# Experimental and Numerical Investigations into Heat Transfer Using a Jet Cooler in High-Pressure Die Casting

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Temperature Measurements

- The experiment starts together with the temperature recordings.
- The water jet cooling is performed with a constant flow rate.
- The heating from the top part of the sample is initiated.
- The thermal steady state is achieved throughout the experiment.
- Temperature cooling curves are saved for the subsequent inverse calculation and post-processing of the heat dissipation rate (also referred to as the cooling power).

#### 2.2. Inverse Heat Conduction Problem in 2D

_{Ni}in each discrete node Ni ∈{1, …, I} by solving the least-square problem (Equation (2)):

#### 2.3. Verification of the IHCP Solver Using a Numerical Experiment

- There is no error regarding the temperature measurements (the exact values are defined at the exact locations);
- There is no error in defining the boundary conditions;
- There is no error in selecting the thermophysical properties;
- There is no error in specifying discretization and solution.

#### 2.4. IHCP Calculations with Experimental Data

^{−2}K

^{−1}. Similar to the previous section with the numerical experiment, the IHCP calculation perfectly converges at the same rate towards the experimental data, i.e., reaching $F\approx 0$ (Table 4). In addition, the temperature difference between the measured and the simulated temperature is provided for each thermocouple #1–#6 and each experiment in Table 4. More details about solver and discretization settings, as well as the parameters of BOBYQA minimizer, can be found in [26]. The measured and calculated uncertainties agree very well with the analysis discussed in [30].

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

F | objective or target function (K^{2}) |

HTC | heat transfer coefficient (Wm^{−2}K^{−1}) |

k | thermal conductivity (Wm^{−1}K^{−1}) |

q | heat flux (Wm^{−2}) |

T | temperature (K) |

Ω | space domain (m) |

Abbreviations | |

BOBYQA | Bounded optimization by quadratic approximation |

BC | Boundary condition |

CFD | Computational fluid dynamics |

Dievar | High-performance chromium–molybdenum–vanadium-alloyed hot work tool steel |

HPDC | High-pressure die casting |

NLopt | Nonlinear optimization toolbox |

OpenFOAM | Open-source CFD package |

RANS | Reynolds-averaged Navier–Stokes (equations) |

PDE | Partial differential equation |

PID | Proportional–integral–derivative controller |

Subscripts | |

i | summation index |

I | number of thermocouples |

N | a point on the curved boundary of the jet cooler generated by the orthogonal projection of the thermocouple tip, i.e., the point P |

P | measuring point of thermocouple |

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**Figure 1.**Die insert used in the study: (

**a**) experimental configuration with dimensions; (

**b**) schematics of the simulated inverse heat conduction problem in 2D with the thermocouples numbered from 1 to 6; (

**c**) a photo of an experimental setup.

**Figure 2.**The steady-state temperatures recorded with the thermocouples #1–6 in the experiments Exp. #1–4. Vertical distance is aligned with geometry.

**Figure 3.**The computational domain geometry and mesh (5000 cells) used (

**a**) in the IHCP solution and (

**b**) in the numerical experiment; (

**c**) corresponding velocity field in the numerical experiment; (

**d**) the algorithm of the IHCP solver as used in the present study.

**Figure 4.**Distribution of (

**a**) HTC, (

**b**) surface temperatures, and (

**c**) heat flux density as a function of the curve length for the numerical experiment (solid black lines) and IHCP (dashed red curves). A vertical black line marks the transition from the hemispherical tip to the straight annular section.

**Figure 5.**Distribution of (

**a**) HTC, (

**b**) surface temperatures, and (

**c**) heat flux density as a function of the curve length obtained from the IHCP calculations using the experimental data from Table 1. A vertical black line marks the transition from the hemispherical tip to the straight annular section.

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

Flow rate | L/h | 90 | 120 | 230 | 300 |

Inlet temperature | °C | 69.6 | 69.7 | 70.2 | 70.2 |

Outlet temperature | °C | 70.8 | 70.5 | 71.0 | 70.7 |

Cooling power | W | 120 | 120.0 | 213 | 190 |

Copper temperature | °C | 600 | 600 | 550 | 550 |

Steady-state readings | #1 | 173.8 | 171.1 | 229.0 | 211.2 |

from thermocouples (°C): | #2 | 139.1 | 136.7 | 173.4 | 162.1 |

#3 | 107.2 | 105.3 | 127.7 | 120.9 | |

#4 | 80.6 | 79.5 | 84.8 | 82.6 | |

#5 | 73.1 | 72.5 | 73.8 | 72.9 | |

#6 | 70.9 | 70.5 | 70.5 | 70.1 |

Parameter Name | Value |
---|---|

ddtScheme | steadyState |

grad (T) | Gauss linear |

Laplacian (k,T) | Gauss linear Corrected |

interpolation | linear |

snGradSchemes | orthogonal |

under-relaxation | none |

linear solver | PCG FDIC (tolerance 1 × 10^{−14}) |

nNonOrthogonalCorrectors | 1 |

minimizer | BOBYQA (bounded quadratic) |

maxTime (s) | 300 |

initial HTC (Wm^{−2}K^{−1}) | 10,000 |

lower bound of HTC (Wm^{−2}K^{−1}) | 0 |

upper bound of HTC (Wm^{−2}K^{−1}) | 100,000 |

**Table 3.**Numerical experiment vs. IHCP calculation: temperatures at the thermocouple tips (the red circle markers in Figure 1b).

Thermocouples: | #1 | #2 | #3 | #4 | #5 | #6 |
---|---|---|---|---|---|---|

Temperatures in | ||||||

num. experiment (°C) | 190.447 | 143.354 | 113.892 | 87.237 | 77.483 | 71.516 |

IHCP calculation (°C) | 190.447 | 143.353 | 113.891 | 87.237 | 77.483 | 71.516 |

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

Goal function $F$ (K^{2}) | 0.242 | 0.135 | 0.0109 | 1.22 × 10^{−7} | |

${T}_{Pi}-{T}_{i}$ | #1 | −0.093 | −0.099 | −0.00034 | −0.000033 |

for thermocouples (K): | #2 | 0.202 | 0.215 | 0.00115 | 0.000062 |

#3 | −0.144 | −0.146 | −0.00261 | −0.000128 | |

#4 | −0.112 | 0.055 | 0.01609 | 0.000026 | |

#5 | −0.247 | 0.115 | −0.05003 | 0.000007 | |

#6 | 0.314 | 0.203 | 0.09037 | −0.000318 |

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**MDPI and ACS Style**

Bohacek, J.; Mraz, K.; Krutis, V.; Kana, V.; Vakhrushev, A.; Karimi-Sibaki, E.; Kharicha, A.
Experimental and Numerical Investigations into Heat Transfer Using a Jet Cooler in High-Pressure Die Casting. *J. Manuf. Mater. Process.* **2023**, *7*, 212.
https://doi.org/10.3390/jmmp7060212

**AMA Style**

Bohacek J, Mraz K, Krutis V, Kana V, Vakhrushev A, Karimi-Sibaki E, Kharicha A.
Experimental and Numerical Investigations into Heat Transfer Using a Jet Cooler in High-Pressure Die Casting. *Journal of Manufacturing and Materials Processing*. 2023; 7(6):212.
https://doi.org/10.3390/jmmp7060212

**Chicago/Turabian Style**

Bohacek, Jan, Krystof Mraz, Vladimir Krutis, Vaclav Kana, Alexander Vakhrushev, Ebrahim Karimi-Sibaki, and Abdellah Kharicha.
2023. "Experimental and Numerical Investigations into Heat Transfer Using a Jet Cooler in High-Pressure Die Casting" *Journal of Manufacturing and Materials Processing* 7, no. 6: 212.
https://doi.org/10.3390/jmmp7060212