Searching for irregular pin-fin shapes for high temperature applications using deep learning methods

https://doi.org/10.1016/j.ijthermalsci.2020.106746Get rights and content

Abstract

Pin-fins have been widely used in cooling channels to enhance internal heat transfer. For over thirty years, the literature has been using regular pin-fin shapes or identical pin-fins arrays. However, it was expected that an efficient pin-fin channel should have irregular pin-fin shapes and localized changing shapes along the streamwise direction. These new degrees of freedom for pin-fins were not well explored due to the lack of data processing method in the past. With the aid of the advanced deep learning techniques arising in these years, this study proposed a new optimization approach for pin-fins using the pix2pix networks and the Genetic Algorithms. A simulation dataset with 300 random spline pin-fin shapes was generated using Computational Fluid Dynamics. Two surrogated models were trained and tested to predict the temperature distributions on the external surfaces and pressure distributions in the middle section of the channel. Five optimized geometries were generated using different combinations of cooling objectives and pressure objectives. Based on the comprehensive results proliferated by the machine leaning methods, detailed sensitive analysis and response analysis were conducted for each input parameter. The optimized results indicated several general suggestions for pin-fin designs under different objectives: (a) square pin-fins could fit better with the cooling requirement and pressure constraints for the most upstream regions, (b) enlarging the opening area of middle stream pin-fins could elevate the uniformity of temperature, (c) streamlined pin-fins helped reduce the pressure drop. This effort was expected to provide a reference to explore cooling channel configurations geometries within a larger degree of freedom.

Introduction

Pin-fins are typical turbulence generators used in heat exchangers or cooling channels. Through the wakes and wall shears created by the cylinders, pin-fins can produce intensive mixing inside the channels and bring in a substantial heat transfer enhancement on the fluid/solid interfaces. The special advantage that distinguish pin-fins from other turbulence generators is its flexibility to be located in narrow channels with limited height. In the meantime of enhancing heat transfer, pin-fins could also act as a good structural support for cooling channels. These advantages consequently result in many implementations of pin-fins in hot section components such as gas turbine blades and chip cooling systems.

Numerous studies have been conducted on pin-fins in the past few decades. The major focus was on the impact of geometric parameters on heat transfer and pressure drop. For a conventional pin-fin arrays, the main parameters included pin-fin diameters, pin-fin height, streamwise pitches, spanwise pitches, while there were also many subsidiary parameters that should be considered when arranging pin-fins in a cooling channel, such as channel width, channel shrinking and flow turning. Early studies in the literature, such as the effort by Metzger et al. [1]and Van Fossen [2], aimed at quantifying the impact of geometric parameters and developing correlations for Nusselt number and friction factors, which helped the design of cooling channels. Chyu et al. [3] further analyzed the effect of pin-endwall fillet on heat transfer and pressure drop, and reported that the existence of fillet generally reduced the heat transfer of pin-fin channels. In the last twenty years, the development of 2-D measurement techniques (e.g. infrared imaging [4] or thermochromic liquid crystals [5]) continuously enriched the knowledge of thermos-fluid phenomena in pin-fin channels. These high dimension data helped the researchers to evaluate the uniformity of heat transfer in pin-fin channels, and pushed the investigations to delicate spatial and local details. However, the 2-D data were not compatible for empirical regressions with analytical formula, and for over thirty years, designing of pin-fin channels have been based on 0-D correlations.

The cross section shape of pin-fins was an important factor that affected the heat transfer and pressure loss of a cooling channel. Conventionally, the shape of pin-fins was described by regular geometries, such as rectangle, triangle, cylinder, elliptic, tear drops or airfoil-shape. The study of Lyall et al. [6]considered the variation of aspect ratios of pin-fins and suggested that a smaller spanwise pitch between pin-fins could result in a higher heat transfer coefficient. Siw et al. [7] experimentally tested triangular-shaped pin-fin channels and semicircular pin-fin channels, and observed more uniform heat transfer in triangular shaped pin-fin channels and lower pressure drop in circular pin-fin channels. Jin et al. [8] numerically compared a series of pin-fin shapes, including circular, elliptic, oblong, teardrop, lancet and NACA shapes, and concluded that the NACA pin-fins showed the lowest friction factor while maintaining high heat transfer coefficients. Lorenzini et al. [9] designed a local cluster of micro pin-fins to enlarge the surface area at the location of hot spots in an integrated circuit architecture. Results indicate the capability of pin-fin clusters in reducing the maximum temperature of a local hot spot. Sarvey et al. [10] integrated the concept of non-uniform pin-fin distributions and irregular pin-fin shapes into a cooling configuration for microelectronic systems. The experimental investigations included cylindricall, hydrofoil micropin-fins as well as local clustered arrays, which further identified the combination of hydrofoil with clustered arrays to be a good structure to reduce the hot spot temperature. A most recent effort presented by Feng et al. [11] studied the potential of gradient distribution of pin-fins and annular-cavity shapes in an integrated circuits chip cooling system. Results demonstrate a good characteristic of the gradient distributed annular-cavity pin fins in increasing heat transfer area and eliminating flow dead zone, which further resulted in maximum reduction of local hotspot temperature and total thermal resistance. The above literature indicated three major principles: (a) irregular cross section of pin-fins could bring in more mixing and vortices to enhance the heat transfer, (b) a well-designed irregular pin-fins could probably induce a lower flow resistance than cylindrical pin-fins, c) a localized size distribution of pin-fins could help eliminating high temperature in specific local regions. In a practical design process, heat transfer and pressure drop should be carefully balanced, while in most cases, additional heat conduction induced by the pin-fins also needed to be counted in a cooling design. As such, the design of pin-fin channels was a multi-objective and conjugate problem.

Although previous efforts formed a systematic comparison among different pin-fin shapes under different conditions, very few studies on pin-fins have considered searching for optimized pin-fin shapes using optimization algorithms. Moreover, the published results did not investigate pin-fin shapes within a sufficiently large degrees of freedom (such as spline cross section or 3-D shapes) to expand the performance boundary of pin-fin channels. Ranut et al. [12] []presented a multi-objective optimization approach for pin-fin pipe heat exchangers using the Genetic algorithm (GA) and the Kriging surrogate modeling method. Heat transfer data and pressure drop data were modelled using the Kriging interpolation method. The optimization process could be initialized using 20 samples. Different pipe shapes were obtained under different objective functions. One reason that obstructed high degrees of freedom was the lack of efficient data modeling techniques to support optimization algorithms which typically required evaluation of over 103 individual geometries. With the growth of computational capability and data regression technology, optimization of high complexity objects such as pin-fin channels became possible. An attractive avenue of achieving this scope was integrating the advanced deep learning techniques with optimization algorithms.

The deep learning techniques arising from 2012, with the inventing of the Back Propagation algorithm and the Rectified Linear Unit loss function, have shown great advantages in data modeling in many living and industrial field, such as image recognizing and system controlling. Among the successful models, convolutional neural networks (CNN) appeared to be a good model category to regress flow field or temperature field in a 3-D space or on a 2-D surface. As for a pin-fin cooling channel, the most important data was located at the solid/fluid interfaces or several cross sections, e.g. the heat transfer coefficient distribution on the interfaces, the temperature distribution on the internal/external surfaces of the cooled solid part, or the flow field along the streamwise direction. Conventional analytic equations or correlations were not capable to regress such high dimensional data due to their simple function formulas. Distinguishing from analytic methods, a neural networks model could establish mapping between any domain subsets based on the “Universal approximation theorem” [13]. For a geometry designing problem, a mapping between the geometry and the temperature field could be an option. With a CNN model, such mapping task could be converted into an image translation problem, by expressing the geometric features and required physics field as images. Previous research example of the authors [14,15] indicated that the 2-D temperature distribution on a perforated surface could be accurately captured by a CNN model [16], which kept the required temperature information on the entire surface while only consuming several micro seconds for prediction.

It should be noted that the CNNs were a very big category of machine learning models, and many of them was focusing on image recognizing or image segmentation, which was not closely relevant to a thermos-fluid problem. Based on the experience of the literature and the authors’ previous studies [[17], [18], [19]], it was recommended to use a pix2pix model for the data regression tasks in a pin-fin channel. These previous studies introduced the concept of “image translation” into the field of transportation phenomena, which showed a good performance of the pix2pix model to regress heat transfer/cooling data. The pix2pix model was a special model framework that belonged to the Conditional Generative Adversarial Neural Networks (cGAN) [[19], [20], [21]] and CNNs. The pix2pix model integrated two most recent concepts of machine learning: (a) the U-net, which was responsible for translating one set of images/fields into another set of images/fields by convolution down sampling, convolution up sampling and skipping connections; (b) adversarial training, which equivalently provided an automatic loss function mechanism for the networks that ensured a maximum likelihood between the training images and the predicted images. Pioneer work that introduced pix2pix models to transportation phenomena was published by Farimani et al. [19], which validated the capability of the model in multiple tasks including flow field, heat conduction and mass transferring. Such machine learning methods fit well with the objective of the present study, i.e. predicting cooling performance and pressure drop for any given pin-fin shape.

The present study aimed at optimizing the shape and arrangement of a pin-fin array in an air cooling channel serving at high pressure and high temperature conditions. A workflow integrating Computational Fluid Dynamics (CFD), pix2pix and GA was adopted to search for optimized geometries that could cool the solid domain uniformly while consuming acceptable pressure. Conjugated heat transfer simulations were conducted for over 300 geometries to provide datasets for the machine learning approach. A pix2pix model was established to regress the 2-D temperature and pressure distributions, which served as an ultra-fast surrogated model for the GA process. Results were obtained for several objective functions, including averaged temperature level, temperature uniformity and pressure drop. The outcome of this effort was expected to provide suggestions to the industry regarding how to design non-cylindrical pin-fins according to the cooling demands and pressure constraints.

Section snippets

Geometries and parameter matrix

The selected prototype geometry was a cooling channel which contained three rows of pin-fins. The external solid wall of the cooling channel had a thickness of t = 1 mm and a length of L = 10 mm. The geometry was periodic along the width direction(Y), with a period of W = 3 mm. The height of the channel (along the Z direction) was H = 3 mm. For the convenience of defining flow boundary conditions, the fluid domain was extended at the inlet and outlet for a length of 30 mm. The original point of

The GA optimization setups

The pin-fin locations and pin-fin shapes of the cooling channel were optimized using a GA approach integrating with the surrogated model obtained from the machine learning process. The GA code was provided by the Python Geatpy package created by Jazzbin et al. [23]. The variable boundaries are listed in Table 4. The encoding method was “Binary Gray”. The population size of the GA optimization was 100. The maximum number of generation was set at 200. The sampling method was random selection

Regression results of the Pix2Pix model and model analysis

The pix2pix model was trained using a NVIDIA Quadro RTX 4000 GPU processor. The algorithm to solve the trainable parameters in the network was the Adam optimization algorithm. The initial learning rate was set at 1 × 10−5. The weight of the adversarial loss term in Eq. (3) was 1. The weight of the (first-ordered norm error) L1 loss term was 100 for the generator. The batch size of each training step was 1. The number of training samples was 200 and the number of testing samples was 100. For

Optimization results of the pin-fin arrays and discussion

With the optimization strategies proposed by this study, five optimized geometries were generated and re-evaluated via CFD. Each optimization process evaluated 2 × 104 points and consumed around 1.5 h. Fig. 8 compared the CFD-predicted temperature and pressure of the optimized geometries and the 200 training samples initially evaluated for regression. The horizontal axis was μ(T*) + 3σ(T*) while the vertical axis was Δp*. The optimized points were marked with a square frame, and each

Conclusion

The present study integrated the advanced deep networks with CFD and GA, and established an optimization workflow to search for efficient irregular pin-fin shapes for cooling channels working under high temperatures. Two pix2pix models were trained with 200 samples and tested by 100 samples to regress the non-linear mapping between the surface middle cross section geometry and the pressure/temperature distributions, respectively. With the aid of the deep learning models, detail analyses were

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work is supported by National Science Foundation of China No. 51906139 and the Shanghai Sailing Program (19YF1423200).

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