameters using a PSO Algorithm
A. J. PEREZ-RODRIGUEZ
Departamento de Mecatrónica y Electromecánica
Instituto Tecnológico Metropolitano
Cra. 74d #732, Medellín, Antioquia
COLOMBIA
J. SIERRA-DEL RIO
Departamento de Mecatrónica y Electromecánica
Instituto Tecnológico Metropolitano
Cra. 74d #732, Medellín, Antioquia
COLOMBIA
L. F. GRISALES-NOREÑA
Facultad de Ingeniería
Institución Universitario Pascual Bravo
Cl. 73 ## 73a-226, Medellín, Antioquia
COLOMBIA
S. GALVIS
Departamento de Mecatrónica y Electromecánica
Instituto Tecnológico Metropolitano
Cra. 74d #732, Medellín, Antioquia
COLOMBIA
Abstract: - Small-scale hydropower generation can satisfy the needs of communities located near natural
sources of flowing water. The operating conditions of a Michell–Banki Turbine (MBT) are relatively easier to
meet than those of other types of turbine, making it useful in places where other devices are not suitable.
Moreover, MBT efficiency is almost invariable with respect to flow rate conditions. Nevertheless, such
efficiency commonly ranges between 70% and 85%, which is lower than that of other water turbines like
Turgo, Pelton, or Francis turbine. The objective of this work is to determine the maximum theoretical
efficiency of an MBT and its associated geometrical parameters by implementing Particle Swarm Optimization.
The results show a higher effectiveness of the mathematical formulation compared with other cases from
literature and show the performance of the optimization method proposed in this study in terms of solution and
processing time. Finally, a maximum MBT efficiency of 93.3% was achieved.
Key-Words: - PSO, cross-flow turbine, efficiency, hydropower, metaheuristic optimization, Velocity triangles
Received: June 3, 2021. Revised: December 20, 2021. Accepted: January 22, 2022. Published: February 8, 2022.
1 Introduction
Large-scale electricity generation from renewable
sources may contribute to the economic development
of Non-Interconnected Zones (NIZs). However, as a
result of high generation and distribution costs,
supplying electricity to those areas from big facilities
is infeasible [1]. For that reason, their inhabitants
resort to conventional generation technologies such
as carbon-based fuels, which have been associated
with genotoxic and carcinogenic risks to human
health [2]. Small Hydro Power Plants (SHPPs) have
been presented as low-cost, low environmental
impact solutions to supply electricity to NIZs; more
specifically, Michell–Banki Turbines (MBTs) have
drawn great interest because they are easy to install,
manufacture, and maintain, avoiding considerable
civil works to store water and safety devices used in
other types of turbines such as Pelton, Turgo, and
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Francis [3]. MBTs are used as energy recovery
systems in Water Distribution Networks (WDNs),
with better performance and lower costs than current
systems that implement Pressure Release Valves
(PRVs) and Pumps As Turbines (PATs) [4], which
means that MBTs offer wide applicability as low-
cost solutions to harness energy.
The efficiency of MBTs usually ranges between
70% and 85%, and it does not vary significantly
when flow rate conditions change [5]. Furthermore,
MBTs require relatively limited flow rate () and
head () (Figure 1), which represents the main
advantage of this technology.
Fig. 1: Operating conditions of different types of
water turbines [6].
Different mathematical formulations have been
developed in order to size the elements of the highest
impact on efficiency of MBTs and characterize the
flow conditions inside the turbomachine. In 1949, C.
A. Mockmore and F. Merryfield presented the first
mathematical model with experimental validation
that could be used to size an MBT according to the
conditions of the site, defined by flow rate and head
[7]. Later, in order to improve the MBT efficiency by
changing its operating conditions, different
experimental studies concluded that flow rate and net
head have no significant effect on the efficiency of
the turbine, unlike the geometric parameters that
constitute its runner and nozzle [8][9]. According to
other authors, the velocity ratio, determined by the
ratio between the tangential velocity of the runner
and the velocity of the water at the nozzle outlet,
represents an important factor in the configuration of
an MBT; in those cases, maximum efficiencies were
found at velocity ratios between 0.5 and 0.6 [8][10].
Nevertheless, the velocity ratio should be established
according to the flow conditions in order to avoid
cavitation when the flow hits the runner at second
stage [11].
In addition to their operating parameters, the
geometric configuration of MBTs has been studied,
adopting different methodologies, to improve their
efficiency by varying the geometry of the nozzle,
runner, and blades. Numerical (Computer Fluid
Dynamics, CFD) and experimental simulations have
been implemented to determine the influence of the
geometric parameters of the nozzle and the runner on
MBT efficiency; for example, when a guide vane is
used, the performance of the nozzle changes[12],
showing an improvement of 5.33%. Other authors
have proposed the implementation of two nozzles in
the turbine to enhance its efficiency [13] and
presented mathematical formulations that can define
the ideal curvature of the nozzle, specifically the
back wall that redirects water to the runner, ensuring
uniform velocity and angle of attack along the runner
inlet. The substantial influence of the geometry of the
nozzle on MBT efficiency is mainly determined by
the angle of attack of the fluid with respect to the
runner [14][15].
Regarding the design of the runner, the geometry
of the blades represents an important geometric
factor, which has been demonstrated to have an
influence between 31.7% and 86% on MBT
efficiency [16]–[18]. Moreover, the number of
blades, usually between 15 and 45, plays an
important role in the performance of an MBT [19]. In
most cases, the maximum recommended number of
blades is directly related to their geometry, especially
their thickness: the thicker the blades, the lower their
number, and vice versa. Finally, the position of the
blades is determined by the angles of attack (α) of
water, the external angle of the blade (β), and the
aspect ratio of the runner, defined as the ratio
between its inner and outer diameters; such variables
influence the efficiency of an MBT, which ranges
between 60% and 90% [20]–[22].
Experimental results provide a real description of
the operation of an MBT with preestablished flow
conditions; they enable researchers to calculate
operating correlations based on empirical modeling
and thus confirm the effect of geometric or operating
parameters on the performance of the turbine.
However, experimental analyses would be infeasible
in this case because changing the geometric
parameters of an MBT an indefinite number of times
would entail excessively high implementation costs.
CFD can overcome that limitation by implementing
numerical solutions of the proposed domains and
varying the geometric parameters in order to estimate
their effect on MBT efficiency [23]. Although they
are efficient, fast, and lower cost than their
experimental counterparts, numerical models
sometimes require great computational capacity in
order to estimate the physical phenomena that take
place in different dynamic flow systems.
For that reason, mathematical formulations and
optimization techniques have been employed in
recent years to reduce processing times and find
adequate solutions to design problems, thus limiting
the complexity and computational effort associated
with the estimation of the effect of geometric and
operating variables on MBT efficiency [20]. Among
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the optimization methods employed in the
specialized literature, metaheuristic techniques have
drawn great attention [24] and provided high-quality
solutions to engineering problems [24]–[27] (e.g.,
designing power generation turbines [28], [29]) with
satisfactory design results and reduced computational
loads. Particle Swarm Optimization (PSO) is one of
the most commonly employed metaheuristic
techniques in the literature to solve continuous
problems [24], which is the case analyzed in this
work. PSO is based on the behavior of groups of
animals looking for food sources, and it offers low
mathematical and programming complexity [30].
This technique has also been implemented to
determine the configuration of α and β in order to
find the best hydraulic performance of the MBT [20],
[21]. In those cases, however, the mathematical
formulations proposed by the authors assumed the
inlet and outlet blade angles to be equal. For that
reason, their mathematical models do not adequately
represent the physics of the model under analysis.
Therefore, this work highlights the importance of
proposing new MBT design methodologies that
adequately represent the physics of the problem and
implement efficient solution methods that entail a
low computational cost. This study has two
objectives: (i) to determine the α and β angles that
guarantee the maximum MBT efficiency by
implementing a PSO method based on the moment of
momentum equation and (ii) to validate the results
obtained with respect to experimental data reported
in the literature.
Section 3 presents a brief theoretical framework
of the operation of an MBT and the physical
behavior of water inside the turbine. Section 4
describes the optimization method proposed in this
work to solve the problem. Subsequently, Section 5
details the methodology and the parameters
implemented to address the problem. Section 6
reports the results, and Section 7 draws the
conclusions.
2 Theoretical Framework
An MBT is a crossflow turbine composed of a
runner, a nozzle, a guide vane, and a casing as shown
in Figure 2. The runner is a wheel defined by inner
and outer diameters that harnesses 70% of the energy
in the water stream during the first stage of the
process (it means, the first time the water makes the
runner work) and 30% during the second stage (it
means, the second time the water makes the runner
work) [14]. Multiple related parameters have been
studied in the literature to improve the efficiency of
such turbine: the angle of attack of water (defined by
the internal and external angles of the blade with
respect to the tangent of the runner), the design of the
upper casing of the nozzle (which guarantees that the
flow enters the runner uniformly), the number of
blades, and the geometric profile of the blades.
a)
b)
Fig. 2: a) Michell–Banki Turbine assembly. b)
Energy transfer in the first and second stages [19].
2.1 Geometrical Formulation
Different models can be used to size and determine
the geometry of the components of an MBT based on
flow rate and net head conditions. The design of the
runner is commonly derived from the velocity
triangle described in Figure 3, where regions 1 and 2
represent the inlet and outlet flow through the runner
during the first stage respectively; in turn, regions 3
and 4 correspond to the inlet and outlet flow in the
second stage, respectively. This geometric analysis
will be subsequently used for the physical
formulation of the behavior of water inside the
turbine.
a)
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b)
Fig. 3: a) Fluid jet trajectory at different locations. b)
Velocity triangles at locations 1-4. [12].
In a typical MBT design, the blades are assumed
to be arcs of circumferences, which are defined by
their and parameters and the runner inner-to-
outer–diameter ratio. If angle is assumed to be
equal to 90°, it is not necessary to select other
parameters (such as the opening angle of the blades
or their radius) because they can be described as a
function of other values, according to [7].
Fig. 4: Geometric parameters of the blades in the
turbine (adapted from [14]).
From Figure 4, it can be obtain
(1)
Likewise, can be described in the same terms.
(2)
The equations above imply that every variable of the
turbine can be described in terms of the inner and
outer diameters of the runner and angle .
2.1.1 Sub-subsection
To develop the physical formulation of the behavior
of water inside the turbine, this study implemented
the four assumptions below. They may not
correspond to the real behavior of the system but to
the conditions needed for an optimal performance.
1. The fluid is considered homogeneous,
uncompressible, and in a steady state.
2. The energy contributions due to the
difference in gravitational potential between the
water inlet and outlet in the turbine are negligible.
3. The angle at which water leaves the turbine
in the first stage is equal to the angle at which water
enters the turbine in the second stage.
4. The loss coefficients equal 1.
MBT efficiency (), defined as the ratio between
the outlet power of the turbine and the inlet power of
water, is described by equation (3).
(3)
Where is the turbine efficiency, is the torque
produced by the water, is the water density and
is gravitational acceleration. For the optimization
process, torque and angular velocity is calculated
using the formulation by M.A. Chavez-Galarza [22].
The function below determines the torque
applied by the water to the turbine runner using the
moment of momentum equation in the control
volume in Figure 5.
a)
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b)
c)
Fig. 5: Proposed boundary conditions.
(4)
Where is the location (1, 2, 3 or 4) from figure
3, is the water arc, is the momentum-correction
factor = 1, and are the external runner radius,
and are the internal runner radius, is the
angle between the velocity of the water and the
tangent of the runner and is the absolute velocity.
If Assumptions 2 and 3 are applied to Equation
(4) to cancel out the second and third terms of the
Equation and is defined, a flow rate and a torque
equation can be presented as
(5)
(6)
Water velocity can be calculated using the
energy conservation principle.
(7)
Where is the loss coefficient associated with
friction, which varies between 0.92 and 0.98 [22];
however, in this study, it equals 1 due to Assumption
4.
Replacing (6) and (7) in (3), we can obtain an
equation that determines MBT efficiency as a
function of both geometric and operating parameters:
(8)
Where is the tangent velocity = .
The experimental results in the specialized
literature revealed that the operating conditions,
especially and , do not exert a great influence on
MBT efficiency. For that reason, Equation (8) was
simplified in order to obtain a numerical expression
of efficiency as a function of the geometric
parameters associated with the runner and the nozzle.
Based on the velocity triangle in Figure 6 (which
relates the fluid dynamics to the runner inlet and
outlet), we obtained Equations (9) to (12).
Fig. 6: Velocity triangles from runner inlet to outlet
[22].
(9)
(10)
(11)
(12)
Where is the angle between the inlet of the
blade and the tangent of the runner, is the angular
velocity of the turbine, and is the absolute
velocity.
Afterward, by replacing (9), (10), (11), and (12)
in (8), it can be obtained Equation (13), which can be
used to determine MBT efficiency as a function of
the geometric parameters and only.
(13)
Equation (13) is thus the objective function in this
optimization process to determine the maximum
efficiency of the turbine.
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3 PSO optimization
Developed by Eberhart and Kennedy in 1995, a PSO
algorithm replicates the behavior groups of animals
(flocks of birds or schools of fish) searching for food
sources [30]. Each particle represents an animal
randomly placed in a solution space limited by
constraints defined for a particular situation. PSO
methods are characterized by the way particles move
over the solution space: every single particle is
affected by the maximum value obtained by itself as
much as the maximum value obtained by the entire
swarm. Furthermore, such methods can control the
progress of the particles with a random component
that prevents the algorithm from being trapped in
local optima.
Importantly, the coordinates that define the
position of the particles over the solution space
correspond to the values of the parameters used to
solve the objective equation. This technique has two
versions: continuous and binary algorithms. This
work implemented the continuous variant due to the
nature of the equation and its variables, which are
represented by real numbers.
The iterative process of the method is applied
after the definition of the number of particles (P)
used to create a P-size set where it is possible to store
the position of every particle denoted by vector .
The latter contains the variables (coordinates of the
particle) to be optimized in the problem, as in
Equation (14). Thus, a set of values associated with
the optimal solution to the problem is found at the
end of the iterative process. Likewise, the velocity
vector contains the velocity of every particle, as
shown in expression (15). Both vectors vary with
every interaction, having the particles move closer to
the maximum point in the domain. Additionally,
such values are assigned between maximum and
minimum allowable limits, which are relevant
constraints determined for every case.
(14)
(15)
Where denotes the number of dimensions or
parameters to be optimized.
The position assigned to every particle, , is
determined by the position of the particle at the
previous iteration, , and the assigned velocity,
, being the current iteration of the particle.
(16)
Variable is obtained from the values assigned
to each variable at the previous iteration and the
implementation of two adaptation functions. By
analyzing the objective function of each particle,
such functions can be used to identify the best
position of the i-th particle () and the position of
the best solution in the swarm of particles ()
[31] in the following way:
(17)
Where and are the cognitive learning ratio
(individual) and social ratio (group), respectively; ,
an inertial coefficient; and and , random
numbers evenly distributed between 0 and 1.
Parameters and denote the relative importance
of the memory (position) of the particle itself and the
memory of the swarm, respectively [32].
Below is the flowchart of the PSO algorithm.
Fig. 7: Flowchart of the PSO algorithm (adapted
from [31]).
4 Methodology
The parameters to be optimized in this study are
angles β1 and α1 in Figure 3. With such angles, it is
possible to calculate the tangential velocity of the
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turbine using its velocity triangle. The objective
function of this problem was defined by Equation
(13), which aims at maximizing MBT efficiency as a
function of variables and . The constrains
associated with the problem are defined in (18) and
(19). The minimum and maximum values of are
15° and 24°, respectively [3]. The lower limit of is
15° because the efficiency in Equation (13) is only
negative if , and the maximum limit at 45°
enables a solution space wide enough to find the
maximum value of the equation. Table 1 presents
other terms used in this study to implement the
algorithm.
(18)
(19)
Table 1. Parameters used in the optimization
algorithm.
Parameter
Value
1.5
0.9
0.4
Iterations
30
Particles
30
The parameters of the optimization algorithm
listed above are not the only ones that can produce an
accurate solution. Nevertheless, with the values in
Table 1, a solution to the problem was found in the
shortest time, always ensuring a standard deviation
below 1x10-5 in the results obtained after the
algorithm was applied 10 times.
5 Results
In order to validate the accuracy of Equation (13),
Table 2 shows other numerical and experimental
efficiency results reported by different authors, as
well as the error rates of the mathematical
formulation proposed here and the method in [21]
with respect to such results. From left to right, Table
2 presents authors and references; reported and
; efficiency obtained experimentally or
numerically; efficiency calculated using the method
in [21]; efficiency calculated by Equation (13)
proposed in this work; and absolute error rate, which
is a comparison of the efficiency obtained by [21]
and Equation (13) with the reported results.
Table 2. Numerical and experimental results in the literature compared to calculations using the method in
[21] and Equation (13) in this work.
Author
(°)
(°)
Reported
efficiency
Efficiency
by [21]
Efficiency
by Eq.
(13)
(%) Error
rate by [21]
(%) Error
rate by Eq.
(13)
A. J. Dakers and G. Martin
[33]
22
30.0
0.69
0.8118
0.7224
17.65
4.70
W. Johnson et al [34]
16
39.0
0.80
0.8765
0.8453
9.56
5.67
Y. Nakase et al[35]
15
39.0
0.82
0.8854
0.8263
7.98
0.77
S. Khosrowpanah et al. [36]
16
39.0
0.80
0.8765
0.8453
9.56
5.67
A. A. Fiuzat and B. Akerker
[37]
24
39.0
0.89
0.7862
0.8263
11.66
7.16
V. R. Desai and N. M. Aziz
[8]
22
39.0
0.88
0.8118
0.8597
7.75
2.31
H. G. S. Totapally and N.
M. Aziz [38]
22
39.0
0.90
0.8118
0.8597
9.80
4.48
V. Sammartano et al. [39]
22
38.9
0.86
0.8118
0.8597
5.60
0.04
Y.C. Ceballos et al. [19]
22
40.0
0.86
0.8118
0.8585
5.60
0.18
Average
9.46
3.44
Standard
Deviation
3.66
2.67
In Table 2, the maximum absolute errors produced
by Equation (13) and the methodology proposed in
[21] were 7.16% and 17.65%, respectively, in
relation to the efficiencies reported in the
specialized literature. However, the performance of
the two mathematical formulations in Table 2 is
different. The method proposed in this paper
produces a reduction of 64% in the average error
rate (3.44) compared to that of the mathematical
formulation in [21] (9.46). Thus, the proposed
formulation demonstrates to be an excellent tool for
calculating MBT efficiency in a numerical form.
In addition, this work implemented a PSO algorithm
to solve the mathematical formulation presented in
Section 5, where and equal 15° and 28.186°,
respectively. Finally, a maximum MBT efficiency
of 93.3% was obtained. This theoretical result is
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better than the output reported in the references used
in this study. In this case, the simulation was carried
out on an Intel® core™ I 7-5500U desktop
computer with 8GB of RAM using the software
MATLAB R2019b®.
6 Conclusions
This paper described the development of a new
methodology for calculating the optimal angle of
attack and blade position of an MBT in order to
achieve its maximum efficiency. To improve its
hydraulic efficiency, the authors proposed a
mathematical formulation based on the moment of
momentum equation and a PSO algorithm as
solution method. Such formulation produced a
maximum uncertainty of 7.16% and a standard
deviation of 2.67% compared to the literature;
additionally, it reduced the absolute average error by
64% compared to the method in [21].
Angle was found to have a considerable
influence on MBT efficiency, and its ideal value is
15° for an optimal performance of the turbine in the
proposed domain, from 15° to 24°. The ideal angle
is 28.186°. A theoretical efficiency of 93.3% was
obtained with those parameters.
The PSO method was successfully adapted to
the objectives of this study, and it found an optimal
value in a very short time (compared to CFD
calculations) when it solved the efficiency equation
900 times. In conclusion, an optimization method
and a computer with limited specifications can
obtain fast results, which is not possible with CFD
techniques.
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A. J. Perez-Rodriguez, J. Sierra-Del Rio,
L. F. Grisales-Noreña, S. Galvis
E-ISSN: 2224-3461
52
Volume 17, 2022
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
J. Sierra-Del Rio and S. Galvis contributed with the
knowledge of the Banki turbine and the
mathematical formulation.
A. Pérez-Rodríguez and L. F. Grisales-Noreña
contributed with the knowledge of PSO method and
programation.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2022.17.6
A. J. Perez-Rodriguez, J. Sierra-Del Rio,
L. F. Grisales-Noreña, S. Galvis
E-ISSN: 2224-3461
53
Volume 17, 2022
