# Entropy Assessment on Direct Contact Condensation of Subsonic Steam Jets in a Water Tank through Numerical Investigation

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## Abstract

**:**

## 1. Introduction

## 2. Geometry Model

## 3. Mathematical Model

#### 3.1. Mixture Model

#### 3.1.1. Continuity Equation

_{m}and

**v**

_{m}are the density and mass-averaged velocity for mixture, which are defined respectively as follows:

_{q}is the volume fraction of phase q.

#### 3.1.2. Momentum Equation

_{m}is the viscosity of mixture, defined as:

#### 3.1.3. Energy Equation

_{q}is the total energy of phase q; κ

_{eff}is the effective thermal conductivity, which can be denoted as:

_{q}is the thermal conductivity of phase q; κ

_{t}is the turbulent thermal conductivity, determined by the turbulence closure model; S

_{Eq}is the volumetric heat source for phase q.

#### 3.2. Turbulence Model

_{t,m}is the turbulent viscosity of mixture, defined as:

_{k,m}is the production of turbulent kinetic energy, and it is computed from:

#### 3.3. Phase Change Model

_{sat}(p

_{g}) is the saturated temperature corresponding to the pressure p

_{g}, and T

_{sat}(p

_{l}) is the saturated temperature corresponding to the pressure p

_{l}.

_{p}is the diameter of dispersed phase, and α

_{p}is volume fraction of dispersed phase. Therefore, the volumetric condensation rate and evaporation rate are given by:

#### 3.4 Entropy Generation Analysis Model

_{i}

^{s}is the entropy flux in i

^{th}direction, and ${\dot{S}}_{{}_{gen}}^{\prime \prime \prime}$ is the entropy generation rate.

#### 3.4.1. Convective Terms

#### 3.4.2. Entropy Generation by Dissipation

#### 3.4.3. Entropy Generation by Heat Transfer

#### 3.4.4. Entropy Generation by Inner Phase Change

#### 3.4.5. Time-Averaged Transport Equation for Entropy

- (1)
- The exact turbulent dissipation approximately equals to the production of density ρ
_{m}and the turbulent dissipation rate ε, therefore, the entropy generation rate due to turbulent dissipation reads as:$${\dot{S}}_{gen,D}^{\prime \prime \prime}=\frac{{\rho}_{m}\epsilon}{\overline{{T}_{m}}}$$ - (2)
- Use the Boussinesque-like approach [31], and then the entropy generation due to fluctuating temperature gradients is:$${\dot{S}}_{gen,C\prime}^{\prime \prime \prime}=\frac{{a}_{t}}{a}\frac{{\kappa}_{m}}{\overline{{T}_{m}^{2}}}{\left(\frac{\partial \overline{{T}_{m}}}{\partial {x}_{i}}\right)}^{2}$$
_{t}the turbulent thermal diffusivity. Assume that the turbulent thermal diffusivity is related as thermal diffusivity through:$$\frac{\kappa}{a}=\frac{{\kappa}_{t}}{{a}_{t}}={\rho}_{m}{c}_{pm}$$Then, the entropy generation due to mean temperature gradients and entropy generation due to fluctuating temperature gradients can be combined as the entropy generation due to heat transfer:$${\dot{S}}_{gen,C}^{\prime \prime \prime}=\frac{\left(\kappa +{\kappa}_{t}\right)}{\overline{{T}_{m}^{2}}}{\left(\frac{\partial \overline{{T}_{m}}}{\partial {x}_{i}}\right)}^{2}=\frac{{\kappa}_{eff}}{\overline{{T}_{m}^{2}}}{\left(\frac{\partial \overline{{T}_{m}}}{\partial {x}_{i}}\right)}^{2}$$From the derivation above, the local volumetric entropy generation is concluded as follows:$${\dot{S}}_{gen}^{\prime \prime \prime}=\underset{heat\text{\hspace{0.17em}}transfer}{\underbrace{\frac{{\kappa}_{eff}}{\overline{{T}_{m}^{2}}}{\left(\frac{\partial \overline{{T}_{m}}}{\partial {x}_{i}}\right)}^{2}}}+\underset{viscous\text{\hspace{0.17em}\hspace{0.17em}}dissipation}{\underbrace{\frac{{\mu}_{m}}{\overline{{T}_{m}}}\left\{2{\left(\frac{\partial \overline{{u}_{i,m}}}{\partial {x}_{i}}\right)}^{2}+{\left(\frac{\partial \overline{{u}_{i,m}}}{\partial {x}_{j}}+\frac{\partial \overline{{u}_{j,m}}}{\partial {x}_{i}}\right)}^{2}\right\}}}+\underset{\begin{array}{l}turbulent\text{\hspace{0.17em}}\\ dissipation\end{array}}{\underbrace{\frac{{\rho}_{m}\epsilon}{\overline{{T}_{m}}}}}+\underset{inner\text{\hspace{0.17em}}phase\text{\hspace{0.17em}}change}{\underbrace{\frac{L\cdot \overline{{J}_{net}}}{\overline{{T}_{m}}}}}$$

## 4. Computation Set-Up

#### 4.1. Simulation Details

^{−3}, while the residual for solution of energy should be below 1 × 10

^{−6}; (2) the entropy generation rate for all two phase domains hardly change with iterations in one time steps.

^{2}s for vapor is enforced at the inlet section, the temperature and pressure for vapor is 374.15 K and 3 kPa. The pressure outlet with value 0 Pa (gauge pressure) is employed for mixture at the outlet section. The rest boundaries are set to be adiabatic walls with no velocity slip.

Phases | Density (kg/m^{3}) | Specific Heat Capacity (J/kg K) | Viscosity (Pa · s) | Thermal Conductivity (W/m K) |
---|---|---|---|---|

Vapor | Incompressible ideal-gas | Polynomial * | 1.34 × 10^{−5} | 0.0261 |

Water | 998.2 | 4182 | 1.003 × 10^{−3} | 0.6 |

_{p}(T) = A

_{0}+ A

_{1}T+ A

_{2}T

^{2}+ A

_{3}T

^{3}+ A

_{4}T

^{4}, A

_{i}(i = 0,1,2,3,4) are the default coefficients in ANSYS FLUENT.

#### 4.2. Grid Independent Verification

^{+}is required to locate in the interval (11.225~200) for the adoption of standard wall functions, then the mesh near wall region is adapted to ensure 30 < Y

^{+}< 50. Totally, three computational grids consisted with 38,6972, 633,452, and 914,597 nodes are constructed. Before the simulation, the effect of grid density on the results is investigated. Figure 3a,b show the transverse distribution of longitudinal velocity at selected locations (z

_{1}= 265 mm, z

_{2}= 280 mm) with the three different grids above. All three grids can predict similarly the velocity profile but the medium size mesh, i.e., 633,452 nodes (≈ 80 × 80 × 100) is adopted considering the balance of accuracy and efficiency, as is shown in Figure 4.

**Figure 3.**Transverse distribution for longitudinal velocity at selected locations (z

_{1}= 265 mm, z

_{2}= 280 mm) for different grid density (grid 1: black, 386,972 nodes; grid 2: red, 633,452 nodes; grid 3: blue, 914,597 nodes. v

_{max}is the max velocity among the three grids).

## 5. Results and Discussion

#### 5.1. Verification & Validation

**Figure 5.**The comparisons of simulation steam shape and the experimental observations at different time; (

**a**) t = 0.0 s; (

**b**) t = 0.5 s.

^{+}is the dimensionless transverse location, and Z

^{+}is the dimensionless distance departure from the steam pipe exit (X

^{+}= x/d

_{0}, Z

^{+}= z/d

_{0}, d

_{0}is the inner diameter of steam pipe). Both the results show steep value in the vicinity of the jet, and smooth temperature change in the outward zone. Besides, the CFD prediction agrees quantitatively well with the numerical results derived by Takase et al. [35] (within 5%) in the central location, whilst it follows the same trend of the results by Takase et al. [35] with marginal deviation within 14% in the outer location.

**Figure 6.**The comparisons of the transverse temperature distribution at selected longitudinal locations between published data and present CFD predictions.

#### 5.2. Numerical Results

#### 5.2.1. Velocity Profile

_{z}for selected longitudinal positions (z = 0.26 m and z = 0.30 m) at different time are displayed in Figure 8, which indicate that V

_{z}of steam in the central location near pipe exit decreases rapidly after it reaches its maximum value. The observations correspond to the results obtained by Dahikar et al. [7] in their experiments and CFD simulation. Besides, it can be observed that the velocity in the centerline is not at the maximum all the time, and sometimes it increases in the jet zone deviated from the center line. Furthermore, the velocity changes sharply with the variation of transverse location near the region of pipe exit, while it changes smoothly in the outward zone, due to finite spreading width of jets in the pool water.

**Figure 7.**Velocity streamline in the x = 0 plane at different time. (

**a**) t = 4 ms; (

**b**) t = 8 ms; (

**c**) t = 12 ms; (

**d**) t = 16 ms; (

**e**) t = 44 ms; (

**f**) t = 120 ms.

**Figure 8.**Transverse profile of V

_{z}for selected longitudinal position at different time; (

**a**) z = 0.26 m; (

**b**) z = 0.30 m.

#### 5.2.2. Temperature Field

^{2}s) and low temperature (374.15 K) of steam.

**Figure 9.**Contours plot of temperature in the x = 0 plane at different times. (

**a**) t = 4 ms; (

**b**) t = 8 ms; (

**c**) t = 12 ms; (

**d**) t = 16 ms; (

**e**) t = 44 ms; (

**f**) t = 120 ms.

**Figure 10.**Transverse profile of temperature for selected longitudinal position at different time; (

**a**) z = 0.26 m; (

**b**) z = 0.30 m.

#### 5.2.3. Plume Shape

**Figure 11.**Contours plot of vapor void fraction in the x = 0 plane at different time. (

**a**) t = 4 ms; (

**b**) t = 8 ms; (

**c**) t = 12 ms; (

**d**) t = 16 ms; (

**e**) t = 44 ms; (

**f**) t = 120 ms.

#### 5.2.4. Mass Transfer

**Figure 12.**Instantaneous condensation rate in the x = 0 plane at different time. (

**a**) t = 4 ms; (

**b**) t = 8 ms; (

**c**) t = 12 ms; (

**d**) t = 16 ms; (

**e**) t = 44 ms; (

**f**) t = 120 ms.

#### 5.3. Entropy Generation

**Figure 13.**Total EGR per unit volume in the x = 0 plane at different time. (

**a**) t = 4 ms; (

**b**) t = 8 ms; (

**c**) t = 12 ms; (

**d**) t = 16 ms; (

**e**) t = 44 ms; (

**f**) t = 120 ms.

Time/(ms) | EGR_Heat Transfer/(W/K) | EGR_Viscous/(W/K) | EGR_Turbulence/(W/K) | EGR_Inner Phase Change /(W/K) | Total EGR /(W/K) |
---|---|---|---|---|---|

2 | 0.04728 | 0.00016 | 12750.01000 | 0.74853 | 12750.8080 |

4 | 0.52859 | 0.00031 | 1361.27000 | 2.90208 | 1364.7051 |

8 | 12.97281 | 0.00090 | 109.47290 | 5.96174 | 128.4084 |

12 | 27.41757 | 0.00099 | 28.85747 | 8.35380 | 64.6298 |

16 | 29.43516 | 0.00087 | 12.32487 | 9.37835 | 51.1393 |

20 | 31.36492 | 0.00103 | 7.30897 | 6.73753 | 45.4124 |

24 | 30.30458 | 0.00086 | 4.67426 | 9.81291 | 44.7926 |

28 | 30.89448 | 0.00086 | 3.51669 | 9.73798 | 44.1500 |

32 | 31.55728 | 0.00087 | 2.87975 | 9.67914 | 44.1170 |

36 | 32.04326 | 0.00087 | 2.49488 | 9.65655 | 44.1956 |

40 | 32.34219 | 0.00087 | 2.24267 | 9.65505 | 44.2471 |

44 | 32.49960 | 0.00087 | 2.06793 | 9.66349 | 44.2319 |

64 | 31.01836 | 0.00079 | 0.89096 | 9.67850 | 41.5886 |

140 | 32.90304 | 0.00065 | 0.53704 | 9.93461 | 43.3753 |

180 | 30.86176 | 0.00062 | 0.42636 | 9.48106 | 40.7698 |

**Figure 14.**The variation of contributions of four kinds of irreversibility to total entropy generation with time.

## 6. Conclusions

- (1)
- (2)
- Three distinct stages of DCC are discriminated clearly at the present conditions, i.e., initial stage, developing stage and oscillatory stage. In the initial stage, the plume shows no fixed shape. In the developing stage, the plume begins to act as an elliptical boundary, and the size of the plume grows quickly. In the oscillatory stage, the plume shape becomes ellipsoidal shape with disturbed structure.
- (3)
- The local volumetric EGR in the initial stage is much larger than those in other stages, but the region possessing considerable entropy generation rate is smaller than other stage. The decrease of EGR proves that the process conform to increasingly economical energy utilization.
- (4)
- The largest proportion in total EGR is occupied by turbulence fluctuation in the initial stage, and then it decreases apparently in the following time, meanwhile, the contributions of heat transfer irreversibility and inner phase change irreversibility to the local entropy generation increase, which makes DCC process become heat dominant in the developing and the oscillatory stage. The variation of EGR can be used to characterize the the dissipation and proceeding of DCC process.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | interfacial area per unit volume, m ^{2}/m^{3} |

C_{μ} | parameter in the turbulent model |

C_{1ε} | parameter in the turbulent model |

C_{2ε} | parameter in the turbulent model |

d_{p} | diameter of dispersed phase, m |

E | total energy, J |

F_{dr} | interaction force between phases, N/m ^{3} |

g | gravitational acceleration vector, m/s ^{2} |

j | evaporation-condensation flux, kg/m ^{2}·s |

J | volumetric phase change rate, kg/m ^{3}·s |

k | turbulent kinetic energy, m ^{2}/s^{2} |

L | latent heat, J/kg |

M | molar mass, kg/mol |

p | pressure, Pa |

T | temperature, K |

v | specific volume, m ^{3}/kg |

v | mean velocity, m/s |

## Greek Letters

α | volume fraction |

γ | factor characterizing intensity of evaporation and condensation, m ^{3}/s |

ε | turbulent energy dissipated per unit mass, m ^{2}/s^{3} |

κ_{eff} | effective thermal conductivity, W/m·K |

μ | viscosity, kg/m·s |

ρ | density, kg/m ^{3} |

## Subscripts and Superscripts

c | condensation |

e | evaporation |

g | vapor |

l | liquid |

m | mixture |

q | qth phase |

sat | saturated state |

T | transpose matrix |

+ | condensation process |

- | evaporation process |

## Abbreviations

CFD | computational fluid dynamics |

DCC | direct contact condensation |

EGR | entropy generation rate |

HTC | heat transfer coefficient |

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**MDPI and ACS Style**

Ji, Y.; Zhang, H.-C.; Tong, J.-F.; Wang, X.-W.; Wang, H.; Zhang, Y.-N.
Entropy Assessment on Direct Contact Condensation of Subsonic Steam Jets in a Water Tank through Numerical Investigation. *Entropy* **2016**, *18*, 21.
https://doi.org/10.3390/e18010021

**AMA Style**

Ji Y, Zhang H-C, Tong J-F, Wang X-W, Wang H, Zhang Y-N.
Entropy Assessment on Direct Contact Condensation of Subsonic Steam Jets in a Water Tank through Numerical Investigation. *Entropy*. 2016; 18(1):21.
https://doi.org/10.3390/e18010021

**Chicago/Turabian Style**

Ji, Yu, Hao-Chun Zhang, Jian-Fei Tong, Xu-Wei Wang, Han Wang, and Yi-Ning Zhang.
2016. "Entropy Assessment on Direct Contact Condensation of Subsonic Steam Jets in a Water Tank through Numerical Investigation" *Entropy* 18, no. 1: 21.
https://doi.org/10.3390/e18010021