# Hydrodynamic and Heat Transfer Study of a Fluidized Bed by Discrete Particle Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Model Verification

_{s}is the specific interfacial area, given by a

_{s}= 6(1 − ε

_{g})/d

_{p}. The details of the analytical solution of Equations (12) and (13) can be found in Bird et al. [18].

## 4. Simulation Settings

_{mf}= 0.262 m/s). The particles were initialized at a temperature of 345 K and stacked on a cubic lattice at the bottom of the bed, after which a constant gas stream with T

_{g,0}= 330 K was supplied through the gas supply at the bottom of the bed. Each particle had a constant heat source of 0.0209 W/(K·m), which is the same as that used by Li et al. [13]. All simulations were performed for 10 s.

## 5. Results

#### 5.1. Effect of Superficial Velocity and Aspect Ratio on Bed Hydrodynamic Behavior

_{0}/W.

_{0}. When the superficial gas velocity increases, the differences of averaged voidage between different aspect ratios increase, which is in line with the observations from Figure 4.

_{g,0}and H

_{0}/W on the solids motion, the temporally and spatially averaged particle velocity was obtained from the last 8 s of simulation data. The time-averaged vertical particle velocities at z = H

_{0}, as a function of superficial gas velocity and aspect ratio, are depicted in Figure 6. It can be seen that particle velocity is affected by superficial gas velocity as follows: both the up-flow in the core and the down-flow in the annulus are enhanced. This can be attributed to enhanced bubble action. The bed hydrodynamics is also influenced by the aspect ratio, as indicated by the altered shape of the time-averaged vertical velocity. When the initial bed height increases, the up-flow in the core region becomes larger. This reveals that a higher aspect ratio promotes intensified upward solids motion.

#### 5.2. Effect of Superficial Velocity and Aspect Ratio on Heat Transfer Behavior

_{0}/W, the bed mass increases, and hence, the associated overall heat source in the bed also increases.

_{0}, the bed is better-mixed and hence becomes more isothermal. Moreover, at higher aspect ratios, the relative influence of entrance effects is reduced and particles become better-mixed, which further reduces the standard deviation of the particle temperature.

_{g,0}and H

_{0}/W on the thermal behavior, the probability density function (PDF) of the temperature was calculated. Prior to this, all particle temperatures were non-dimensionalized by subtracting the inlet gas temperature (T

_{p}-T

_{g,0}) and dividing by the driving force, i.e., the particle melting point (T

_{p,m}= 380 K) minus the inlet gas temperature (T

_{p,m}-T

_{g,0}). Dimensionless temperatures of 0 and 1 correspond to a particle temperature equal to the inlet gas temperature and the melting point of the solid material, respectively. Subsequently, the dimensionless temperatures were used as an input to create a probability density function (PDF). This was done by dividing the dimensionless temperature range between 0 and 1 into 25 equal-sized bins. During a simulation period of 0.25–2 s, the number of particles in each of the temperature bins was counted (Li et al., [13]). When u

_{g,0}was increased, the particle temperature PDF changed considerably (see Figure 9). The mean value of the temperature dropped as u

_{g,0}was increased, implying that more heat is removed as more gas is supplied. When the aspect ratio increased, the mean value of dimensionless particle temperature increased, and the dimensionless temperature exceeded 1 at H

_{0}/W = 2.0. This can be explained by the fact that at higher initial bed height, there are more particles and, consequently, a stronger heat source in the bed, which will further increase the equilibrium temperature and dimensionless particle temperature.

#### 5.3. Effect of the Heat Source Term on Heat Transfer Behavior

_{0}/W, inlet effects are dominant, leading to long tails towards low temperatures for both particle classes. From the comparison between simulation results and equilibrium temperatures, we observe that there still are differences, and the differences become larger at lower superficial velocity. This is because the assumption of a thermally ideally mixed bed is not valid. This also illustrates the added value of fluidized bed CFD-DEM simulations for the inspection of hot and cold zones in the bed.

## 6. Conclusions

_{g,0}, the particles become better-mixed, reducing the standard deviation. Furthermore, the time-averaged particle velocity profiles strongly depend on u

_{g,0}and H

_{0}/W. Enhanced particle circulation with increasing superficial velocity is observed, whereas more particles are transported upwards in the central region for increasing initial bed heights. The fluidized bed is more isothermal when the superficial gas velocity and aspect ratio increase. The diversity of particles with and without heat production also has an apparent effect on heat transfer behavior. Because of the very good heat transfer characteristics, the heat transfer from very active particles via the gas phase to the non-active particles is very effective. The gas and particle temperatures predicted by CFDEM simulations agree reasonably well with the equilibrium temperatures.

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

C_{p} | Heat capacity, J·kg^{−1}·K^{−1} |

d_{p} | Particle diameter, m |

g | Gravitational acceleration constant, m·s^{−2} |

h | Effective interfacial heat transfer coefficient, W·m^{−2}·K^{−1} |

k | Fluid thermal conductivity, W·m^{−1}·K^{−1} |

N_{p} | Particle number, - |

Nu_{p} | Particle Nusselt number, - |

Pr | Prandtl number, - |

Q | Source term for the interphase heat transfer exchange, W·m^{−3} |

q | Heat production, W |

Re_{p} | Particle Reynolds number, - |

t | Time, s |

T | Temperature, K |

Greek symbols | |

Δξ | Width of grid cell, m |

Δψ | Depth of grid cell, m |

Δζ | Height of grid cell, m |

ε | Volume fraction, - |

ρ | Density, kg·m^{−3} |

µ | Dynamic viscosity, kg·m^{−1}·s^{−1} |

σ | Standard deviation of particle temperature, K |

Subscripts | |

p | Particle phase |

g | Gas phase |

## References

- Kunii, D.; Levenspiel, O. Fluidization engineering. In Butterworth-Heinemann Series in Chemical Engineering; Butterworth-Heinemann Limited: Oxford, UK, 1991. [Google Scholar]
- Basu, P.; Nag, P. An investigation into heat transfer in circulating fluidized beds. Int. J. Heat Mass Transf.
**1987**, 30, 2399–2409. [Google Scholar] [CrossRef] - Zhou, Z.; Yu, A.; Zulli, P. Particle scale study of heat transfer in packed and bubbling fluidized beds. AIChE J.
**2009**, 55, 868–884. [Google Scholar] [CrossRef] - Zhou, Z.; Yu, A.; Zulli, P. A new computational method for studying heat transfer in fluid bed reactors. Powder Technol.
**2010**, 197, 102–110. [Google Scholar] [CrossRef] - Borodulya, V.A.; Ganzha, V.L.; Teplitskii, Y.S.; Epanov, Y.G. Heat transfer in fluidized beds. J. Eng. Phys.
**1985**, 49, 1197–1202. [Google Scholar] [CrossRef] - Valenzuela, J.; Glicksman, L. An experimental study of solids mixing in a freely bubbling two-dimensional fluidized bed. Powder Technol.
**1984**, 38, 63–72. [Google Scholar] [CrossRef] - Limtrakul, S.; Boonsrirat, A.; Vatanatham, T. DEM modeling and simulation of a catalytic gas–solid fluidized bed reactor: A spouted bed as a case study. Chem. Eng. Sci.
**2004**, 59, 5225–5231. [Google Scholar] [CrossRef] - Zhou, H.; Flamant, G.; Gauthier, D. DEM-LES of coal combustion in a bubbling fluidized bed. Part i: Gas-particle turbulent flow structure. Chem. Eng. Sci.
**2004**, 59, 4193–4203. [Google Scholar] [CrossRef] - Patil, A.V.; Peters, E.; Kolkman, T.; Kuipers, J. Modeling bubble heat transfer in gas–solid fluidized beds using DEM. Chem. Eng. Sci.
**2014**, 105, 121–131. [Google Scholar] [CrossRef] - Deen, N.G.; Annaland, M.V.S.; Van Der Hoef, M.; Kuipers, J. Review of discrete particle modeling of fluidized beds. Chem. Eng. Sci.
**2007**, 62, 28–44. [Google Scholar] [CrossRef] - Van Der Hoef, M.A.; Annaland, M.V.S.; Deen, N.G.; Kuipers, J. Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy. Annu. Rev. Fluid Mech.
**2008**, 40, 47–70. [Google Scholar] [CrossRef] - Hoomans, B.; Kuipers, J.; Briels, W.; Van Swaaij, W. Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: A hard-sphere approach. Chem. Eng. Sci.
**1996**, 51, 99–118. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Annaland, M.V.S.; Kuipers, J.; Deen, N.G. Effect of superficial gas velocity on the particle temperature distribution in a fluidized bed with heat production. Chem. Eng. Sci.
**2016**, 140, 279–290. [Google Scholar] [CrossRef] - Beetstra, R.; Van Der Hoef, M.; Kuipers, J. Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations. Chem. Eng. Sci.
**2007**, 62, 246–255. [Google Scholar] [CrossRef] - Syamlal, M.; Gidaspow, D. Hydrodynamics of fluidization: Prediction of wall to bed heat transfer coefficients. AIChE J.
**1985**, 31, 127–135. [Google Scholar] [CrossRef] - Gunn, D.J. Transfer of heat or mass to particles in fixed and fluidized beds. Int. J. Heat Mass Transf.
**1978**, 21, 467–476. [Google Scholar] [CrossRef] - Annaland, M.V.S.; Deen, N.G.; Kuipers, J. Numerical simulation of gas–liquid–solid flows using a combined front tracking and discrete particle method. Chem. Eng. Sci.
**2005**, 60, 6188–6198. [Google Scholar] [CrossRef] - Bird, R.; Stewart, W.; Lightfoot, E. Transport Phenomena; John Wiley and Sons: Hoboken, NJ, USA, 2001. [Google Scholar]

**Figure 4.**Snapshots of instantaneous voidage patterns for varying aspect ratios: (

**a**) H

_{0}/W = 0.5, (

**b**) H

_{0}/W = 1, (

**c**) H

_{0}/W = 2, and different superficial gas velocities: 0.4, 0.5 and 0.6 m/s (from left to right).

**Figure 5.**Temporally and spatially averaged bed voidage as a function of superficial velocity and aspect ratio.

**Figure 6.**Profiles at different heights of time-averaged vertical particle velocity for varying aspect ratios: (

**a**) H

_{0}/W = 0.5, (

**b**) H

_{0}/W = 1, (

**c**) H

_{0}/W = 2. Note that all profiles are shown at z = H

_{0}.

**Figure 7.**Snapshots of particle temperature as a function of aspect ratio: (

**a**) H

_{0}/W = 0.5, (

**b**) H

_{0}/W = 1, (

**c**) H

_{0}/W = 2, and different superficial gas velocities: 0.4, 0.5 and 0.6 m/s (from left to right).

**Figure 9.**Dimensionless particle temperature probability density function (PDF) for varying aspect ratios: 0.5, 1 and 2 (top to bottom) and different gas superficial velocities.

**Figure 10.**Snapshots of gas temperature as a function of aspect ratio: (

**a**) H

_{0}/W = 0.5, (

**b**) H

_{0}/W = 1, (

**c**) H

_{0}/W = 2, and different superficial gas velocities: 0.4, 0.5 and 0.6 m/s (from left to right).

**Figure 11.**Snapshots of the temperature of two types of particles as a function of aspect ratio: (

**a**) H

_{0}/W = 0.5, (

**b**) H

_{0}/W = 1, (

**c**) H

_{0}/W = 2, and different superficial gas velocities: 0.4, 0.5 and 0.6 m/s (from left to right).

**Figure 12.**Dimensionless temperature PDF of all particles, inactive particles and active particles for different gas superficial velocities and varying aspect ratios: 0.5, 1 and 2.

Property (Unit), Symbol | Value |
---|---|

Gas density (kg/m^{3}), ρ_{g} | 1000 |

Inlet gas temperature (K), T_{g,0} | 373 |

Gas viscosity (Pa s), µ_{g} | 1.0 × 10^{−3} |

Gas heat capacity (J/kg/K), C_{pg} | 4187 |

Gas thermal conductivity (W/m/K), k_{g} | 0.5 |

Particle diameter (m), d_{p} | 3.95 × 10^{−3} |

Particle density (kg/m^{3}), ρ_{p} | 8400 |

Initial temperature of gas and particles (K) | 273 |

Particle heat capacity (J/kg/K), C_{pp} | 385 |

Setting | Value |
---|---|

Width of the bed (m) | 0.1 |

Depth of the bed (m) | 0.1 |

Initial particle bed height (m) | 0.5 |

Particle number | 78,125 (25 × 25 × 125) |

Δx = Δy = Δz (m) | 0.004 |

Time step of particle phase (s) | 0.0001 |

Time step of gas phase (s) | 0.002 |

Setting | Value |
---|---|

Width of the bed (m) | 0.08 |

Depth of the bed (m) | 0.01 |

Initial particle bed height (m) | 0.04/0.08/0.16 |

Particle number | 37,225/74,450/148,900 |

Δx = Δy = Δz (m) | 0.0025 |

Time step of particle phase (s) | 1.0 × 10^{−5} |

Time step of gas phase (s) | 2.0 × 10^{−4} |

Property (Unit), Value | Value |
---|---|

Gas density (kg/m^{3}), ρ_{g} | 1.49 |

Inlet gas temperature (K), T_{g,0} | 330 |

Gas viscosity (Pa s), µ_{g} | 1.0 × 10^{−5} |

Gas heat capacity (J/kg/K), C_{pg} | 1670 |

Gas thermal conductivity (W/m/K), k_{g} | 2.09 × 10^{−2} |

Particle diameter (m), d_{p} | 9.95 × 10^{−4} |

Particle density (kg/m^{3}), ρ_{p} | 667 |

Particle heat capacity (J/kg/K), C_{pp} | 1670 |

Normal coefficient of restitution (particle–particle), e | 0.6 |

Youngs Modulus | 5.0 × 10^{5} |

Poisson ratio | 0.45 |

Non-Uniform Cases | Uniform Cases | |||||
---|---|---|---|---|---|---|

$\raisebox{1ex}{${\mathit{H}}_{0}/\mathbf{W}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{u}}_{\mathit{g},0}$}\right.$ | T_{g} | T_{p1} | T_{p2} | (T_{p1}-T_{p2})/(T_{p1}-T_{g,in}) | T_{g} | T_{p} |

0.5/0.4 | 346 | 346 | 367 | −1.28 | 346 | 346 |

0.5/0.5 | 343 | 343 | 362 | −1.5 | 343 | 343 |

0.5/0.6 | 341 | 341 | 359 | −1.7 | 341 | 341 |

1.0/0.4 | 362 | 362 | 383 | −0.64 | 362 | 363 |

1.0/0.5 | 356 | 356 | 375 | −0.75 | 356 | 356 |

1.0/0.6 | 351 | 351 | 370 | −0.85 | 351 | 352 |

2.0/0.4 | 394 | 394 | 415 | −0.32 | 394 | 395 |

2.0/0.5 | 381 | 381 | 401 | −0.37 | 381 | 382 |

2.0/0.6 | 373 | 373 | 391 | −0.42 | 373 | 373 |

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**MDPI and ACS Style**

Mu, L.; Buist, K.A.; Kuipers, J.A.M.; Deen, N.G.
Hydrodynamic and Heat Transfer Study of a Fluidized Bed by Discrete Particle Simulations. *Processes* **2020**, *8*, 463.
https://doi.org/10.3390/pr8040463

**AMA Style**

Mu L, Buist KA, Kuipers JAM, Deen NG.
Hydrodynamic and Heat Transfer Study of a Fluidized Bed by Discrete Particle Simulations. *Processes*. 2020; 8(4):463.
https://doi.org/10.3390/pr8040463

**Chicago/Turabian Style**

Mu, Lijing, Kay A. Buist, J. A. M. Kuipers, and Niels G. Deen.
2020. "Hydrodynamic and Heat Transfer Study of a Fluidized Bed by Discrete Particle Simulations" *Processes* 8, no. 4: 463.
https://doi.org/10.3390/pr8040463