# Banki-Michell Optimal Design by Computational Fluid Dynamics Testing and Hydrodynamic Analysis

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## Abstract

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## 1. Introduction

## 2. The State of the Art on Michell-Banki Parameter Design

_{idr}as the ratio between the power available for production by the machine (P

_{t}) and the input power P

_{in}related to the hydraulic head H

_{in}at the inlet of the machine, and for a given water discharge Q:

_{1}and D

_{2}of the impeller, the angles β

_{1}and β

_{2}of each blade with respect to the tangent direction respectively at the outer and the inner diameters, the blade number and thickness, the angle λ of the arc available for the discharge inlet along the impeller outer circumference, and the ratio W/B, where B is the nozzle width and W is the impeller width. We assume in the first step of the design that the shape of the distributor is able to guarantee an approximately constant angle α all along the outer circumference of the inlet impeller.

_{1}value:

_{1}that avoids any discontinuity in the path of the water particle entering the impeller is the one that satisfies the following condition:

_{r}is the radial component of the particle velocity V and ω is the angular velocity of the rotating reference system. We will show in the following section that Equation (3) is equivalent to Equation (2) when the velocity of the rotating system is half of the component of the particle velocity in the tangent direction.

_{2}is π/2 [10,11,12,13]. This value provides a radial direction to the relative outlet velocity inside the impeller at the inner diameter (D

_{2}), and this leaves to the fluid, in the inner part of the impeller, only the energy of the rotating system, which can be recovered during the next blade crossing. Mockmore and Marryfield [12] derived the expression of the maximum efficiency according to some restrictive hypotheses and to Relationship (3), as a function of the angle α:

_{2}/D

_{1}) corresponds to a stronger curvature of the impeller blades, because the blade angle has to shift from the β

_{2}to the β

_{1}value in a shorter distance. This implies stronger turbulence, but also a shorter distance for energy transfers. Aziz and Totapally suggest that the maximum efficiency is achieved with a diameter ratio of 0.68 [13].

_{b}on turbine efficiency has been explored by experimental and numerical studies [13,14,15,16,17]. The studies showed that an increment in the blade number has a positive impact on the efficiency of the turbine, due to a more regular velocity profile inside the space between each couple of blades. In particular, Aziz and Totapally observed that an increment in efficiency can be obtained by increasing the number of blades from 15 to 35 [13]. On the other hand, a further increase provides larger energy dissipation due to a strong effect of the blades solid walls. Choi et al., using CFD analyses, observed the same trend as Aziz, but the maximum efficiency of the turbine was obtained in the case of an impeller with 30 blades [14].

Design parameters | values | Description of the design parameters |
---|---|---|

D_{2}/D_{1} | 0.68 | Diameter Ratio |

α | 22° | Angle of attack |

β_{2} | 90° | Blade exit angle |

λ | 90° | Inlet discharge angle |

N_{b} | 35 | Number of blades |

## 3. The Proposed Two-Step Design Procedure

_{1}and of the blade attack angle β

_{1}, for given water discharge Q and hydraulic head H, as well as of the nozzle profile. The maximum efficiency attainable in these conditions is derived from the Euler’s equation for rotating machines and from the assumption of zero energy dissipation inside the nozzle. The other impeller parameters are estimated by testing single options by means of CFD analysis. The inner diameter D

_{2}is optimized by evaluating the efficiency of the turbine for different values of the diameter ratio D

_{2}/D

_{1}; the inner radius blade, the attack angle, the number of blades, their shape and the inner/outer diameter ratio are optimized starting from known literature results. Computations can be carried out in cases both of zero downstream pressure (where the air phase is present in the impeller case) and non-zero downstream pressure (where the impeller case is fully pressurized).

_{t}transferred from the current flow to the rotating system, according to the Euler’s equation can be written in the following form:

_{1}, P

_{2}, P

_{3}and P

_{4}. Assuming a β

_{2}blade angle equal to π/2, the sum of the second and third components is equal to:

_{2}= U

_{3}).

_{4}, and assuming that almost all the fluid energy has been either transferred or dissipated before leaving the impeller, the only component affecting the turbine efficiency remains the first one. Thus, Equation (5) could be simplified and becomes:

_{1,t}is the tangential component of the relative velocity $\overrightarrow{W}$. Thus, Equation (6) can be written as:

#### 3.1. First Step: Design of Basic Parameters

_{1}or the α angle by substituting Equation (9) in Equation (7). Assuming that $U=\omega {D}_{1}/2$, we obtain:

_{1}starting from the α angle, or vice versa. Once both D

_{1}and the inlet velocity angle α are known, the blade β

_{1}angle can be computed as the inverse of the tangent function, equal to the ratio between the velocity relative components, that is:

_{0}, its width B and the shape of the wall between its tip and the initial rectangular section with height S

_{0}(see Figure 1). The height S

_{0}is a function of the specific water discharge q (per unit width), and this can be directly computed according to the continuity equation applied at the inlet of the impeller:

_{1}stems mainly from a balance between the search of a low angle α (and a high hydraulic efficiency according to Equation (10)) and the need to limit the width B of the nozzle resulting from Equation (14), because the blade length W (W > B) strongly affects the mechanical stress inside the same blades. Moreover, by reducing the angle α, the ratio between the free inlet surface of the impeller and the total inlet surface of the same impeller drops due to the enhanced effect of the blade thickness.

_{e}and q

_{l}respectively) are equal to:

_{r}is the radial velocity. According to this hypothesis the upper wall radius r(θ) is equal to:

_{o}can be calculated as:

**Figure 3.**Nozzle upper wall shape: (

**a**) geometric scheme; (

**b**) entering and leaving water flow in the nozzle.

#### 3.2. Second Step: Impeller Parameters Optimization

_{2}/D

_{1}affects the efficiency of the cross-flow turbine: an increment of the diameter ratio leads to a reduction in the blade radius, a reduction of the blade surfaces and a shorter distance for energy transfers. In order to select the optimal internal diameter D

_{2}, a sensitivity analysis on the diameter ratio can be carried out by using a fluid dynamic investigation. As will be shown in the methodology testing, the sensitivity analysis often leads to a diameter ratio very close to the 0.68 literature value.

_{2}is set equal to 90° and the blade radius ρ

_{b}and the central angle δcan be computed as follows:

## 4. Fluid Dynamic Investigation by CFX Code

**Figure 6.**3D-computational mesh with a zoom view close to the inlet of the impeller (case with W > B).

_{l}, ρ

_{l}, μ

_{l}and U

_{l}; respectively, the volume fraction, the density, the viscosity and the mean in time value of velocity vector for phase l (l = w, a), that is:

_{i}is an external source term. The momentum equation is instead referred to the “averaged” phase:

_{M}is the momentum of the external source term S; ${\mu}_{eff}$ is the effective viscosity accounting for turbulence and defined as:

_{t}is the turbulence viscosity and ${p}^{\prime}$ is the modified pressure, equal to:

**U**. To close the set of six governing scalar equations [Equations (22) and (23) (two) and Equation (24) (three)], a k – ε turbulence model is applied. The unknowns of the problem are: (1) pressure p; (2) the scalar components U

_{i}(i = 1, 2, 3 of the velocity field); and (3) the volume fraction of each phase (two). The computational domain is divided into two sub-domains: the stator (nozzle and casing) has an inertial reference system; the rotor (impeller) has a non-inertial reference system, integral with the rotation axes of the rotor. In the CFX code, the interface model “general connection” was set, since the reference system changes at the interface of the abovementioned sub-domains. The “transient rotor-stator” option was also selected to take into account the transient effects along the abovementioned interface. Using this option, the interface position was updated at each time step, and the relative position of the grids on each side of the interface changes. At the entrance of the nozzle volume fractions were set at zero for air and one for water. At the base of the production chamber, at the water outlet, a pressure value of 1 atm was imposed as a boundary condition, enabling the possible flow of air from the outside towards the inside of the machine. The same boundary condition was imposed in the nodes along the air vents.

## 5. Impeller Design Testing

_{1}, parameters B, β

_{1}and the profile of the nozzle external wall (parameters S

_{o}and K) were computed according to Equations (11)–(19) reported in step 1 of the procedure. The geometry of the impeller blades (ρ

_{b}and δ) was computed according to Equation (20), reported in step 2 of the procedure, assuming a diameter ratio D

_{2}/D

_{1}equal to 0.68, as suggested in the literature [8,13,18]. It was also assumed that a value α = 22° provides the best equilibrium between hydraulic efficiency and mechanical strength, as also suggested in the literature [10,13]. The other parameters were initially set equal to the values suggested in the literature. The geometry of the resulting cross flow turbine is reported in Table 2.

Parameter | value | Description of the geometrical parameters |
---|---|---|

D_{1} (mm) | 161 | Impeller outer perimeter diameter |

D_{2} (mm) | 109 | Impeller inner perimeter diameter |

N_{b} (-) | 35 | Number of blades |

λ (°) | 90 | Inlet discharge angle |

α (°) | 22 | Attack angle of the outlet nozzle velocity |

β_{1} (°) | 38.9 | Angle between the blade and the outer perimeter of the impeller |

β_{2} (°) | 90.0 | Angle between the blade and the inner perimeter of the impeller |

ρ_{b} (mm) | 27.7 | Radius of blade |

δ (°) | 61.5 | Central angle of blade |

S_{0} (mm) | 47 | Nozzle initial height |

B (mm) | 93 | Nozzle width |

K (-) | 31.5 | Constant in Equation (19) |

W (mm) | 139 | Impeller width |

#### 5.1. 2D-Simulations: Optimum Geometry Tests

_{b}equal to 30, 35 and 40 were investigated. For each simulation, the efficiency of the cross-flow turbine was calculated as the ratio between the power supplied to the impeller (applied torque T times the impeller angular velocity ω) and the power lost by water passing through the turbine (difference between the hydraulic power of the water flow in the inlet and in the outlet of the turbine):

_{inlet}and P

_{outlet}are the hydraulic power of the water flow in the inlet and in the outlet of the turbine:

_{s}= 1.5 s). In the simulation, a time step size $\Delta T$ of 0.001 sec was selected. In Figure 8 the time series of the instantaneous output power (P

_{out}) corresponding to the three abovementioned configurations are shown. The efficiency displays regular oscillations, decreasing along with the number of blades N

_{b}. The time averaged value of the output power P

_{out}is equal to 5.12 KW for 30 blades (Figure 8a), 5.20 KW for 35 blades and 5.07 KW for 40 blades, while the averaged efficiency for the impeller with 30, 35 and 40 blades is, respectively, 87%, 88.4% and 86%. The optimum configuration was the impeller with 35 blades, in agreement with the experimental investigations in [13].

**Figure 8.**Time series of the output power P

_{out}for the impeller with a different number of blades: (

**a**) 30 blades; (

**b**) 35 blades; and (

**c**) 40 blades. The red dotted lines represent the time averaged value of the output power P

_{out}.

_{b}= 35, a numeric simulation was carried out in order to test the efficiency of the impeller with the nozzle designed by using Equations (16–19). To this aim the velocity norm V and the attack angle α along all the inlet of the impeller were computed by removing the idealized boundary condition assigned in the previous tests and solving the problem along the entire computational domain presented in Section 4. This simulation was performed taking into account two physical domains—rotor and stator—as also described in Section 4, with a total simulation time T

_{s}of 1.5 s. For each time step the value of the turbine efficiency η was estimated by using Equations (27a–27b). The plot of the instantaneous output power P

_{out}versus time is reported in Figure 9 and this shows that the output power is about 5.02 KW. The time averaged value of the efficiency η drops slightly to a value of 85.39%, with a limited 3.01% reduction.

**Figure 9.**The time series of the output power P

_{out}for the turbine with 35 blades and the designed nozzle. The black dotted line represents the time averaged value of the output power P

_{out}.

_{s}= 1.5 sec), the tangential velocity V

_{t}($V\mathit{cos}\alpha $), the radial velocity V

_{r}($V\mathit{sin}\alpha $) and the attack angle α evaluated at the middle points between two consecutive blades along all the inlet arc of the impeller (ξ), respectively, as shown in Figure 10 (V

_{t}and V

_{r}) and in Figure 11 (a). The average value of V

_{t}was about 14 m/s, the average value of V

_{r}was about 5 m/s and both remained almost constant along ξ. The attack angle α has an average value of about 21°, very close to the optimal α value suggested in the literature and adopted in the design procedure (step 1 and step 2). Moreover in Figure 12, it can be seen that the velocity norm V did not change significantly inside the nozzle. This validates the hypothesis of negligible energy dissipation.

_{2}/D

_{1}and the corresponding values of the blade radius ρ

_{b}and of the central angle δ are shown in Table 3.

D_{2}/D_{1} | ρ_{b} (mm) | δ (degree) |
---|---|---|

0.58 | 34 | 66 |

0.63 | 31 | 63 |

0.65 | 30 | 63 |

0.68 | 28 | 61 |

0.73 | 24 | 60 |

0.78 | 20 | 58 |

_{2}/D

_{1}= 0.65, close to the value D

_{2}/D

_{1}= 0.68 suggested in the literature [8,13,18]. The efficiency ${\eta}_{a}$ of the cross flow turbine, obtained by selecting a ratio D

_{2}/D

_{1}= 0.65, is 85.6%.

_{2}/D

_{1}= 0.65−0.68) did not significantly change.

_{t}/U. The efficiency curve of the turbine, reported in Figure 15, shows that the peak of efficiency was in the range 1.8 ÷ 2.0. This result is in agreement with the assumption represented by Equation (11). Moreover, the average efficiency of the turbine was greater than 80% for a value of V

_{t}/U varying between 1.2 and 3.0 (corresponding to a water discharge varying between 35 l/s and 90 l/s). In Figure 16, Figure 17 and Figure 18 the distributions of the water volume fraction, the velocity field, and the pressure in the simulated sub-domains for the optimum configuration are shown.

**Figure 14.**The angle of attack α at the impeller’s inlet for different values of the ratio D

_{2}/D

_{1}.

Parameter | Value | Description of the geometrical parameters |
---|---|---|

D_{1} (mm) | 161 | Impeller outer perimeter diameter |

D_{2} (mm) | 104 | Impeller inner perimeter diameter |

N_{b} (-) | 35 | Number of blades |

λ (°) | 90 | Inlet discharge angle |

α (°) | 22 | Angle of attack |

β_{1} (°) | 38.9 | Angle between the blade and the outer perimeter of the impeller |

β_{2} (°) | 90.0 | Angle between the blade and the inner perimeter of the impeller |

ρ_{b} (mm) | 29.8 | Radius of blade |

δ (°) | 62.6 | Central angle of blade |

S_{0} (mm) | 47 | The nozzle initial height |

B (mm) | 93 | Nozzle width |

K (-) | 31.5 | Constant in Equation (16) |

W(mm) | 139 | Impeller width (using W/B = 1.5) |

#### 5.2. 3D-Simulations: Spread Ratio Tests

_{out}calculated for two different ratios W/B, equal respectively to 1.5 and 1.0, are plotted in Figure 21 versus time. The resulting average efficiencies were respectively, 78% and 85%, while the respective power outputs were 4.73 KW and 5.03 KW. This result suggests that, at least in this particular case, the use of a larger stream spread W/B equal to 1.5 does not improve the efficiency of the turbine, in disagreement with the results of Aziz and Desai [10]. Indeed, when the nozzle has the same width as the impeller (W/B = 1) the average efficiency is 85%, against the 78% value obtained for W/B = 1.5. Also observe that the average efficiency obtained for the W/B = 1 value using a 3D approach is very close to the ratio computed using a more simple 2D approach. This validates the overall procedure, based on 2D computations.

**Figure 21.**The instantaneous values of the output power P

_{out}of the turbine for two different values of the W/B ratio: W/B = 1.0 and W/B = 1.5.

## 6. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Sammartano, V.; Aricò, C.; Carravetta, A.; Fecarotta, O.; Tucciarelli, T.
Banki-Michell Optimal Design by Computational Fluid Dynamics Testing and Hydrodynamic Analysis. *Energies* **2013**, *6*, 2362-2385.
https://doi.org/10.3390/en6052362

**AMA Style**

Sammartano V, Aricò C, Carravetta A, Fecarotta O, Tucciarelli T.
Banki-Michell Optimal Design by Computational Fluid Dynamics Testing and Hydrodynamic Analysis. *Energies*. 2013; 6(5):2362-2385.
https://doi.org/10.3390/en6052362

**Chicago/Turabian Style**

Sammartano, Vincenzo, Costanza Aricò, Armando Carravetta, Oreste Fecarotta, and Tullio Tucciarelli.
2013. "Banki-Michell Optimal Design by Computational Fluid Dynamics Testing and Hydrodynamic Analysis" *Energies* 6, no. 5: 2362-2385.
https://doi.org/10.3390/en6052362