Modeling of convection heat transfer of supercritical carbon dioxide in a vertical tube at low Reynolds numbers using artificial neural network

Abstract

Today, many researches have been directed on heat transfer of supercritical fluids; however, since the analysis of heat transfer in these fluids founded by a mathematical model based on the effective parameters is complicated, so in this paper, a group method of data handling (GMDH) type artificial neural network are used for calculating local heat transfer coefficient h_x of supercritical carbon dioxide in a vertical tube with 2 mm diameter at low Reynolds numbers (Re < 2500) by empirical results obtained by Jiang et al. [1].At first, we considered h_x as target parameter and G, Re, Bo^⁎, x⁺ and q_w as input parameters. Then, we divided empirical data into train and test sections in order to accomplish modeling. We instructed GMDH type neural network by 80% of the empirical data. 20% of primary data which had been considered for testing the appropriateness of the modeling were entered into the GMDH network. Results were compared by two statistical criterions (R² and RMSE) with empirical ones. The results obtained by using GMDH type neural network are in excellent agreement with the experimental results.

Introduction

Supercritical fluids have been used in various industries and processes for several years. These applications include increasing the drug solubility in pharmaceutical industry [2], extraction of perfumes and essences from plant material [3], [4], production of ε-caprolactam from cyclohexanone-oxime using a supercritical water microreactor and formation of polymer particles in chemical engineering process [5], [6]. Convection heat transfer in vertical channels is an important research aspect of heat transfer which has been extensively studied in engineering applications. Although, vertical channels have principally simple geometry, but because of buoyancy effects, non-uniform fluid thermophysical properties and fluid velocity variations due to axial changes of bulk temperature, heat transfer process may be so complicated. Quick changes of fluid properties in temperatures near T_pc (T_pc is defined as a temperature, in which C_p has the highest value) induces mutual effects between temperature and fluid fields that results in an irregular buoyancy which affects flow of the average field. These conditions produce many changes in heat transfer properties of supercritical fluids compared to fluids in normal pressures. But we may abandon the effects of the fluid thermophysical properties changes, when the studies are performed on the fluid in constant pressure and temperature [7].

In the last few years, many investigations have been performed on convection heat transfer of supercritical carbon dioxide in channels. He et al. [8] modeled convection heat transfer of supercritical carbon dioxide in a vertical tube with an internal diameter of 0.984 mm and a heated length of 0.055 m and pressures 8.5 and 9.5 MPa and in Reynolds number Re ranged between 9000 and 30,000 numerically. They compared the results of this simulation with empirical ones and observed that in some cases, their results coincided with empirical results but in other cases their simulation details indicated dramatic differences with empirical results. Song et al. [7] studied heat transfer of supercritical carbon dioxide in a vertical tube with 4.4 mm and 9 mm diameters and 2 m heated length in different mass and wall heat fluxes and 8.12 MPa pressure. They obtained wall temperatures and heat transfer coefficient for various inputs situations. In addition, many researchers such as Asinari [9], Guardo et al. [10], Shan et al. [11], Cheng et al. [12] and Sharabi et al. [13] also have performed investigations on heat transfer of supercritical carbon dioxide.

Regarding the high complexity of heat transfer of supercritical fluids, as well as, disability in their exact theoretical analysis, empirical experiments can be effective to identify this complex mechanism and we may recognize the effect of important factors on this process by empirical experiment results. Results of empirical data are useful to evaluate and to complete the theoretical investigations and it is possible to provide a mathematical model by these results. Researchers were always interested in system identification and modeling of complicated processes by input and output data. In fact, methods of system identification in order to model and anticipate complex and unknown systems are used in many aspects by input and output data.

In the recent few years, theoretical research about information processing has been increasingly developed in order to use it in applied aspects, particularly for problems that are not soluble or easily solved. This interest has been specially displayed much more in the development of intelligent systems which are based on the empirical data. Artificial neural networks are among the systems which transfer the knowledge and rules exist beyond the empirical data into the network structure by their processing. Because artificial neural network don't consider any presuppositions about statistical distribution and characteristics of the data, they are practically more efficient than common statistical methods. On the other hand, they use a non-linear approach to create a model, so when encountered with the complicated and non-linear data, these networks may express such a data much more accurately as a defined model. In recent years, many investigators such as Taymaz and Islamoglu [14], Karadağ and Akgöbek [15], Tahavvor and Yaghoubi [16], Varol et al. [17] have performed researches to apply the artificial neural network for modeling of the engineering heat transfer processes.

Theoretically, it is necessary that mathematical relations are defined between input and output data in order to model a system. Considering the concept of the systems, to find such mathematical models are so difficult. On the other hand, methods of Norm calculations which are known as fuzzy logic, neural networks and genetic algorithms are highly capable of controlling and identifying non-linear complex systems. Many researches have been performed about the use of evolutionary approaches as effective tools for system identification. Among these, group method of data handling is a self-organizing method in which complex models are gradually formed on the basis of multiple primary data of evolved inputs and outputs. Thus, using GMDH type neural network there is no need to know mathematical introduction. In other words, we may use GMDH type neural network for modeling without any professional information about systems. The main objective of GMDH type neural network is to establish an analytical function which is formed based on a feedforward network in which any element is a second order of transfer function and its coefficients are obtained by returning approaches [18].

Considering the complexity of analytical and numerical methods as well as very high costs of empirical experiments, artificial intelligence methods are the appropriate choice to model heat transfer of supercritical carbon dioxide. In this study, a model was created for convection heat transfer of supercritical carbon dioxide in a vertical tube at low Reynolds numbers by GMDH type neural network. In this model, local heat transfer coefficient h_x was the target variable (output parameter) and mass flux (G), Reynolds number (Re), buoyancy number (Bo^⁎), dimensionless axial coordinate (x⁺) and heat flux on the inner tube surface (q_w) were chosen as designing variables (input parameters).