The pulsating heat pipe is a very promising heat dissipation device to address the challenge of higher heat-flux electronic chips, as it is characterised by excellent heat transfer ability and flexibility for miniaturisation. To boost the application of PHP, reliable heat transfer performance evaluation models are especially important. In this paper, a heat transfer correlation was firstly proposed for closed PHP with various working fluids (water, ethanol, methanol, R123, acetone) based on collected experimental data. Dimensional analysis was used to group the parameters. It was shown that the average absolute deviation (AAD) and correlation coefficient (r) of the correlation were 40.67% and 0.7556, respectively. For 95% of the data, the prediction of thermal resistance and the temperature difference between evaporation and condensation section fell within 1.13 K/W and 40.76 K, respectively. Meanwhile, an artificial neural network model was also proposed. The ANN model showed a better prediction accuracy with a mean square error (MSE) and correlation coefficient (r) of 7.88e-7 and 0.9821, respectively.

With the rapid development of the electronics industry, chips are becoming increasingly compact, generating a larger amount of heat in an ever-smaller physical size. This additional heat must be efficiently dissipated by the thermal network. Otherwise, the chips will suffer significantly from lower efficiency, short life-span, and even physical damage [

Reliable prediction of the performance of PHP has been a hot research topic as it plays an essential role to promote its application. Generally, there are two basic methods to predict performance of the PHP, classic heat transfer correlations and ANN network models. For classic heat transfer correlations, typical heat transfer dimensionless numbers, like the Kutateladze number, Morton number, Jackob number, Bond number, and Prandtl number, are correlated to study the influence of various geometric, property and operating parameters. In 2003, Khandekar et al. [

The existing heat transfer correlations and ANN models provide remarkable solutions to estimate the performance of the PHP. However, the proposed heat transfer correlations were commonly regressed from the specific range of the experimental data, and with relative low accuracy. More effort is still required to improve the accuracy of these models, and further validate the suitable application range. Despite the excellent accuracies of the ANN models, their application is highly dependent on the collected data and with certain parameters as the input. In this paper, the impact of typical geometrical parameters, properties of working fluids, and operating parameters were considered by dimensional analysis first to obtain the dimensionless numbers. A heat transfer correlation was proposed based on these dimensionless numbers. Following that, an ANN model was also presented to correlate the thermal resistance of PHP, taking advantage of its high accuracy.

For performance prediction, the most important factor is to figure out the relation between the heat flux and the related geometrical, property, and operating parameters,

To analyse the impact of various parameters on the performance of PHP, the dimensional analysis was used to get the dimensionless parameters. This method is conducted to group the influence of different parameters and generalise various operational conditions. The impacts of geometric, property, and operating parameters were considered. The correlation was presented as,

The dimensions of _{l}, _{p}, _{i}, _{e}, ^{-1}^{-2}^{3}, ^{-3}, ^{-3}^{-1}^{2=-1}^{-1}, ^{-2}^{2}^{-2}^{-1}, ^{-3}, ^{-2}, respectively. There are ten parameters, and four basic dimensions in _{l},

Thus, we get the correlation

It is well-known that the physical properties change with temperature, and a reference temperature is needed to calculate them. To evaluate the thermal-physical properties in the

To obtain the performance prediction models, the experimental data for the bottom-heated PHP with various inner diameters, filling ratios, turn numbers, evaporation lengths, and working fluids were gathered from the literature [

Parameters | Ranges |
---|---|

Working fluids | Water, ethanol, methanol, R123, acetone |

Evaporation section length/(mm) | 8~100 |

Filling ratio | 0.2~0.9 |

Inner diameter/(mm) | 0.8~2.45 |

Number of turns | 2~20 |

Heat flux/(W∙m^{-2}) |
494~134160.2 |

Based on the experimental data and dimensional analysis presented above, the following heat transfer correlation was obtained to predict the performance of PHP based on least square method.

The prediction performance of the heat transfer correlation is shown in

It is noteworthy to further analyse the prediction characteristics of the correlation

The influence of thermal resistance on the prediction of Δ^{2}.

A possible reason which may contribute to the great deviations of the correlation for higher thermal resistance and lower heat flux is when the heat flux is in a lower level, the PHP experiences a condition of pre-startup or quasi-startup, in which status the operating of PHP is not stable, and is characterised by different flow patterns compared to the stable oscillation at a higher heat flux level [

A new, fully connected ANN model was defined to correlate the performance of PHP, to take advantage of excellent it accuracy in non-linear analysis. The collected experimental data were divided into training, validation, and testing data sets randomly at the ratio of 70%, 15%, and 15% respectively. To avoid too complex an ANN model, only one hidden layer was used. The parameters considered in above section were used as the input of ANN model. In an ANN model, the outputs of each node in

Therefore, the output of the ANN model was presented by,

For all input data, the following function was used to pre-processes the input data,

The transfer function is used to pre-process the data. The following sigmoid function was used,

The error of the ANN model can be evaluated by,

The linear function was implemented for the output layer. Due to great stability, flexibility, and adaptability of Levenberg- Marquardt algorithm, it was selected as the learning algorithm of the ANN model. The prediction performance of the ANN model is usually described by correlation efficient (r) and mean square error (MSE), and they are defined by the following equations [

A trial-and-error method was implemented to find the optimal hidden layer neuron number. The analysis result is shown in

The predictions for the training, validation, testing, and total data set are further shown in

Node | Hidden layer weights | Hidden |
Output |
Output |
||||||
---|---|---|---|---|---|---|---|---|---|---|

_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
||||||

1 | –1.248 | –3.632 | –0.1192 | 1.235 | 1.356 | 0.1696 | –1.476 | –3.939 | –2.206 | 0.9240 |

2 | –2.195 | –2.037 | –0.3265 | –3.480 | –4.121 | 0.2727 | 7.667 | 3.185 | 1.146 | |

3 | –3.092 | –0.9589 | –0.4393 | 0.8211 | –0.5356 | 0.3144 | 1.478 | 1.145 | –1.637 | |

4 | 0.9958 | –2.089 | –1.581 | –2.450 | –4.904 | –0.06707 | 4.561 | 1.457 | –2.719 | |

5 | –0.5800 | 1.093 | –4.606 | –3.526 | –4.279 | –0.1185 | 5.356 | 2.816 | –2.347 | |

6 | –0.0728 | 0.09350 | –1.346 | –1.154 | 2.619 | –0.0179 | –1.913 | –0.134 | –1.878 | |

7 | 1.241 | 1.530 | –0.3851 | –1.470 | –3.255 | –0.1292 | 1.940 | 0.5034 | –1.120 | |

8 | –0.7466 | 1.611 | –3.850 | –1.032 | 1.775 | –0.00359 | –2.568 | 0.0328 | 0.6827 | |

9 | 0.3775 | 0.01606 | 3.352 | 1.300 | 3.228 | 0.04450 | –3.026 | –1.116 | –3.333 | |

10 | –5.753 | 2.455 | –1.088 | 0.1840 | 3.278 | 0.2349 | 0.1813 | –1.753 | –1.390 | |

11 | –3.271 | –0.7660 | –1.067 | –0.2324 | –1.677 | 0.2798 | 0.5518 | 1.334 | 1.901 | |

12 | 0.8520 | 2.641 | 0.2677 | 0.4395 | –1.836 | –0.1138 | 1.599 | 3.627 | –5.279 |

The influence of thermal resistance on the prediction of Δ

In this paper, the dimensional analysis was conducted to group various influence parameters to predict the heat transfer performance of PHP with various working fluids (water, ethanol, methanol, R123, acetone). A heat transfer correlation was firstly presented based on the collected data. The coefficient correlation and AAD of the correlation were 0.755 and 47.65%, respectively. Further analysis suggested that most great deviation occurred for higher thermal resistance (>2 K/W) or lower heat flux (<40000 W/m^{2}). For 95% of the data, the prediction of thermal resistance and temperature difference between evaporation and condensation section were 1.13 K/W and 40.76 K, respectively. In addition, an ANN model was also proposed to better predict the performance of PHP, considering its great advantage in solving and correlating complex non-linear problems. The proposed ANN model has 7 nodes, 12 nodes, and 1 node in input layer, hidden layer, and output layer, respectively. It was found that the prediction agreed with the experimental data very well, with the MSE and R at 7.88e-7 and 0.9821, respectively. Furthermore, the ANN model also showed good accuracy in predicting the Δ

Despite that, further studies regarding the internal heat transfer characteristics and oscillation flow patterns of the PHP in various conditions still need to be conducted. Revealing them will help to build more accurate models and boost the application of PHP.

bias

covariance

_{p}

specific heat (J/(kg∙K))

_{i}

inner diameter (m)

gravity (m/s^{2})

length (m)

number of turns

Power (W)

heat flux (W/m^{2})

thermal resistance (K/W)

correlation coefficient

temperature difference (K)

temperature (K)

input

output

data value

thermal conductivity (W/(m∙K))

dynamic viscosity (Pa∙s)

_{i}

dimensionless number

_{l}

liquid density (kg/m^{3})

surface tension (N/m)

filling ratio

weight

evaporation section

experiment

predicted results

_{2}O

_{3}–water nanofluids