# Heat Transfer Models for Dense Pulverized Particle Jets

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Heat Transfer Modeling in Lagrangian Parcels

## 3. Synthetic Particle Cluster

^{®}v7 [46] for the simulations of the heat-up of the particle clusters. chtMultiRegionFoam couples fluid and solid regions explicitly in space and time. This approach is referred to as loose coupling in the literature [47] and requires suitable time stepping and mesh resolution [48]. The boundary conditions of the fluid are representative for the pulverized particle jet entering the raceway zone of blast furnaces [21,49]. Temperatures were kept constant at 2500 K, outlet pressure was set to 5 bar

_{(a)}, while the inlet velocity was varied according to Table 1. The initial particle clusters temperature is 400 K.

^{−6}and 7.5 × 10

^{−4}s for the investigated cases. The overall mean cell size of the fluid grid is between 20 and 30 $\mathsf{\mu}$m, with significant refinement near the particles. The time step is of order 10

^{−7}to 10

^{−8}seconds for the 0.5 and 13 m/s relative velocities. Therefore, the chosen mesh resolution and time step sizes suit DNS. The grid independence of the cluster simulations is discussed in Appendix A.1. Thermal radiation is modeled by the finite volume discrete ordinates model (fvDOM) [41] with 16 discrete rays (two azimuthal angles in $\pi $/2 on X-Y plane and two polar angles in $\pi $ between the Z and X-Y plane). Gas-phase radiation contribution is approximated by grey mean absorption [50]. According to Narayanaswamy and Chen [51], Song et al. [52], near-field effects between the solid particles can be disregarded for the investigated cases. The OpenFOAM

^{®}fvDOM implementation has been validated in the literature [53,54,55], while the flow and convective heat transfer is validated using single sphere simulations, which are given in Appendix A.1.

## 4. Single Sphere Reference Cases

- size equivalent case (SE)
- mass equivalent case (ME),
- surface area equivalent case (AE), and
- surface area and mass equivalent case (AME).

^{3}to ensure mass equivalence.

## 5. Cluster Heat Transfer Models

#### 5.1. Projected Surface Area Correction

#### 5.2. Convective Heat Transfer Model

#### 5.2.1. Multiple Linear Regression Approach (MLR)

#### 5.2.2. Lévêque Approach (GLE)

#### Drag Coefficient

#### Nusselt Number

#### 5.3. Comparison with Random Cluster Simulations

#### 5.4. Comparison of Nusselt Correlations from Literature

## 6. Euler–Lagrangian Validation Simulations

#### 6.1. Convective Heat Transfer Model Validation

#### 6.2. Projected Surface Area Correction Validation

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Symbols

## Latin Symbols

symbol | unit | description |

A | [m^{2}] | area |

c_{p} | [Jkg^{−1}K^{−1}] | specific heat capacity |

d | [m] | diameter |

d_{32} | [m] | Sauter diameter |

e | [Jkg^{−1}] | specific internal energy |

F_{f} | [N] | drag force |

f (φ) | [-] | radiation correction function |

G | [Wm^{−2}] | incident radiation |

h_{conv} | [Wm^{−2}K^{−1}] | convective heat transfer coefficient |

I | [Wm^{−2}] | radiative intensity |

k | [-] | shape parameter of the Weibull distribution |

L | [m] | length scale |

m | [kg] | mass |

N_{p} | [-] | number of particles |

Nu | [-] | Nusselt number |

Pr | [-] | Prandtl number |

Q | [W] | integral heat flux |

R | [-] | coefficient of determination |

Re_{P} | [-] | particle Reynolds number |

t | [s] | time |

U | [ms^{−1}] | velocity |

x | [-] | distribution parameter |

x_{f} | [-] | frictional drag force share |

## Greek Symbols

symbol | unit | description |

ε | [-] | emissivity |

κ | [Wm^{−1}K^{−1}] | thermal conductivity |

λ | [-] | scale parameter of the Weibull distribution |

µ | [Pas] | dynamic viscosity |

ν | [m^{2}s^{−1}] | kinematic viscosity |

φ | [-] | void fraction |

ρ | [kgm^{−3}] | density |

σ | [Wm^{−2}K^{−4}] | Stefan-Boltzmann constant |

ξ | [-] | drag coefficient |

ξ_{f} | [-] | drag coefficient due to fluid friction |

## Subscripts

symbol | description |

C | cluster |

conv | convective |

GLE | Generalized Lévêque equation |

h | hydraulic |

∞ | bulk fluid |

MLR | multiple linear regression |

P | particle |

R | projected/illuminated |

rad | radiative |

rel | relative |

S | surface |

## Appendix A. Grid Independence Study

#### Appendix A.1. Synthetic Cluster

Scaling Factors | ${\mathit{Re}}_{\mathit{P}}$ = 2.058 | |||||
---|---|---|---|---|---|---|

id | a | b | c | ${\mathit{\xi}}_{\mathit{sim}}$ (-) | Error (% of 4) | Error (% of 6) |

1 | 2.66 | 1.25 | 1.83 | 1.649 | 10.49 | 24.48 |

2 | 2.66 | 2.5 | 1.83 | 1.503 | 0.66 | 13.41 |

3 | 2.66 | 3.75 | 1.83 | 1.492 | 0.03 | 12.63 |

4 | 2.66 | 5 | 1.83 | 1.492 | - | 12.66 |

5 | 7 | 3.75 | 5.33 | 1.358 | - | 2.47 |

6 | 12 | 3.75 | 10 | 1.325 | - | - |

Cell Size | ${\mathit{\xi}}_{\mathit{sim}}$ (-) | Deviation (%) | ${\mathit{Nu}}_{\mathit{P},\mathit{sim}}$ (-) | Deviation (%) |
---|---|---|---|---|

−50% | 2.2341 | −0.62 | 0.3283 | −0.43 |

−20% | 2.2257 | −0.25 | 0.3288 | −0.58 |

base | 2.2202 | - | 0.3269 | - |

+20% | 2.2122 | 0.36 | 0.3250 | 0.58 |

+50% | 2.2059 | 0.65 | 0.3242 | 0.83 |

#### Appendix A.2. Representative Particle

Scaling Factors | ${\mathit{Re}}_{\mathit{P}}$ = 2.058 | ${\mathit{Re}}_{\mathit{P}}$ = 51.469 | |||||||
---|---|---|---|---|---|---|---|---|---|

id | a | b | c | ${\mathit{\xi}}_{\mathit{sim}}$ (-) | Error (% of 5) | ${\mathit{\xi}}_{\mathit{th}}$ (-) | ${\mathit{\xi}}_{\mathit{sim}}$ (-) | Error (% of 5) | ${\mathit{\xi}}_{\mathit{th}}$ (-) |

1 | 15 | 4 | 11 | 15.353 | 6.65 | 15.153 | 1.579 | 2.38 | 1.569 |

2 | 19 | 6 | 13 | 14.765 | 2.57 | 1.556 | 0.88 | ||

3 | 23 | 8 | 15 | 14.586 | 1.33 | 1.549 | 0.42 | ||

4 | 27 | 10 | 17 | 14.474 | 0.55 | 1.546 | 0.21 | ||

5 | 37 | 15 | 22 | 14.394 | - | 1.542 | - |

Cell Size | ${\mathit{\xi}}_{\mathit{sim}}$ (-) | Deviation (%) | ${\mathit{\xi}}_{\mathit{th}}$ (-) | Error (%) | ${\mathit{Nu}}_{\mathit{P},\mathit{sim}}$ (-) | Deviation (%) | ${\mathit{Nu}}_{\mathit{P},\mathit{th}}$ (-) | Error (%) | |
---|---|---|---|---|---|---|---|---|---|

−50% | 1.530 | 1.21 | 1.567 | 2.51 | 5.687 | −1.1 | 5.279 | −7.72 | |

−20% | 1.545 | 0.24 | 1.567 | 1.55 | 5.639 | −0.25 | 5.279 | −6.82 | |

base | 1.548 | - | 1.567 | 1.31 | 5.625 | - | 5.279 | −6.55 | |

+20% | 1.550 | −0.11 | 1.567 | 1.21 | 5.618 | 0.12 | 5.279 | −6.45 | |

+100% | 1.553 | −0.28 | 1.567 | 1.04 | 5.612 | 0.23 | 5.279 | −6.30 |

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**Figure 1.**Synthetic rhomboid-shaped particle cluster with a void fraction ($\varphi $) of 0.741 (120 $\mathsf{\mu}\mathrm{m}$ distance between particle centers).

**Figure 2.**Horizontal cross-section of the synthetic particle cluster. 2 × x is the distance between the particle centers, which is varied to obtain clusters with different void fraction. Cuts through all Cartesian axis planes look identical. The scaling factors of the simulation domain are a = 12, b = 3.75, c = 10.

**Figure 3.**Comparison of mean cluster temperature (solid) and range of individual particle temperatures (shaded area) during the heat-up at 0.5 m/s relative velocity ($R{e}_{P}$ = 0.29), as defined in Table 1. red ( ): convective and radiative region coupling; blue ( ): convective region coupling.

**Figure 4.**Comparison of mean cluster temperature (solid) and range of individual particle temperatures (shaded area) during the heat-up at 13 m/s relative velocity ($R{e}_{P}$ = 7.50), as defined in Table 1. red ( ): convective and radiative region coupling; blue ( ): convective region coupling.

**Figure 5.**Horizontal cross-section of the size equivalent sphere cases. Cuts through all Cartesian axis planes look identical. The scaling factors of the simulation domain are a = 23, b = 8, c = 14.5.

**Figure 6.**Comparison of cluster and equivalent sphere temperature profiles (

**a**) 0.5 m/s (

**b**) 13 m/s relative velocity. Hatched areas represent cluster temperature profiles. SA, ME, AE, and AME are the size, mass, surface area, and surface area and mass equivalent case, respectively.

**Figure 7.**Comparison of specific radiative (

**a**) and convective (

**b**) integral heat flux towards the particle cluster versus the cluster void fraction for different particle Reynolds numbers ($R{e}_{P}$).

**Figure 8.**Comparison of specific radiative (

**a**) and convective (

**b**) integral heat flux towards the particle cluster versus the particle Reynolds number for different void fractions ($\varphi $).

**Figure 11.**Comparison of ${d}_{h}/L$ correlation given by Equation (19) ( ) with simulation data (●) and ${d}_{h}=2/3\varphi /{(1-\varphi )}^{2/3}$ [61] ( ) versus the cluster void fraction (

**a**) and particle Reynolds number (

**b**). The relative errors of Equation (19) compared to CFD-simulated data is presented in (

**c**).

**Figure 12.**Comparison of drag coefficient correlation given by Equation (20) ( ) with simulation data (●) and the drag coefficient according to Brauer [62] ( ) versus the cluster void fraction (

**a**) and particle Reynolds number (

**b**). The relative errors of Equation (20) compared to CFD-simulated data is presented in (

**c**).

**Figure 13.**Comparison of Nusselt number calculated using Equations (17) and (20), ${x}_{f}$ = 2/3 and Equation (19) ( ) with values determined from simulations (●) and Equation (5) ( ) versus the cluster void fraction (

**a**) and particle Reynolds number (

**b**). The relative errors of Equation (17) compared to CFD-simulated data is presented in (

**c**).

**Figure 14.**Comparison of (

**a**) the projected surface area correction calculated using Equation (13) ( ) (

**b**) the Nusselt number calculated using Equation (16) ( ) and Equation (17) ( ), (

**c**) the drag coefficient according to Equation (20) ( ), and (

**d**) ${d}_{h}/L$ calculated using Equation (19) ( ) with values determined from simulations (●) versus the cluster particle Reynolds number.

**Figure 15.**Comparison of various Nusselt number correlations from the literature (Achenbach [33] ( ), Gnielinski [32] ( ), Gunn [65] ( ), Sun et al. [24] ( ), Municchi and Radl [25] ( )) with the multiple linear regression (MLE) of Equation 16 ( ) and the Lévêque (GLE) of Equations (17)–(19) ( ) correlations versus void fraction (

**a**) and particle Reynolds number (

**b**). Red: (

**a**) lower Re

_{P}, (

**b**) lower void fraction; Blue: (

**a**) higher Re

_{P}, (

**b**) higher void fraction.

**Figure 16.**Comparison of resolved cluster mean temperatures ( ) and Lagrangian parcel temperatures using the multiple linear regression (MLR) ( ) and the Lévêque (GLE) ( ) Nusselt correlation for relative velocities of 0.5 m/s (

**a**) and 13 m/s (

**b**). Size equivalent ( ) and surface and mass equivalent ( ) reference cases use the Whitaker heat transfer model.

**Figure 17.**Comparison of resolved cluster mean temperatures ( ) and Lagrangian parcel temperatures using the radiative heat transfer correction and the Nusselt correlation ( ) and the Lévêque approach ( ) for relative velocities of 0.5 m/s (

**a**) and 13 m/s (

**b**). Size equivalent ( ) and surface and mass equivalent ( ) reference cases use the Whitaker heat transfer model.

Case | ${\mathit{U}}_{\mathbf{rel}}$ | ${\mathbf{Re}}_{\mathit{P}}$ | $\mathit{\varphi}$ | Distance 2 × x |
---|---|---|---|---|

id | (m/s) | (-) | (-) | ($\mathsf{\mu}\mathbf{m}$) |

T1 | 0.5 | 0.29 | 0.552 | 100 |

T2 | 13 | 7.50 | 0.552 | 100 |

T3 | 0.5 | 0.29 | 0.741 | 120 |

T4 | 13 | 7.50 | 0.741 | 120 |

T5 | 0.5 | 0.29 | 0.935 | 190 |

T6 | 13 | 7.50 | 0.935 | 190 |

Patch | U | p | T | I |
---|---|---|---|---|

inlet | fixed value | zero gradient | fixed value | gray body |

outlet | zero gradient | fixed value | zero gradient | gray body |

wall | slip | zero gradient | zero gradient | gray body |

gas-solid interface | no slip | zero gradient | coupled | gray body |

**Table 3.**Gas-phase composition and thermophysical properties and solid thermophysical properties used in the simulations.

Gas-Phase Composition (kg/kg) | |
---|---|

H_{2}O | 0.01 |

N_{2} | 0.72 |

O_{2} | 0.27 |

Bulk Gas-Phase Thermophysical Properties | |

density ($\rho $) | 0.69 kg m^{−3} |

specific heat capacity (${c}_{p}$) | 1300
J kg^{−1}K^{−1} |

thermal conductivity ($\kappa $) | 0.133 W m^{−1}K^{−1} |

dynamic viscosity ($\mu $) | 7.213× 10^{−5} Pa s |

Solid Thermophysical Properties | |

density ($\rho $) | 1100 kg m^{−3} |

emissivity ($\epsilon $) | 1 |

specific heat capacity (${c}_{p}$) | 1660 J kg^{−1}K^{−1} |

thermal conductivity ($\kappa $) | 1.241 W m^{−1}K^{−1} |

Case | ${\mathit{U}}_{\mathbf{rel}}$ | ${\mathit{d}}_{\mathit{P}}$ | ${\mathbf{Re}}_{\mathit{P}}$ | m |
---|---|---|---|---|

id | (m/s) | ($\mathsf{\mu}\mathbf{m}$) | (-) | (kg) |

SE1 | 0.5 | 60 | 0.29 | 1.24 × 10^{−10} |

SE2 | 13 | 60 | 7.50 | 1.24 × 10^{−10} |

ME1 | 0.5 | 211.8 | 1.02 | 5.47 × 10^{−9} |

ME2 | 13 | 211.8 | 26.46 | 5.47 × 10^{−9} |

AE1 | 0.5 | 397.8 | 1.91 | 3.63 × 10^{−8} |

AE2 | 13 | 397.8 | 49.70 | 3.63 × 10^{−8} |

AME1 | 0.5 | 397.8 | 1.91 | 5.47 × 10^{−9} |

AME2 | 13 | 397.8 | 49.70 | 5.47 × 10^{−9} |

Case | ${\mathit{U}}_{\mathbf{rel}}$ | ${\mathbf{Re}}_{\mathit{P}}$ | $\mathit{\varphi}$ | Solid Loading | Distance 2 × x |
---|---|---|---|---|---|

id | (m/s) | (-) | (-) | Ratio (-) | ($\mathsf{\mu}\mathbf{m}$) |

ST1–ST5 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.477 | 1737 | 95 |

13, 25 | 7.5, 14.42 | ||||

ST6–ST10 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.552 | 1288 | 100 |

13, 25 | 7.5, 14.42 | ||||

ST11–ST15 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.663 | 805 | 110 |

13, 25 | 7.5, 14.42 | ||||

ST16–ST20 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.741 | 556 | 120 |

13, 25 | 7.5, 14.42 | ||||

ST21–ST25 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.796 | 407 | 130 |

13, 25 | 7.5, 14.42 | ||||

ST26–ST30 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.867 | 243 | 150 |

13, 25 | 7.5, 14.42 | ||||

ST31–ST35 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.909 | 159 | 170 |

13, 25 | 7.5, 14.42 | ||||

ST36–ST40 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.935 | 111 | 190 |

13, 25 | 7.5, 14.42 | ||||

ST41–ST45 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.969 | 51 | 243 |

13, 25 | 7.5, 14.42 | ||||

ST46–ST50 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.994 | 10 | 415 |

13, 25 | 7.5, 14.42 | ||||

ST51–ST55 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.997 | 5 | 523 |

13, 25 | 7.5, 14.42 | ||||

ST56–ST60 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 0.999 | 1 | 892 |

13, 25 | 7.5, 14.42 | ||||

ST61–ST65 | 0.5, 1, 5, | 0.29, 0.58, 2.88, | 1 | - | - |

13, 25 | 7.5, 14.42 |

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**MDPI and ACS Style**

Bösenhofer, M.; Pichler, M.; Harasek, M. Heat Transfer Models for Dense Pulverized Particle Jets. *Processes* **2022**, *10*, 238.
https://doi.org/10.3390/pr10020238

**AMA Style**

Bösenhofer M, Pichler M, Harasek M. Heat Transfer Models for Dense Pulverized Particle Jets. *Processes*. 2022; 10(2):238.
https://doi.org/10.3390/pr10020238

**Chicago/Turabian Style**

Bösenhofer, Markus, Mario Pichler, and Michael Harasek. 2022. "Heat Transfer Models for Dense Pulverized Particle Jets" *Processes* 10, no. 2: 238.
https://doi.org/10.3390/pr10020238