# Numerical Study and Structural Optimization of Vehicular Oil Cooler Based on 3D Impermeable Flow Model

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, length of 90 mm, layer number of 11, the model accuracy was 0.6%, as the optimal structure parameters, the heat transfer increase by 47% and with the total pressure drop increased by only 30%.

## 1. Introduction

## 2. Equivalent Theory

#### 2.1. Non-Uniform Permeable Flow Model

**u**is the velocity vector, m/s, ρ is the density of the fluid, kg/m

^{3}; p is the pressure,

**I**is identity orthogonal matrix, ε

_{p}is the porosity,

**κ**is the penetration matrix, m

^{2}; Q

_{m}is the quality of the source term;

**F**is volume force, kg/(m

^{2}·s

^{2}).

#### 2.2. Local Thermal Non-Equilibrium Model

_{p}in the transient term of energy Equation (5) becomes effective volume heat capacity at constant pressure, which is defined as follows:

## 3. Numerical Model of Oil Cooler

#### 3.1. Heat Exchange Unit Model

**κ**in two different directions of x direction and y direction, so as to establish the permeability matrix $\mathsf{\kappa}=\left[\begin{array}{ccc}{\kappa}_{x}& 0& 0\\ 0& {\kappa}_{y}& 0\\ 0& 0& {\kappa}_{z}\end{array}\right]$ of x, y and z anisotropically. This permeability matrix can be used to equivalent flow resistance on both sides of oil and water in the overall model of the oil cooler [27].

#### 3.2. Grid Dependence Analysis

#### 3.3. Nonlinear Fitting Correlation

**κ**can be obtained through calculation of H direction and Z direction respectively, as shown in Equation (7). In macro heat exchanger model,

**κ**and β were the key parameters to characterize the flow resistance of the oil cooler.

#### 3.4. Establishment of Equivalent Model

#### 3.5. Boundary Conditions and Thermophysical Parameters

## 4. Experimental Verification

#### 4.1. Experimental Rig Construction and Error Analysis

#### 4.2. The Results Discussed

## 5. Result and Discussion

#### 5.1. Flow Heat Transfer Performance Analysis

#### 5.2. Performance at Different Cross-Sectional Areas

#### 5.3. Performance at Different Flow Path Lengths

#### 5.4. Performance with Different Number of Flow Channel Layers

#### 5.5. Optimization of Structural Parameters

^{2}, it is 142% higher than the original parameter, when the length is 90 mm, it is 119% higher than the minimum parameter, and when the number of layers is 11, it is 134% higher than the original parameter. Therefore, the optimal structural parameters are proposed as follows: cross-sectional area is $3\times {10}^{-4}$ mm

^{2}, length is 90 mm, number of layers is 11. According to the calculation in Figure 14a,b, under this parameter, the heat transfer is increased by 47%, and with the total pressure drop increased by only 30%.

## 6. Conclusions

- (1)
- First, a multi-scale coupling method based on unit heat transfer model is proposed to simulate the flow and heat transfer performance of heat exchanger. The flow of the whole heat exchanger is simulated by the non-uniform seepage flow model, and the heat transfer is simulated by the local thermal non-equilibrium model.
- (2)
- Next, a vehicular oil cooler is used to verify the effectiveness of this method. By comparing with the experimental results, the maximum error of this equivalent simulation model for flow and heat transfer under different working conditions is 9.2%, which proves the validity of the equivalent model.
- (3)
- Finally, the flow and heat transfer performance under different structural parameters was studied. At the same time, the best structural parameters could applicable to the present oil cooler are proposed, namely: cross-sectional area of $3\times {10}^{-4}$ mm
^{2}, length of 90 mm, number of layers is 11. Comparing with the original structure, the heat transfer performance is increased by 47%, while the total pressure drop increased by only 30%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

μ | Dynamic viscosity |

u | Velocity vector |

$\rho $ | Fluid density |

p | Pressure |

I | Identity orthogonal matrix |

ε_{p} | Void fraction |

κ | Permeability of porous media |

${C}_{p}$ | Specific heat capacity |

${T}_{in}$ | Inlet temperature |

${Q}_{m}$ | Quality of the source |

F | Volume force |

κ | Porosity matrix |

${C}_{F}$ | Dimensionless Faux-Hemmel |

${\theta}_{s}$ | Volume fraction of a solid |

${\mathrm{k}}_{s}$ | Thermal conductivity of solids |

${\mathrm{k}}_{\mathrm{f}}$ | Thermal conductivity of a liquid |

${T}_{out}$ | Outlet temperature |

T | Temperature |

Q | Heat exchange amount, total heat exchange of heat exchanger |

${\rho}_{s}$ | Solid density |

${\rho}_{\mathrm{f}}$ | Liquid density |

m | Mass quality |

K | Heat transfer coefficient of fluid |

A | Heat exchange area, cross-sectional area |

$\Delta T$ | Temperature difference between the inlet and outlet of the hot side of the oil cooler |

Nu | Nusselt number |

Re | Reynolds number |

Pr | Prandtl number |

${v}_{m}$ | Average velocity of the fluid in the flow channel |

L | Length of flow channel |

$\Delta p$ | Pressure difference on both sides of the flow passage |

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**Figure 2.**Three dimensional geometry structure of staggered fin unit model. (

**a**) geometric model; (

**b**) simulation model.

**Figure 7.**The pressure drop of test and simulation varies with the flow rate of oil side and cold side.

**Figure 10.**Study of the influence factor of oil cooler: (

**a**) f-factor; (

**b**) variance; (

**c**) j-factor; (

**d**) j/f.

**Figure 11.**Study of the influence factor of oil cooler:(

**a**) f-factor; (

**b**) variance; (

**c**) j-factor; (

**d**) j/f.

**Figure 12.**Velocity distribution diagram of oil cooler and flow channel velocity distribution diagram: (

**a**) 4-layer heat exchanger speed distribution diagram; (

**b**) 11-layer heat exchanger temperature distribution diagram; (

**c**) Velocity distribution of flow channel with 4 layers; (

**d**) Velocity distribution of flow channel with 11 layers.

**Figure 13.**Study of the influence factor of oil cooler:(

**a**) f-factor; (

**b**) variance; (

**c**) j-factor; (

**d**) j/f.

**Figure 14.**Comparison of heat exchange between basic and improved parameters: (

**a**) Heat exchange after basic parameter setting; (

**b**) Heat exchange after improved parameter setting.

Number | Oil Inlet Temperature (°C) | Cold Test Inlet Temperature (°C) | Oil Flow Rate (kg/min) | Cold Side Flow Rate (kg/min) |
---|---|---|---|---|

1 | 100 | 70 | 12.25 | 12.75 |

2 | 100 | 70 | 12.25 | 15.75 |

3 | 100 | 70 | 6.75 | 8.75 |

4 | 100 | 70 | 6.75 | 12.75 |

5 | 100 | 70 | 6.75 | 16.00 |

6 | 100 | 70 | 10.00 | 16.00 |

7 | 100 | 70 | 10.00 | 13.00 |

8 | 100 | 70 | 10.00 | 8.25 |

9 | 100 | 70 | 12.75 | 8.25 |

10 | 130 | 90 | 12.00 | 15.25 |

11 | 130 | 90 | 12.00 | 12.25 |

12 | 130 | 90 | 12.00 | 8.25 |

13 | 130 | 90 | 9.50 | 8.00 |

14 | 130 | 90 | 10.00 | 12.50 |

15 | 130 | 90 | 10.00 | 15.50 |

16 | 130 | 90 | 6.50 | 15.50 |

17 | 130 | 90 | 6.50 | 12.50 |

18 | 130 | 90 | 6.50 | 8.00 |

Density /kg/m^{3} | $y=2\times {10}^{-6}{x}^{3}-0.002{x}^{2}-0.4554x+1074.6$ ${R}^{2}=0.9999$ |

Constant pressure heat capacity/J/(kg °C) | $y=2\times {10}^{-5}{x}^{3}-0.0051{x}^{2}+4.207x+3256.5$ ${R}^{2}=0.9989$ |

Dynamic viscosity/Pa · s | $y=-9\times {10}^{-14}{x}^{5}+8\times {10}^{-11}{x}^{4}-2\times {10}^{-8}{x}^{3}+3\times {10}^{-6}{x}^{2}-0.0003x+0.0084$ ${R}^{2}=0.9989$ |

Coefficient of thermal conductivity/W/(m · °C) | $y=7\times {10}^{-7}{x}^{2}-0.0003x+0.4423$ ${R}^{2}=0.9991$ |

Density /kg/m^{3} | $y=-8\times {10}^{-5}{x}^{2}-0.5779x+898.75$ ${R}^{2}=1$ |

Constant pressure heat capacity/J/(kg °C) | $y=0.0014{x}^{2}+4.078x+1801.4$ ${R}^{2}=1$ |

Dynamic viscosity/Pa · s | $y=-1\times {10}^{-10}{x}^{5}+6\times {10}^{-8}{x}^{4}-1\times {10}^{-5}{x}^{3}+0.0009{x}^{2}-0.0356x+0.5894$ ${R}^{2}=0.9972$ |

Coefficient of thermal conductivity/W/(m · °C) | $y=2\times {10}^{-8}{x}^{2}-1\times {10}^{-4}x+0.1464$ ${R}^{2}=1$ |

The Length of the Channel | Channel Width | Oil Domain Channel Height | Water Channel Height | Number of Layer |
---|---|---|---|---|

90 mm | 60 mm | 6 mm | 6 mm | 6 |

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**MDPI and ACS Style**

Fu, J.; Hu, Z.; Zhang, Y.; Lu, G.
Numerical Study and Structural Optimization of Vehicular Oil Cooler Based on 3D Impermeable Flow Model. *Sustainability* **2022**, *14*, 7757.
https://doi.org/10.3390/su14137757

**AMA Style**

Fu J, Hu Z, Zhang Y, Lu G.
Numerical Study and Structural Optimization of Vehicular Oil Cooler Based on 3D Impermeable Flow Model. *Sustainability*. 2022; 14(13):7757.
https://doi.org/10.3390/su14137757

**Chicago/Turabian Style**

Fu, Jiahong, Zhecheng Hu, Yu Zhang, and Guodong Lu.
2022. "Numerical Study and Structural Optimization of Vehicular Oil Cooler Based on 3D Impermeable Flow Model" *Sustainability* 14, no. 13: 7757.
https://doi.org/10.3390/su14137757