# Energy Analysis of an Industrial Nozzle with Variable Outlet Conditions during Compressible and Transient Airflow

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

^{®}toolbox [27]. The computational model used in the simulations is based on the geometry of an actual functioning nozzle in heavy industry. Furthermore, the physical model and the phenomena involved are derived from the working cycle of the pneumatic pulsator system. For this reason, the principle of operation is briefly outlined below. The system visualisation is shown in Figure 1 together with an example of the location on a silo.

**5**. The system working cycle begins with activating the quick-acting valve located in the head

**4**. Then, the air is guided through the nozzle

**1**to the silo

**3**. The momentum of the expanded air creates the pneumatic impact that acts on the loose material

**2**. The working cycle of the pneumatic pulsator system stops when the pressure between the pressure accumulator and silo (environment) becomes equal.

^{2}and at the outlet −77.5 cm

^{2}.

#### 2.1. Governing Equations

#### 2.1.1. Mass Balance

#### 2.1.2. Momentum Balance

#### 2.1.3. Energy Balance

#### 2.2. Turbulence Model

- the generation of turbulence kinetic energy due to the mean velocity gradients:$${P}_{k}=-\rho \overline{\overrightarrow{{u}^{\prime}}\overrightarrow{{u}^{\prime}}}\nabla \overrightarrow{u},$$
- the moduli of the mean rate-of-strain tensor ${S}_{k}$ and ${S}_{\epsilon}$:$$S\equiv \sqrt{2(\overrightarrow{S}:\overrightarrow{S})},$$
- the viscous destruction:$${Y}_{M}=2\nu \overline{{\left({\nabla}^{2}\overrightarrow{{u}^{\prime}}\right)}^{2}},$$

#### 2.3. Boundary Conditions

#### 2.4. Solver Validation & Error Estimation

## 3. Exergy Destruction

#### 3.1. Energy

#### 3.2. Exergy

## 4. Results & Discussion

#### 4.1. Flow Parameters Results

#### 4.2. Energy Analysis Results

^{−5}s, and it was the time step of the numerical data writing.

#### 4.3. Exergy at Different Ambient Conditions

## 5. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

FVM | Finite Volume Method |

FEM | Finite Element Method |

LES | Large Eddy Simulation |

RANS | Reynolds Averaged Navier-Stokes |

GCI | Grid Convergence Index |

CFL | Courant-Friedrichs-Levy convergence criterion |

Nomenclature | |

A | boundary area, $\left[{\mathrm{m}}^{2}\right]$ |

${A}_{s}$ | Sutherland coefficient, $\left[\mathrm{kg}\xb7{\mathrm{s}}^{-1}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1/2}\right]$ |

${c}_{p}$ | specific heat at constant pressure, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}\right]$ |

${c}_{v}$ | specific heat at constant volume, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}\right]$ |

E | energy, $\left[\mathrm{J}\right]$ |

$\dot{E}$ | energy rate, $\left[\mathrm{W}\right]$ |

e | specific internal energy, $\left[\frac{\mathrm{J}}{\mathrm{kg}}\right]$ |

$Ex$ | exergy, $\left[\mathrm{J}\right]$ |

$\dot{Ex}$ | exergy rate, $\left[\mathrm{W}\right]$ |

h | specific enthalpy, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\right]$ |

k | kinetic energy of turbulence, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\right]$ |

$Ma$ | Mach number, $\left[-\right]$ |

$\dot{m}$ | mass flux, $\left[\mathrm{kg}\xb7{\mathrm{s}}^{-1}\right]$ |

$\overrightarrow{n}$ | normal vector, $\left[-\right]$ |

p | pressure, $\left[\mathrm{Pa}\right]$ |

$\overrightarrow{q}$ | heat flux, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-2}\right]$ |

R | gas constant, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}\right]$ |

s | specific entropy, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}\right]$ |

T | temperature, $\left[\mathrm{K}\right]$ |

${T}_{s}$ | Sutherland temperature, $\left[\mathrm{K}\right]$ |

t | time, $\left[\mathrm{s}\right]$ |

$\overrightarrow{u}$ | velocity vector, $\left[\mathrm{m}\xb7{\mathrm{s}}^{-1}\right]$ |

V | control volume, $\left[{\mathrm{m}}^{3}\right]$ |

$\gamma $ | adiabatic index, $\left[-\right]$ |

$\epsilon $ | dissipation rate of kinetic energy of turbulence, $\left[\mathrm{W}\xb7{\mathrm{kg}}^{-1}\right]$ |

$\lambda $ | thermal conductivity, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1}\right]$ |

$\mu $ | dynamic viscosity, $\left[\mathrm{Pa}\xb7\mathrm{s}\right]$ |

$\nu $ | kinematic viscosity, $\left[{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}\right]$ |

$\rho $ | density, $\left[\mathrm{kg}\xb7{\mathrm{m}}^{-3}\right]$ |

$\psi $ | compressibility, $\left[{\mathrm{s}}^{2}\xb7{\mathrm{m}}^{-2}\right]$ |

## References

- Staruszkiewicz, A. Krótka historia pojȩcia energii. Foton
**2011**, 114, 4–8. (In Polish) [Google Scholar] - Wegener, F.D.A. A True Proteus: A History of Energy Conservation in German Science and Culture, 1847–1914. Ph.D. Thesis, Universiteit Utrecht, Utrecht, The Netherlands, 2009. [Google Scholar]
- Dincer, I.; Rosen, M.A. Exergy: Energy, Environment and Sustainable Development; Newnes: London, UK, 2012; p. 571. [Google Scholar]
- Szargut, J. Exergy Method: Technical and Ecological Applications; WIT Press: Southampton, UK, 2005; p. 189. [Google Scholar]
- Alekseev, G.N. Energy and Entropy; Mir Publishers: Moscow, Russia, 1986; p. 200. [Google Scholar]
- Yang, Y.; Zhu, X.; Yan, Y.; Ding, H.; Wen, C. Performance of supersonic steam ejectors considering the nonequilibrium condensation phenomenon for efficient energy utilisation. Appl. Energy
**2019**, 242, 157–167. [Google Scholar] [CrossRef] [Green Version] - Schmandt, B.; Herwig, H. Diffuser and Nozzle Design Optimization by Entropy Generation Minimization. Entropy
**2011**, 13, 1380–1402. [Google Scholar] [CrossRef] - Kirmaci, V.; Kaya, H.; Cebeci, I. An experimental and exergy analysis of a thermal performance of a counter flow Ranque–Hilsch vortex tube with different nozzle materials. Int. J. Refrig.
**2018**, 85, 240–254. [Google Scholar] [CrossRef] - Adhikari, R.C.; Wood, D.H. A new nozzle design methodology for high efficiency crossflow hydro turbines. Energy Sustain. Dev.
**2017**, 41, 139–148. [Google Scholar] [CrossRef] - Weilenmann, M.; Noiray, N. Experiments on sound reflection and production by choked nozzle flows subject to acoustic and entropy waves. J. Sound Vib.
**2021**, 492, 115799. [Google Scholar] [CrossRef] - Ishak, M.H.; Ismail, F.; Mat, S.C.; Abdullah, M.Z.; Abdul Aziz, M.S.; Idroas, M.Y. Numerical Analysis of Nozzle Flow and Spray Characteristics from Different Nozzles Using Diesel and Biofuel Blends. Energies
**2019**, 12, 281. [Google Scholar] [CrossRef] [Green Version] - Hemidi, A.; Henry, F.; Leclaire, S.; Seynhaeve, J.M.; Bartosiewicz, Y. CFD analysis of a supersonic air ejector. Part II: Relation between global operation and local flow features. Appl. Therm. Eng.
**2009**, 29, 2990–2998. [Google Scholar] [CrossRef] [Green Version] - Huet, M.; Emmanuelli, A.; Ducruix, S. Influence of viscosity on entropy noise generation through a nozzle. J. Sound Vib.
**2021**, 510, 116293. [Google Scholar] [CrossRef] - Quaatz, J.F.; Giglmaier, M.; Hickel, S.; Adams, N.A. Large-eddy simulation of a pseudo-shock system in a Laval nozzle. Int. J. Heat Fluid Flow
**2014**, 49, 108–115. [Google Scholar] [CrossRef] - Ota, M.; Udagawa, S.; Inage, T.; Maeno, K. Interferometric Measurement in Shock Tube Experiments. In Interferometry-Research and Applications in Science and Technology; Padron, I., Ed.; InTech: New York, NY, USA, 2012; Chapter 11; pp. 226–244. [Google Scholar]
- Inage, T.; Tsuchikura, S.; Ota, M.; Maeno, K. Three-dimensional laser interferometric CT (LICT) measurement of shock wave interaction around a circular cylinder. Flow Meas. Instrum.
**2013**, 31, 102–106. [Google Scholar] [CrossRef] - Honma, H.; Ishihara, M.; Yoshimura, T.; Maeno, K.; Morioka, T. Interferometric CT measurement of three-dimensional flow phenomena on shock waves and vortices. Shock Waves
**2003**, 13, 179–190. [Google Scholar] [CrossRef] - Bergersen, A.W.; Mortensen, M.; Valen-Sendstad, K. The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”. Int. J. Numer. Methods Biomed. Eng.
**2019**, 35, e3150. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Szargut, J. Egzergia: Poradnik Obliczania i Stosowania [in Polish]; Wydawnictwo Politechniki Śla̧skiej: Gliwice, Poland, 2007; p. 129. [Google Scholar]
- Bejan, A. Entropy Generation Through Heat and Fluid Flow; Wiley: Hoboken, NJ, USA, 1982; p. 248. [Google Scholar]
- Hassan, H.Z. Evaluation of the local exergy destruction in the intake and fan of a turbofan engine. Energy
**2013**, 63, 245–251. [Google Scholar] [CrossRef] - Kröner, D.; LeFloch, P.G.; Thanh, M.D. The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. ESAIM Math. Model. Numer. Anal.
**2008**, 42, 425–442. [Google Scholar] [CrossRef] - Rohrbach, Z.J.; Buresh, T.R.; Madsen, M.J. Modeling the exit velocity of a compressed air cannon. Am. J. Phys.
**2012**, 80, 24–26. [Google Scholar] [CrossRef] - Wołosz, K.J. Exergy destruction in the pneumatic pulsator system during one working cycle. Energy
**2018**, 146, 124–130. [Google Scholar] [CrossRef] - Wołosz, K.J.; Wernik, J. Heat generation calculation on the basis of numerical simulation results of supersonic airflow in a nozzle. Chem. Eng. Trans.
**2014**, 39, 1363–1368. [Google Scholar] [CrossRef] - Wołosz, K.J.; Wernik, J. On the heat in the nozzle of the industrial pneumatic pulsator. Acta Mech.
**2016**, 227, 1111–1122. [Google Scholar] [CrossRef] [Green Version] - Weller, H.G.; Tabor, G.; Jasak, H.; Fureby, C. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys.
**1998**, 12, 620. [Google Scholar] [CrossRef] - Petrila, T.; Trif, D. Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics; Springer: Boston, MA, USA, 2005. [Google Scholar]
- Pozrikidis, C. Fluid Dynamics: Theory, Computation, and Numerical Simulation; Springer: Boston, MA, USA, 2009. [Google Scholar]
- Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef] [Green Version] - Tu, J.; Yeoh, G.H.; Liu, C. Computational Fluid Dynamics. A Practical Approach; Elsevier: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Wołosz, K.J.; Wernik, J. Comparative Study on Heat Generation During Compressible Airflow Through Heatproof Nozzles. Chem. Eng. Trans.
**2018**, 70, 151–156. [Google Scholar] [CrossRef] - Utyuzhnikov, S. Robin-type wall functions and their numerical implementation. Appl. Numer. Math.
**2008**, 58, 1521–1533. [Google Scholar] [CrossRef] [Green Version] - Giglmaier, M.; Quaatz, J.F.; Gawehn, T.; Gülhan, A.; Adams, N.A. Numerical and experimental investigations of pseudo-shock systems in a planar nozzle: Impact of bypass mass flow due to narrow gaps. Shock Waves
**2014**, 24, 139–156. [Google Scholar] [CrossRef] - Courant, R.; Friedrichs, K.; Lewy, H. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann.
**1928**, 100, 32–74. [Google Scholar] [CrossRef] - Freitas, C.J. The issue of numerical uncertainty. Appl. Math. Model.
**2002**, 26, 237–248. [Google Scholar] [CrossRef] - Ferziger, J.H.; Perić, M. Computational Methods for Fluid Dynamics; Springer: Berlin, Germany, 2002. [Google Scholar]
- Szumowski, A.; Selerowicz, W.; Piechna, J. Dynamika Gazów; Wydawnictwa Politechniki Warszawskiej: Warszawa, Poland, 1988. (In Polish) [Google Scholar]
- Costa, V.A.F. On the exergy balance equation and the exergy destruction. Energy
**2016**, 116, 824–835. [Google Scholar] [CrossRef] - Bejan, A. Fundamentals of exergy analysis, entropy generation minimization, and the generation of flow architecture. Int. J. Energy Res.
**2002**, 26, 545–565. [Google Scholar] [CrossRef] - Szargut, J.; Ziębik, A. Podstawy Energetyki Cieplnej; PWN: Warszawa, Poland, 1998. (In Polish) [Google Scholar]

**Figure 1.**Visualization of the pneumatic pulsator system, and their location on silo walls:

**1**—nozzle (outlined investigated object);

**2**—loose material;

**3**—silo (cross-section);

**4**—head;

**5**—pressure accumulator. Blue arrows show possible direction of impact.

**Figure 3.**Time distribution of inlet static pressure for both investigated cases on the background of the total pressure. Reproduced from [32].

**Figure 5.**Results obtained by the author during current study for a qualitative comparison with experimental taken from [34].

**Figure 8.**Area-averaged velocity magnitude considering inlet, outlet and each outlet channel of the nozzle.

Inlet | Outlet | % | Inner Channel | % | Outer Channel | % | |
---|---|---|---|---|---|---|---|

Energy [J] | 10,399.0 | 10,322.1 | 99.26% | 3928.8 | 37.37% | 6393.3 | 61.48% |

Exergy [J] | - | 4395.0 | 42.26% | 1621.1 | 15.59% | 2773.9 | 26.67% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wołosz, K.J.
Energy Analysis of an Industrial Nozzle with Variable Outlet Conditions during Compressible and Transient Airflow. *Energies* **2022**, *15*, 841.
https://doi.org/10.3390/en15030841

**AMA Style**

Wołosz KJ.
Energy Analysis of an Industrial Nozzle with Variable Outlet Conditions during Compressible and Transient Airflow. *Energies*. 2022; 15(3):841.
https://doi.org/10.3390/en15030841

**Chicago/Turabian Style**

Wołosz, Krzysztof J.
2022. "Energy Analysis of an Industrial Nozzle with Variable Outlet Conditions during Compressible and Transient Airflow" *Energies* 15, no. 3: 841.
https://doi.org/10.3390/en15030841