# A Novel Hybrid Approach for Numerical Modeling of the Nucleating Flow in Laval Nozzle and Transonic Steam Turbine Blades

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Two-Phase Flow

#### 2.1. Conservation Equations Governing the Main Flow

#### 2.1.1. Equation of State

#### 2.1.2. Wetness Fraction

#### 2.1.3. Nucleation Equation for the Critical Radius

#### 2.1.4. Relations of Droplet Growth

- (a)
- The number of produced droplets and the average radius at the inlet of the new calculation element is defined by the average radius at the outlet of the previous element. In the nucleation region, the number of new droplets and their radius is determined in the new calculation element. Both new and old sets of droplets can grow along the calculation element, knowing the number of droplets in each of the sets, different methods including surface or volume averaging can be used to calculate the average radius at the outlet of the new calculation element.
- (b)
- In the region after the nucleation, i.e. the wet region, the flow tends towards equilibrium or saturation, the number of droplets is constant and the droplets can only grow. After calculating the droplet growth in each calculation element, any of the mentioned averaging methods can be used to obtain the average radius, which in this research, to reduce the calculations the surface averaging method is used.

## 3. CUSP’s Method

## 4. Inverse Method

#### 4.1 The Mathematics of the Problem

#### 4.2. Convergence of the Solution

## 5. Algorithm for Combining the Scalar and CUSP Finite Volume Methods with the Inverse Method

## 6. Results

#### 6.1. Moore Nozzle Type A

#### 6.2. Moore Nozzle Type B

#### 6.3. Young Nozzle Type C

#### 6.4. Barschdorff Nozzle

#### 6.5. Mid-Section Turbine Blade

#### 6.6. Young Nozzle Type L

#### 6.7. Mid-Section Turbine Blade

#### 6.8. Tip-Section Turbine Blade

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

$A$ | The element area (m^{2}) |

$\mathrm{CFL}$ | Courant number |

$e$ | Internal energy or inverse method error (kJ/kg) |

${F}_{p}$ | Flux vector of the pressure in the CUSP |

${F}_{x},\text{}{F}_{y}$ | Flux vector |

$\Delta G$ | Gibbs energy (kJ) |

$\mathrm{G}$ | Vapor-phase Symbol |

$h$ | Enthalpy (kJ/kg) |

${h}_{0}$ | The total enthalpy |

$J$ | Nucleation rate (No. of Droplet/m^{3}·s) |

$kn$ | Knudsen number |

$k$ | Boltzmann’s constant |

$\mathrm{L}$ | Heat latent or Liquid phase symbol or symbols left by CUSP (kJ) |

$L\left(u,v\right)$ | Switch function of artificial dissipation |

$\overline{l}$ | Mean free path of vapor molecules (m) |

$m$ | Mass flow or the mass of a molecule (kg or kg/s) |

$M$ | Mach number |

$\mathrm{MP}$ | Mid passage (center line) |

$N$ | Number of droplets per unit mass |

$P$ | Static pressure (kPa) |

${P}_{0}$ | Stagnation pressure (kPa) |

$\overrightarrow{P}$ | Unknown parameters |

$\mathrm{PS}$ | Pressure side |

${\overrightarrow{P}}_{\mathrm{est}}$ | An estimate of the unknown parameter |

$r$ | Radius Droplet (m) |

$q$ | Condensation factor |

$Q$ | Displacement flux |

$\mathrm{R}$ | Gas constant or symbol in the right of CUSP (kJ/(kg·K)) |

$\mathrm{SS}$ | Suction side |

$S$ | Function calculator inverse method |

${S}_{x}$${S}_{y}$ | x and y directions of the vector elements |

$T$ | Static temperature (K) or inverse method output |

${T}_{0}$ | Stagnation temperature (K) |

${T}_{\mathrm{G}}$ | Steam temperature (K) |

${T}_{\mathrm{L}}$ | Droplet temperature (K) |

${T}_{\mathrm{s}}$ | Saturation temperature (K) |

$u,v$ | Components of Velocity (m/s) |

$\Delta V$ | Elements Size (m^{3}) |

$\mathrm{V}$ | Velocity (m/s) |

$w$ | Flux |

$x,y$ | In order to coordinate the flow perpendicular to it |

$X$ | Sensitivity matrix |

$z$ | Convergence parameter on CUSP’s method |

## Greek Symbols

$\alpha $ | Heat transfer coefficient of droplet with vapor |

$\sigma $ | Surface tension (N/m) |

$\nu $ | Kinematic viscosity or correction factor in the nucleation equation (m^{2}/s) |

$\lambda $ | Thermal conductivity (W/m·K) |

$\mathsf{\Omega}$ | Diagonal matrix to reduce the deviation |

$\rho $ | Density (kg /m^{3}) |

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**Figure 2.**Stage algorithm of hybrid method (scalar (original Jameson) + CUSP + inverse) for modeling the two-phase nucleating flow to obtain the optimized convergence parameter $z$.

**Figure 3.**Stage algorithm of hybrid finite volume method (scalar (original Jameson) + CUSP) for modeling the two-phase nucleating flow in Laval nozzle, mid and tip-section blades of a steam turbine with using the optimized convergence parameter ($z=2.667$).

**Figure 4.**Grid computing standard for Moore nozzle type A [40].

**Figure 5.**Comparing the distribution of static pressure with inlet stagnation pressure on the center line stream (MP) along the nozzle results of the scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with experimental data in supersonic outlet flow for Moore nozzle type A [40].

**Figure 6.**Distribution variation of logarithmic radius along the nozzle in central line stream (MP) results of scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with the experimental droplet radius in supersonic outlet flow for Moore nozzle type A [40].

**Figure 7.**Grid computing standard for Moore nozzle type B [40].

**Figure 8.**Comparing the distribution of static pressure with inlet stagnation pressure on the center line stream (MP) along the nozzles results of the scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with experimental data in supersonic outlet flow for Moore nozzle type B [40].

**Figure 9.**Comparing distribution variation of logarithmic radius along the nozzle in central line stream (MP) results of scalar and hybrid (scalar + CUSP+ inverse) numerical methods and comparing with the experimental radius in supersonic outlet flow for Moore nozzle type B [40].

**Figure 11.**Comparing the distribution of static pressure with inlet stagnation pressure on the center line stream (MP) along the nozzles results of the scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with experimental data in supersonic outlet flow for Young nozzle type C [43].

**Figure 13.**Comparing the distribution of static pressure with inlet stagnation pressure on the center line stream along the nozzles results of the scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with experimental data in supersonic outlet flow for Barschdorff nozzle [44].

**Figure 14.**Distribution variation of logarithmic radius along the nozzle in central line stream (MP) results of scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with the experimental radius in supersonic outlet flow for Barschdorff nozzle [44].

**Figure 16.**Comparing the distribution of static pressure with inlet stagnation pressure along the blades on suction side (SS) results of the scalar and hybrid (scalar + CUSP + inverse) numerical methods and comparing with experimental data in supersonic outlet flow for dry steam in mid-section turbine blade [1,12].

**Figure 17.**The impact value of convergence parameter $z$ values improvement on the change of mass flow percent with inverse method.

**Figure 18.**The percent of CUSP method in hybrid method (scalar + CUSP) for flow modeling between mid-section turbine blade use of inverse method.

**Figure 20.**Comparing the distribution of static pressure with inlet stagnation pressure on the center line stream (MP) along the nozzle results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods and comparing with experimental data for supersonic outlet flow in Young nozzle type L [10,43]. Note: The Laval nozzle is adiabatic, only the convergent duct is reversible.

**Figure 21.**Comparing the distribution percent variation of stagnation pressure with inlet stagnation pressure on the center line stream (MP) along the nozzles results of the Scalar and hybrid (Scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in Young nozzle type L.

**Figure 22.**Distribution variation of wetness fraction on the center line stream (MP) along the nozzles results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in Young nozzle type L.

**Figure 23.**Distribution variation of nucleation rate on the center line stream (MP) along the nozzles results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in Young nozzle type L.

**Figure 25.**Distribution variation percent of mass flow on the center line stream (MP) along the nozzles results of the Scalar and hybrid (Scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in Young nozzle type L.

**Figure 26.**Distribution of static pressure with inlet stagnation pressure along the blades on the suction side and the pressure results of numerical hybrid (scalar + CUSP, $z=2.667$) method and comparing with experimental data for supersonic outlet flow in mid-section blade [12].

**Figure 27.**Distribution of static pressure with inlet stagnation pressure along the blades on suction side results of the scalar and hybrid (scalar + CUSP + $z=2.667$) numerical methods and comparing with experimental data for supersonic outlet flow in mid-section blade [12].

**Figure 28.**Comparing the distribution percent variation of stagnation pressure with inlet stagnation pressure along blade in central line stream (MP) results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in mid-section blade.

**Figure 29.**Distribution variation of wetness fraction along the suction side results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in mid-section blade.

**Figure 30.**Distribution variation of nucleation rate on suction side results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in mid-section blade.

**Figure 31.**Distribution variation of logarithmic radius droplets along the suction sides and pressure stream results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods and comparing with experimental data for supersonic outlet flow in mid-section blade [12].

**Figure 32.**Distribution variation percent of mass flow results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in mid-section blade.

**Figure 33.**Distribution variation of logarithmic residual of density results for the hybrid (Scalar + CUSP, $z=2.667$) method (

**a**) for supersonic outlet flow in mid-section blade; (

**b**) with 100 additional iterations after convergence for supersonic outlet flow in mid-section blade.

**Figure 35.**Distribution of static pressure with inlet stagnation pressure along the blades on the suction pressure side and central line stream results of the hybrid (Scalar + CUSP, $z=2.667$) numerical method and comparing with experimental data for supersonic outlet flow in tip-section blade [45].

**Figure 36.**Comparing distribution of static pressure with inlet stagnation pressure along the blades in central line (MP) results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods and comparing with experimental data for supersonic outlet flow in tip-section blade [45].

**Figure 37.**Comparing the distribution percent variation of stagnation pressure with inlet stagnation pressure along blade in central line stream (MP) results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in tip-section blade.

**Figure 38.**Results of the numerical hybrid (scalar + CUSP, $z=2.667$) method between blades for supersonic outlet flow in tip-section blade (

**a**) Contours of Mach; (

**b**) Contours of wetness fraction.

**Figure 39.**Distribution variation of wetness fraction along the suction side (SS) and central stream (MP) results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in tip-section blade.

**Figure 40.**Distribution variation of nucleation rate on suction side results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in tip-section blade.

**Figure 41.**Distribution variation of logarithmic radius drops along the suction sides and central stream results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods and comparing with experimental data in supersonic outlet flow on tip-section blade [45].

**Figure 42.**Distribution variation percent of mass flow results of the scalar and hybrid (scalar + CUSP, $z=2.667$) numerical methods for supersonic outlet flow in tip-section blade.

Geometry | Conditions | Mesh Size | Inlet Condition | $\mathbf{\u2206}{\mathit{T}}_{\mathit{S}\mathit{u}\mathit{p}\mathit{e}\mathit{r}\mathit{c}\mathit{o}\mathit{o}\mathit{l}\mathit{e}\mathit{d}}$ $\left(\mathbf{K}\right)$ | Pressure Ratio | |
---|---|---|---|---|---|---|

${\mathit{P}}_{0\mathit{i}\mathit{n}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$ | ${\mathit{T}}_{0\mathit{i}\mathit{n}}\left(\mathbf{K}\right)$ | $\frac{{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}}{{\mathit{P}}_{0\mathit{i}\mathit{n}}}$ | ||||

Young Nozzle Type (L) | 2D, Steady state, Supersonic outlet, Inviscid, Adiabatic Two-phase Flow | 30 × 220 | 320 | 544 | 6.7 | 0.43 |

Mid-Section Turbine Blade | 30 × 250 | 172 | 654 | 8 | 0.48 | |

Tip-Section Turbine Blade | 30 × 250 | 100.8 | 642 | 5.7 | 0.431 |

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## Share and Cite

**MDPI and ACS Style**

Yousefi Rad, E.; Mahpeykar, M.R.
A Novel Hybrid Approach for Numerical Modeling of the Nucleating Flow in Laval Nozzle and Transonic Steam Turbine Blades. *Energies* **2017**, *10*, 1285.
https://doi.org/10.3390/en10091285

**AMA Style**

Yousefi Rad E, Mahpeykar MR.
A Novel Hybrid Approach for Numerical Modeling of the Nucleating Flow in Laval Nozzle and Transonic Steam Turbine Blades. *Energies*. 2017; 10(9):1285.
https://doi.org/10.3390/en10091285

**Chicago/Turabian Style**

Yousefi Rad, Edris, and Mohammad Reza Mahpeykar.
2017. "A Novel Hybrid Approach for Numerical Modeling of the Nucleating Flow in Laval Nozzle and Transonic Steam Turbine Blades" *Energies* 10, no. 9: 1285.
https://doi.org/10.3390/en10091285