# Enhancing the Thermal Performance of Slender Packed Beds through Internal Heat Fins

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Methods

#### 2.1.1. Additive Manufacturing Process

#### 2.1.2. Particle Count and Pressure Drop Measurements

#### 2.2. Numerical Methods

- The generation of a random particle packing
- The construction of a CAD description of the packing
- Meshing of the geometry
- The CFD simulation itself

#### 2.2.1. Numerical Packing Generation

#### 2.2.2. Computational Fluid Dynamics

## 3. Results and Discussion

#### 3.1. Experimental Validation

#### 3.2. Detailed Numerical Study

#### 3.2.1. Bed Morphology

^{−1}, whereas it is 431 m

^{−1}for a bed with helical fins and 444 m

^{−1}for a bed with straight fins. This is obvious, because additional particle-wall contacts are created. The radial void fraction distributions, given in Figure 8, show that this effect is limited to the near wall region, which extend until one particle diameter away from the wall. Here, the straight and helical fin designs show a similar increased void fraction in comparison to the reactor without heat fins. At a position of ${r}^{\ast}=\left(\right)open="("\; close=")">R-r$, a discontinuity is observed, leading to a small, but recognizable, increase of void fraction. This corresponds to the position of the fin tips. For the straight fin design, Figure 9 shows, based on a parallel projected view from the top of the reactor, a transmitted light image of all particles that are in contact with the wall. The one sphere near the center of the bed is in contact with the bottom plate of the reactor. It can be seen that the majority of particles are arranged between the two heat fins and only a small number is in contact with the tip if the fins itself. This results in the observed steep increase of void fraction close to the fin tips. However, after a distance of one until one and a half particle diameters, the straight fin design shows the lowest void fraction, while the helical design and the reactor without fins show similar values. In the core of the bed, the reactor without fins has the lowest void fraction, while the straight design shows the highest values. The coincident trends of the radial void fraction profiles near the wall show that there is no increase in particle-wall contacts for the helical design compared to the straight design, although this could have been expected. Furthermore, it can be seen that away from the wall, the impact of fins on bed voidage is low.

#### 3.2.2. Fluid Dynamics

#### 3.2.3. Heat Transfer

## 4. Conclusions

- It is applicable to new and already existing reactors, as the fins are manufactured as replaceable sleeves.
- The morphological and fluid dynamic characteristics (bed voidage and pressure drop) change only moderately.
- The active catalytic surface area changes only slighty. It is reduced by 7% for the helical and below 5% for the straight fin design, respectively.
- Reactor filling and re-filling strategies can stay unaffected.

- Although, some publications are available, indicating the accuracy of particle-resolved CFD when it comes to heat transfer simulation, e.g., [46,47], broader validation is necessary, to make this a reliable and predictive design tool, especially under industrial conditions, e.g., complex particle shapes, turbulent flows, steep gradients, and coupled with catalytic reactions.
- Design parameters, such as number of fins, fin thickness, fin material and fin shape, should be varied to find an optimized fin design that incorporates a beneficial heat transfer characteristic with a reduced impact on bed voidage, active catalytic surface area and pressure drop.
- It needs to be evaluated, if a generalized optimized fin design can be found, that is also applicable to other particle shapes than spheres, or if individual solutions must be developed.
- Possible limits regarding the manufacturability using 3D printing need to also be evaluated.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CAD | Computer aided design |

CFD | Computational fluid dynamics |

DEM | Discrete element method |

FLM | Fused layer modeling |

MFC | Mass flow controller |

PLA | Polyactid |

POCS | Periodic open-cell structures |

PVA | Polyvinylalcohol |

## Nomenclature

${C}_{\mathrm{fr}}$ | rolling friction coefficient | [-] |

${C}_{\mathrm{fs}}$ | static friction coefficient | [-] |

${C}_{\mathrm{n},\mathrm{rest}}$ | normal restitution coefficient | [-] |

${C}_{\mathrm{t},\mathrm{rest}}$ | tangential restitution coefficient | [-] |

D | tube diameter | [m] |

${E}_{\mathrm{eq}}$ | particle equivalent Young’s modulus | [Pa] |

${G}_{\mathrm{eq}}$ | particle equivalent shear modulus | [Pa] |

H | bed height | [m] |

${J}_{1}\left(\right)$ | Bessel function of first kind and first order | [-] |

${K}_{\mathrm{n}}$ | normal spring stiffness | [N m^{−1}] |

${K}_{\mathrm{t}}$ | tangential spring stiffness | [N m^{−1}] |

${M}_{\mathrm{eq}}$ | particle equivalent mass | [kg] |

N | tube-to-particle dimeter ratio | [-] |

${N}_{\mathrm{n},\mathrm{damp}}$ | normal damping coefficient | [-] |

${N}_{\mathrm{n}}$ | normal damping | [N s m^{−1}] |

${N}_{\mathrm{p}}$ | particle count | [-] |

${N}_{\mathrm{t},\mathrm{damp}}$ | tangential damping coefficient | [-] |

${N}_{\mathrm{t}}$ | tangential damping | [N s m^{−1}] |

R | tube radius | [m] |

${R}_{\mathrm{eq}}$ | particle equivalent radius | [m] |

T | temperature | [K] |

${T}_{0}$ | inlet temperature | [K] |

${T}_{\mathrm{c}}$ | core temperature | [K] |

${T}_{\mathrm{m}}$ | average outlet temperature | [K] |

${T}_{\mathrm{w}}$ | wall temperature | [K] |

${V}_{\mathrm{POCS}}$ | volume of the POCS | [m^{3}] |

${V}_{\mathrm{p},\mathrm{total}}$ | volume of all particles | [m^{3}] |

${V}_{\mathrm{total}}$ | empty reactor volume | [m^{3}] |

$\mathit{D}$ | deformation tensor | [s^{−1}] |

${\mathit{F}}_{\mathrm{b}}$ | body forces | [N] |

${\mathit{F}}_{\mathrm{n}}$ | normal contact force | [N] |

${\mathit{F}}_{\mathrm{s}}$ | surface forces | [N] |

${\mathit{F}}_{\mathrm{t}}$ | tangential contact forces | [N] |

${\mathit{I}}_{\mathrm{p}}$ | particle moment of inertia | [kg m^{−2}] |

$\mathit{I}$ | unit tensor | [-] |

${\mathit{M}}_{\mathrm{c}}$ | moment due to contact | [N m] |

$\mathit{T}$ | stress tensor | [Pa] |

${\mathit{r}}_{\mathrm{p}}$ | position vector from particle center of gravity to contact point | [m] |

${\mathit{v}}_{\mathrm{p}}$ | particle velocity | [m s^{−1}] |

$\mathit{v}$ | fluid velocity | [m s^{−1}] |

$\dot{\mathit{q}}$ | conductive heat flux | [W m^{−2}] |

${a}_{1}$ | parameter | [-] |

${a}_{\mathrm{v}}$ | specific surface area | [m^{−1}] |

${c}_{\mathrm{p},\mathrm{f}}$ | fluid specific heat | [J kg^{−1} K^{−1}] |

${d}_{\mathrm{n}}$ | overlap in normal direction | [m] |

${d}_{\mathrm{pore}}$ | pore diameter | [m] |

${d}_{\mathrm{p}}$ | particle diameter | [m] |

${d}_{\mathrm{t}}$ | overlap in tangential direction | [m] |

h | specific enthalpy | [J kg ^{−1}] |

i | contact index | [-] |

${m}_{\mathrm{p}}$ | particle mass | [m^{3}] |

p | pressure | [Pa] |

r | radial coordinate | [m] |

${r}^{\ast}$ | dimensionlesss wall distance | [-] |

t | time | [s] |

${v}_{0}$ | superficial velocity | [m s^{−1}] |

${v}_{\mathrm{n}}$ | normal velocity component of the relative sphere surface velocity | [m s^{−1}] |

${v}_{\mathrm{t}}$ | tangential velocity component of the relative sphere surface velocity | [m s^{−1}] |

z | axial coordinate | [m] |

$\Psi $ | friction factor | [-] |

$\Theta $ | dimensionless temperature | [-] |

${\Theta}_{\mathrm{log},\mathrm{c}}$ | dimensionless logarithmic core temperature | [-] |

${\alpha}_{\mathrm{w}}$ | wall heat transfer coefficient | [W m^{−2} K^{−1}] |

${\mathit{\omega}}_{\mathrm{p}}$ | particle angular velocity | [rad s^{−1}] |

$\delta $ | relativ deviation | [%] |

$\lambda $ | thermal conductivity | [W m^{−1} K^{−1}] |

${\lambda}_{\mathrm{r},\mathrm{eff}}$ | effective radial thermal conductivity | [W m^{−1} K^{−1}] |

$\mu $ | dynamic viscosity | [Pa s] |

$\nu $ | Poisson ratio | [-] |

$\rho $ | fluid density | [kg m^{−3}] |

${\rho}_{\mathrm{f}}$ | fluid density | [kg m^{−3}] |

${\rho}_{\mathrm{p}}$ | particles solid density | [kg m^{−3}] |

$\epsilon $ | void fraction | [-] |

${\epsilon}_{\mathrm{foam}}$ | foam voidage | [-] |

## Dimensionless Numbers

$Bi={\alpha}_{\mathrm{w}}R/{\lambda}_{\mathrm{r},\mathrm{eff}}$ | Biot number |

$R{e}_{\mathrm{p}}={v}_{0}{d}_{\mathrm{p}}{\rho}_{\mathrm{f}}/{\mu}_{\mathrm{f}}$ | particle Reynolds number |

$N{u}_{\mathrm{w}}={\alpha}_{\mathrm{w}}{d}_{\mathrm{p}}/{\lambda}_{\mathrm{f}}$ | wall Nusselt number |

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**Figure 1.**CFD simulation result of the radial void fraction distribution and radial profile of the averaged normalized axial velocity for a packing of spherical particles ($N=8.8$).

**Figure 2.**Recent process intensification concepts for catalytic flow reactors: (

**A**) Complex particle shapes, (

**B**) Periodic open cell structures, (

**C**) Packed foams and (

**D**) Macroscopic wall structures.

**Figure 3.**Visualization of the investigated reactor designs: Tube without heat fins (

**A**), with straight heat fins (

**B**), and with helical heat fins (

**C**). Tube diameter = 25.4 mm; particle diameter = 7 mm; fin depth = 4.2 mm; fin thickness = 1 mm.

**Figure 5.**Experimental setup of the pressure drop measurements: details of the bed structure with 3D printed heat fins (

**left**) and sketch of the experimental setup (

**right**).

**Figure 6.**Predicted specific pressure drop in comparison to experimental results for (

**A**) the reactor without heat fins, and (

**B**) the reactor with helical fins.

**Figure 7.**Parity plot, comparing the specific pressure drop of the reactor without fins, and the one with helical fins. The provided data is based on polynomial fitting of the simulation data according to Equations (23) and (24).

**Figure 10.**Radial profiles of the circumferentially and axially averaged absolute value of (

**A**) axial and (

**B**) tangential velocity, which are normed to the local interstitial velocity.

**Figure 11.**Helical design: Visualization of the intersection points of streamlines with cross-sectional planes at a distance of 10 mm The streamlines were injected in the annular space near the wall at a bed height of 100 mm.

**Figure 12.**Comparison of the axial profile of the core temperature (

**left**) and the averaged temperature (

**right**) for $R{e}_{\mathrm{p}}=105.6$ (

**A**,

**B**), $R{e}_{\mathrm{p}}=528$ (

**C**,

**D**) and $R{e}_{\mathrm{p}}=1056$ (

**E**,

**F**).

Fluid Dynamic Validation Study: | |

Inlet velocity [m/s] (without fins) | 0.21; 0.41; 0.76; 1.04; 1.25; 1.46; 1.58 |

Inlet velocity [m/s] (helical fins) | 0.21; 0.41; 0.77; 1.05; 1.26; 1.47; 1.54 |

Numerical Heat Transfer Study: | |

Inlet velocity [m/s] | 0.22; 1.10; 2.20 |

Inlet temperature [°C] | 20 |

Wall temperature [°C] | 200 |

Thermal conductivity of particles [W/(m K)] | 0.25 |

Thermal conductivity of heat fins [W/(m K)] | 15.10 |

Thermal conductivity of gas phase [W/(m K)] | 0.026 |

Property | Value |
---|---|

Young’s modulus [MPa] | 100.0 |

Poisson ratio [-] | 0.45 |

Static friction coefficient [-] | 0.01 |

Rolling friction coefficient [-] | 0.001 |

Restitution coefficients [-] | 0.5 |

Particle density [kg/m^{3}] | 1100.0 |

**Table 3.**Comparison of the particle count ($H=600\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$) between experimental and DEM results.

Particle Count | |||||
---|---|---|---|---|---|

Exp. #1 | Exp. #2 | Exp. #3 | Exp. #4 | DEM | |

Without fins | 826 | 826 | 828 | 827 | 823 |

Helical fins | 750 | 747 | 741 | - | 745 |

${\mathit{Re}}_{\mathbf{p}}$ | Design | $\mathbf{\Delta}\mathit{p}/\mathbf{\Delta}\mathit{z}$ (Pa/m) | ${\mathit{\delta}}_{\mathbf{\Delta}\mathbf{p}}$ (%) | $\mathbf{\Psi}={\displaystyle \frac{\mathbf{\Delta}\mathit{p}}{{\mathit{\rho}}_{\mathbf{f}}{\mathit{u}}_{0}^{2}}}\xb7{\displaystyle \frac{{\mathit{d}}_{\mathbf{p}}}{\mathit{L}}}\xb7{\displaystyle \frac{{\mathit{\epsilon}}^{3}}{1-\mathit{\epsilon}}}$ (-) | ${\mathit{\delta}}_{\mathbf{\Psi}}$ (%) | ${\mathit{\alpha}}_{\mathbf{w}}$ (W/(m^{2} K)) | ${\mathit{\delta}}_{{\mathit{\alpha}}_{\mathbf{w}}}$ (%) | ${\mathit{\lambda}}_{\mathbf{r},\mathbf{eff}}$ (W/(m K)) | ${\mathit{\delta}}_{{\mathit{\lambda}}_{\mathbf{r},\mathbf{eff}}}$ (%) |
---|---|---|---|---|---|---|---|---|---|

105.6 | No fin | 192.0 | - | 4.17 | - | 29.5 | - | 0.149 | - |

Straight fin | 157.9 | −18 | 4.01 | −4 | 32.0 | +8 | 0.189 | +27 | |

Helical fin | 193.9 | +1 | 5.43 | +30 | 37.5 | +25 | 0.219 | +47 | |

528 | No fin | 2755.4 | - | 2.41 | - | 73.3 | - | 0.461 | - |

Straight fin | 1950.8 | −29 | 1.98 | −18 | 86.9 | +19 | 0.449 | −3 | |

Helical fin | 2804.6 | +2 | 3.14 | +30 | 98.7 | +35 | 0.769 | +67 | |

1056 | No fin | 9721.9 | - | 2.11 | - | 124.3 | - | 0.831 | - |

Straight fin | 6600.2 | −32 | 1.68 | −20 | 140.5 | +13 | 1.082 | +30 | |

Helical fin | 10,072.8 | +4 | 2.82 | +34 | 149.9 | +21 | 1.573 | +89 |

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## Share and Cite

**MDPI and ACS Style**

Jurtz, N.; Flaischlen, S.; Scherf, S.C.; Kraume, M.; Wehinger, G.D.
Enhancing the Thermal Performance of Slender Packed Beds through Internal Heat Fins. *Processes* **2020**, *8*, 1528.
https://doi.org/10.3390/pr8121528

**AMA Style**

Jurtz N, Flaischlen S, Scherf SC, Kraume M, Wehinger GD.
Enhancing the Thermal Performance of Slender Packed Beds through Internal Heat Fins. *Processes*. 2020; 8(12):1528.
https://doi.org/10.3390/pr8121528

**Chicago/Turabian Style**

Jurtz, Nico, Steffen Flaischlen, Sören C. Scherf, Matthias Kraume, and Gregor D. Wehinger.
2020. "Enhancing the Thermal Performance of Slender Packed Beds through Internal Heat Fins" *Processes* 8, no. 12: 1528.
https://doi.org/10.3390/pr8121528