# Application of a Reduced-Dimensional Model for Fluid Flow between Stacks of Parallel Plates with Complex Surface Topography

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature

#### 1.3. Contributions

#### 1.4. Paper Layout

## 2. Models and Methods

#### 2.1. Thermal Storage Module

#### 2.2. Sliced Three-Dimensional Model

#### 2.3. Reduced Two-Dimensional Approximation

#### 2.3.1. Reduced-Dimensional Model

#### 2.3.2. Two-Dimensional Approximation for Sliced Two-Channel Model

#### 2.3.3. Two-Dimensional Approximation for Full Two-Channel Model

#### 2.4. Implementation Details

## 3. Results

#### 3.1. Comparison of Sliced Two-Channel Model

#### 3.2. Full Two-Channel Flow Distribution

#### 3.2.1. Reduced-Dimensional Model for Unsteady Flow

#### 3.2.2. Different Inlet Velocity Profiles

#### 3.2.3. Overall Flow Distribution

#### 3.2.4. Effect of Average Inlet Velocity Magnitude

#### 3.2.5. Effect of Asymmetry in Inlet Velocity

#### 3.2.6. Effect of Inlet Velocity Angles

#### 3.2.7. Effect of Plate Spacing

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CSM | Compact Storage Module |

DOFs | Degrees of freedom |

PCM | Phase change material |

TES | Thermal energy storage |

## Appendix A. Mesh Convergence Analysis

#### Appendix A.1. Sliced Two-Channel Model

**Figure A2.**The pressure drop of the three-dimensional sliced two-channel model varies with the number of DOFs.

#### Appendix A.2. Full Two-Channel Model

## References

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**Figure 1.**Graphical overview of the application of the reduced-dimensional model to simulate the flow between the stacked heat exchanger plates.

**Figure 3.**Boundary conditions and dimensions for a three-dimensional sliced two-channel model. Plates are not modeled.

**Figure 4.**Boundary conditions and dimensions for the sliced two-channel model for the reduced-dimensional model.

**Figure 5.**Height distribution of sliced two-channel model for reduced-dimensional model. (

**a**) Height for momentum equations, $h\phantom{\rule{-1.27501pt}{0ex}}\left({x}_{1},{x}_{2}\right)$; (

**b**) height for mass conservation, $\tilde{h}\phantom{\rule{-1.27501pt}{0ex}}\left({x}_{1},{x}_{2}\right)$.

**Figure 6.**Boundary conditions and dimensions for the full two-channel model for reduced-dimensional model.

**Figure 7.**Height distributions of full two-channel model for reduce-dimensional model. (

**a**) Heights for momentum equation, $h\phantom{\rule{-1.27501pt}{0ex}}\left({x}_{1},{x}_{2}\right)$; (

**b**) heights for mass equation, $\tilde{h}\phantom{\rule{-1.27501pt}{0ex}}\left({x}_{1},{x}_{2}\right)$.

**Figure 8.**Comparison of velocity distributions obtained from full three-dimensional models and reduced-dimensional models for different inlet velocities and different channel thicknesses.

**Figure 9.**Relative error of the velocity at channel outlet between reduced-dimensional model and full three-dimensional model for different inlet velocities and different channel heights.

**Figure 10.**Pressure drop error between reduced-dimensional model and full three-dimensional model for different inlet velocities and different channel heights.

**Figure 11.**Profiles of the inlet velocity for different parameters, where ${U}_{avg}$ controls the magnitude of the profile and ${y}_{c}$ (see Equation (6)) controls the asymmetry of the profile.

**Figure 12.**Velocity and pressure distributions of the full two-channel model obtained from the reduced-dimensional model when ${U}_{avg}=0.189$ m/s and ${y}_{c}=W/2$. (

**a**) Velocity distribution of full channel; (

**b**) pressure distribution of full channel.

**Figure 13.**Velocity profiles at different cut line positions when ${y}_{c}=W/2$ and ${U}_{avg}=0.189$ m/s.

**Figure 14.**Comparison of velocity profiles over the cut lines at the channel inlet ${\mathsf{\Gamma}}_{in}$ and channel outlet ${\mathsf{\Gamma}}_{out}$ for different velocity magnitudes ${U}_{avg}$ at ${y}_{c}=W/2$. For (

**a**), the solid line indicates the inlet velocity and the dashed line indicates the outlet velocity. For (

**b**), the solid line indicates the inlet velocity and the dotted line indicates moving average velocity. (

**a**) Velocity profiles of the channel inlet and channel outlet; (

**b**) channel inlet velocity distribution and the moving average of the channel outlet velocity distributions.

**Figure 15.**Velocity profiles of the channel inlet ${\mathsf{\Gamma}}_{in}$ and channel outlet ${\mathsf{\Gamma}}_{out}$ for different ${y}_{c}$ when ${U}_{avg}=0.189$ m/s. (

**a**) Velocity profiles at the channel inlet for different ${y}_{c}$; (

**b**) channel inlet and outlet velocity profiles when ${y}_{c}=\frac{7W}{8}$.

**Figure 17.**Velocity profiles of channel inlet ${\mathsf{\Gamma}}_{in}$ and channel outlet ${\mathsf{\Gamma}}_{out}$ for different velocity angles $\theta $ when ${U}_{avg}=0.189$ m/s and ${y}_{c}=W/2$. (

**a**) Velocity profiles at channel inlet for different velocity angles $\theta $; (

**b**) channel inlet and outlet velocity profiles when $\theta =\frac{\pi}{3}$.

**Figure 18.**Velocity and pressure distributions of the full channel when ${U}_{avg}=0.159$ m/s, ${y}_{c}=W/2$, $\theta =0$, and the channel spacing height is 6.5 mm. (

**a**) Velocity distribution of full channel; (

**b**) pressure distribution of full channel.

**Figure 19.**Velocity profiles at different cut line positions when $\theta =0$, ${y}_{c}=\frac{W}{2}$, ${U}_{avg}=0.159$ m/s, and the channel spacing height is 6.5 mm.

**Figure 20.**Velocity profiles of the channel outlet ${\mathsf{\Gamma}}_{out}$ for different ${y}_{c}$ when $\theta =0$ and different $\theta $ when ${y}_{c}=\frac{W}{2}$. The channel spacing is 6.5 mm and the average inlet velocity is ${U}_{avg}=0.159$ m/s. (

**a**) Different ${y}_{c}$; (

**b**) different $\theta $.

Parameters | Symbols | Values | Units |
---|---|---|---|

Width | W | 43 | mm |

Height | H | 32 | mm |

Length of inlet | ${L}_{in}$ | 60 | mm |

Length of outlet | ${L}_{out}$ | 60 | mm |

Length of channels | ${L}_{channel}$ | 258 | mm |

Length of transitions | ${L}_{trans}$ | 10 | mm |

Length of dimples | ${L}_{dimple}$ | 9 | mm |

Thickness of channels | ${d}_{channel}$ | 3.5 | mm |

Thickness of dimples | ${d}_{dimple}$ | 1.5 | mm |

Thickness of transitions | ${d}_{trans}$ | 1.5 | mm |

Thickness of plates | ${d}_{plate}$ | 15 | mm |

Density of fluid | $\varrho $ | 1.177 | ${\mathrm{kg}/\mathrm{m}}^{3}$ |

Viscosity of fluid | $\mu $ | $1.846\times {10}^{-5}$ | $\mathrm{kg}/(\mathrm{m}\xb7\mathrm{s}$) |

Average inlet velocity | ${U}_{avg}$ | 0.05 | m/s |

**Table 2.**Dimensions of the full two-channel model and the physical parameters of air at 27 ${}^{\circ}$C [2].

Parameters | Symbols | Values | Units |
---|---|---|---|

Width | W | 426 | mm |

Height | H | 32 | mm |

Length of inlet | ${L}_{in}$ | 60 | mm |

Length of outlet | ${L}_{out}$ | 60 | mm |

Length of channels | ${L}_{channel}$ | 258 | mm |

Length of transitions | ${L}_{trans}$ | 10 | mm |

Length of dimples | ${L}_{dimple}$ | 9 | mm |

Thickness of channels | ${d}_{channel}$ | 3.5 | mm |

Thickness of dimples | ${d}_{dimple}$ | 1.5 | mm |

Thickness of transitions | ${d}_{trans}$ | 1.5 | mm |

Thickness of plates | ${d}_{plate}$ | 15 | mm |

Diameter of small circle | ${D}_{cc}$ | 21 | mm |

Diameter of circle 1 | ${D}_{c1}$ | 78.8 | mm |

Diameter of circle 2 | ${D}_{c2}$ | 90 | mm |

Distance between circle center and edge | ${d}_{c}$ | 123 | mm |

Density of fluid | $\varrho $ | 1.177 | ${\mathrm{kg}/\mathrm{m}}^{3}$ |

Viscosity of fluid | $\mu $ | $1.846\times {10}^{-5}$ | $\mathrm{kg}/(\mathrm{m}\xb7\mathrm{s})$ |

Average inlet velocity | ${U}_{avg}$ | 0.189 | m/s |

**Table 3.**Comparison of pressure drop for different models with different inlet velocities and different channel heights.

${\mathit{U}}_{\mathbf{avg}}$ [m/s] | ${\mathit{d}}_{\mathbf{channel}}$ [mm] | Pressure Drop [Pa] | Relative Error of Velocity [%] | ||
---|---|---|---|---|---|

2D | 3D | Error [%] | |||

0.02 | 3.5 | 0.56 | 0.62 | 9.43 | 0.30 |

0.02 | 4.5 | 0.25 | 0.27 | 7.58 | 1.00 |

0.02 | 5.5 | 0.13 | 0.14 | 5.90 | 1.81 |

0.02 | 6.5 | 0.08 | 0.08 | 4.29 | 2.50 |

0.03 | 3.5 | 0.85 | 0.93 | 9.05 | 0.48 |

0.03 | 4.5 | 0.37 | 0.40 | 7.13 | 1.09 |

0.03 | 5.5 | 0.19 | 0.20 | 5.21 | 1.83 |

0.03 | 6.5 | 0.11 | 0.12 | 3.26 | 2.47 |

0.04 | 3.5 | 1.13 | 1.24 | 8.59 | 0.57 |

0.04 | 4.5 | 0.49 | 0.53 | 6.68 | 1.01 |

0.04 | 5.5 | 0.26 | 0.27 | 4.55 | 1.56 |

0.04 | 6.5 | 0.15 | 0.16 | 2.27 | 2.07 |

0.05 | 3.5 | 1.42 | 1.54 | 8.09 | 0.55 |

0.05 | 4.5 | 0.62 | 0.66 | 6.25 | 0.74 |

0.05 | 5.5 | 0.32 | 0.34 | 3.92 | 1.06 |

0.05 | 6.5 | 0.19 | 0.19 | 1.32 | 1.40 |

**Table 4.**Average velocity and standard deviation of the inlet ${\mathsf{\Gamma}}_{in}$ and outlet ${\mathsf{\Gamma}}_{out}$ of the channel and the moving average of the outlet velocity for different velocity magnitudes.

${\mathit{U}}_{\mathbf{avg}}$ [m/s] | Average Velocity [m/s] | Standard Deviation [-] | ||||
---|---|---|---|---|---|---|

Inlet | Outlet | Mov. avg. | Inlet | Outlet | Mov. avg. | |

0.189 | 1.9345 | 1.9367 | 1.9364 | 0.0998 | 0.1259 | 0.0528 |

0.378 | 3.6759 | 3.6951 | 3.6953 | 0.2590 | 0.2658 | 0.1295 |

0.567 | 5.5148 | 5.5600 | 5.5612 | 0.4778 | 0.4466 | 0.2709 |

**Table 5.**Average velocity and standard deviation of the inlet ${\mathsf{\Gamma}}_{in}$ and outlet ${\mathsf{\Gamma}}_{out}$ of the channel and the moving average of the outlet velocity for different ${y}_{c}$ when ${U}_{avg}=0.189$ m/s.

${\mathit{y}}_{\mathit{c}}$ [mm] | Average Velocity [m/s] | Standard Deviation [-] | ||||
---|---|---|---|---|---|---|

Inlet | Outlet | Mov. avg. | Inlet | Outlet | Mov. avg. | |

W/8 | 1.9392 | 1.9368 | 1.9365 | 0.1254 | 0.1252 | 0.0529 |

W/4 | 1.9366 | 1.9367 | 1.9364 | 0.1115 | 0.1256 | 0.0529 |

W/2 | 1.9345 | 1.9367 | 1.9364 | 0.0998 | 0.1259 | 0.0528 |

3W/4 | 1.9366 | 1.9366 | 1.9364 | 0.1113 | 0.1257 | 0.0530 |

7W/8 | 1.9392 | 1.9367 | 1.9364 | 0.1251 | 0.1253 | 0.0530 |

**Table 6.**Average velocity and standard deviation of the inlet ${\mathsf{\Gamma}}_{in}$ and outlet ${\mathsf{\Gamma}}_{out}$ of the channel and the moving average of the outlet velocity for different $\theta $ when ${U}_{avg}=0.189$ m/s, ${y}_{c}=W/2$.

$\mathit{\theta}$ [rad] | Average Velocity [m/s] | Standard Deviation [-] | ||||
---|---|---|---|---|---|---|

Inlet | Outlet | Mov. avg. | Inlet | Outlet | Mov. avg. | |

0 | 1.9345 | 1.9367 | 1.9364 | 0.0998 | 0.1259 | 0.0528 |

$\pi /6$ | 1.9358 | 1.9367 | 1.9364 | 0.1181 | 0.1258 | 0.0534 |

$\pi /5$ | 1.9366 | 1.9367 | 1.9364 | 0.1277 | 0.1256 | 0.0536 |

$\pi /4$ | 1.9385 | 1.9366 | 1.9363 | 0.1485 | 0.1255 | 0.0542 |

$\pi /3$ | 1.9467 | 1.9366 | 1.9363 | 0.2149 | 0.1256 | 0.0562 |

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**MDPI and ACS Style**

Sun, Y.; Hassan, H.M.A.; Alexandersen, J.
Application of a Reduced-Dimensional Model for Fluid Flow between Stacks of Parallel Plates with Complex Surface Topography. *Fluids* **2023**, *8*, 174.
https://doi.org/10.3390/fluids8060174

**AMA Style**

Sun Y, Hassan HMA, Alexandersen J.
Application of a Reduced-Dimensional Model for Fluid Flow between Stacks of Parallel Plates with Complex Surface Topography. *Fluids*. 2023; 8(6):174.
https://doi.org/10.3390/fluids8060174

**Chicago/Turabian Style**

Sun, Yupeng, Hafiz Muhammad Adeel Hassan, and Joe Alexandersen.
2023. "Application of a Reduced-Dimensional Model for Fluid Flow between Stacks of Parallel Plates with Complex Surface Topography" *Fluids* 8, no. 6: 174.
https://doi.org/10.3390/fluids8060174