On the evaluation of thermal conductivity of nanofluids using advanced intelligent models
Introduction
Nanofluids are colloidal suspension of nanoparticles (NPs) in a base fluid. NPs must have at least one dimension with size between 1 and 100 nm, hence portray extremely large total surface area. This in turn cause unique thermal, mechanical, magnetic, electrical and optical properties [1]. The NPs may be metals (Al, Fe, Ag, Ni, Cu, Au, etc.), metal alloys (Al2Cu, Ag2Al, etc.), oxide ceramics (Al2O3, Fe3O4, ZnO, CeO2), nitride/carbide ceramics (AlN, SiC, etc.), semiconductors (TiO2, SiO2, SnO2, etc.), graphene and graphene oxide nanosheets, single and multi-walled carbon nanotubes (SWCNTs, MWCNTs), diamonds, and other particles [1]. Uniform and stable dispersion of small amount of these nanometer-sized particles into a base fluid can provide highly improved transport and thermophysical properties for the nanofluids. For instance, nanofluids possess significantly increased heat transfer capability due to enhanced TC and thermal diffusivity. Moreover, nanofluids have adjustable TC, viscosity and surface wettability suited for different applications simply through altering particle concentration [2]. Nanofluids require lower pumping power in comparison to pure liquids to reach a specific heat transfer intensification [3]. Exclusive features of nanofluids make these new class of fluids useful in many industrial and engineering applications. Many researches have explored potential application of nanofluids in various filed such as: transportation, nuclear cooling, heating buildings, industrial cooling, pollution reduction, space, solar absorption, energy storage, magnetic sealing, lubricity, drug delivery, antimicrobial activity, drilling, enhanced oil recovery and asphaltene precipitation inhibition [[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]]. Many review articles are available, which summarize the contribution of nanofluids for different applications [3,[16], [17], [18], [19], [20], [21], [22], [23], [24]].
TC is defined as the ability of a material to conduct or transmit heat [3]. High TC is one of the main characteristics of nanofluids as it causes better heat transfer efficiency. Although conventional heat transfer media, e.g. water, oil, propylene glycol and ethylene glycol, are widely used, they often offer inadequate performance for industrial equipment. Solid materials, on the other hand, exhibit superior TC. Consequently, addition of solid particles to traditional heat transfer mediums can enhance their TC, hence, enhance their heat transfer capacity. Maxwell in 1881 recommended the use of a mixture approach suggesting that dispersing of solid particles in liquids can dramatically increase the effective TC [25]. Since then, numerous studies were carried out to explore thermophysical properties of suspension of high TC solid particles in liquids. Early studies focused on enhancement of TC by dispersing solid particles with particle size in millimeter or micrometer ranges in conventional liquids [26,27]. Although these studies reported successful augmentation of TC, many problems such as instability of the mixtures, high potential of severe clogging and corrosion arose. Accordingly, significant increase in the pumping power and operating costs was encountered [28,29]. Developments in nanotechnology led to the emergence of novel heat transfer fluids with optimized thermal performance by Choi in 1995 [30].
Measurement of the TC of nanofluids over all of the desired operation conditions such as different temperatures and NP volume fraction and volume, a large volume of experiments is required. Although experimental measurement of the TC can provide accurate and reliable data, it is time-consuming, expensive and requires the availability of specific equipment and facilities. Consequently, numerous models were developed to predict the TC of nanofluids. Generally, the mathematical models for TC of nanofluids can be classified as theoretical models, empirical correlations and computer-aided models.
The model proposed by Maxwell [25] was likely the first correlation to calculate TC of suspensions knowing the TC of their constituent phases and the volume fraction of the solid phase, as shown in Eq. (1). Maxwell [25] considered very low volume fraction of suspended spherical particles, hence neglected the interaction among particles, and solved the Laplace equation for the steady state temperature outside the spherical particle to obtain the following equation for TC.where Knf is the mean TC of the suspension, Kp is the TC of the particles, Kbf is the TC of the base fluid and φ is the volume fraction of the particles. This correlation is only a first order approximation and can only predict TC of dilute dispersions of micro- or millimeter-sized spherical particles.
Hamilton and Crosser [31] proposed a modified version of Maxwell model through considering an empirical shape factor, n, to extend the applicability of Maxwell's correlation to non-spherical particles. The model is given as follows:and n = 3/ψ, where ψ is the particle sphericity and represent the ratio of the surface area of a sphere having the same volume as the particle to the surface area of the particle.
When the particle concentration is high in the mixture the predictions of Maxwell model is no longer accurate. Bruggemann [32] suggested a method to extend Maxwell model to higher particle volume fractions. Bruggemann assumed a gradual increase in the TC of the medium surrounding a particle from Kbf to the final TC of the mixture, Knf, as the particle volume fraction slowly increases. The correlation proposed by Bruggemann is recommended for calculating various TC's of mixed bodies of isotropic materials. The mathematical expression of this model is [32]:
This model gives the same outcome as the Maxwell and Hamilton and Crosser [31] models for spherical particles with ψ = 1. It is expressed as [33]:
Xue [34] introduced a theoretical model for calculating the effective TC by considering the very large axial ratio and the space distribution of the carbon nanotubes (CNTs). Based on Maxwell’s theory, Xue presented the following equation [34].
Eq. (5) overcomes the inaccuracy of models developed for spherical particles or rotational elliptical particles with small axial ratio.
Rayleigh [35] investigated the TC of a mixture with a uniform particle size distribution for the first time. For a simple cubic array of similar spheres, Rayleigh proposed that at the center of the reference sphere the ambient temperature gradient is equivalent to the summation of the fields generated by the surrounding spheres plus the average temperature gradient. Rayleigh solved the Laplace equation for temperature gradient inside and surrounding a sphere by applying the superposition property. He proposed the following Equation [35]:
Meredith and Tobias [36] extended Rayleigh's model and proposed a model that provides better results at high particle volume fraction. They used a different potential function and took into account the higher terms in the series expansion [36].
In general, a nanofluid TC model should provide the base fluid TC when the particle volume fraction is zero and the NP TC when the particle volume fraction is 1. Based on this simple theory, a general equation (Eq. (8)) called mixture rule is introduced by Landau and Lifshitz [37].
Landau and Lifshitz [37] assumed the mixture of the base fluid and suspended solids as an isotropic and humongous medium and considered the power value (n) in the mixture rule [38,39] equal to 1/3. By taking an average over large volume, Landau and Lifshitz proposed the following equation:
Lichtenecker introduced a logarithmic mixing rule for calculating TC of sols as follows [40]:
Yu and Choi [41] introduced a new mechanism suggesting that a nanolayer to behave as a thermal bridge connecting the solid NP and the base fluid which leads to enhanced thermal conductivities of nanofluids. The effect of this ordered nanolayer was captured by modifying Maxwell model [25] to the following equation [41]:
Timofeeva et al. [42] developed a model for estimating the improved TC of nanofluids based on effective medium theory. They used experimental data of alumina particles dispersion in water and ethylene glycol. Their method approximates the properties of a medium based on the properties and the relative fractions of its components, at low volume fraction of the NPs. In this simple theory, the particles are assumed to be fixed and immobile and the TC of the particles is much higher than that of the fluid.
Based on least-square curve fitting of experimental data, Maiga et al. [43] proposed two empirical correlations for predicting the TC of water- and ethylene glycol-based alumina nanofluids as followsfor water-based nanofluids, andfor ethylene glycol-based nanofluids.
Li and Peterson [44] evaluated the effects of temperature and particle volume fraction on the steady-state effective TC of two distinctive NP suspensions including Al2O3 and CuO. Based on the experimental results they proposed the following expressions [44]:for Al2O3-water suspension, andfor CuO-Water suspension.
Based on experimental data for oil-based nanofluids, Aberoumand et al. [45] developed a correlation relating the TC of an organosol to the volume fraction of the particles (0%≤φ ≤ 2%), temperature and NP type. They included the thermal conductivity of the NPs as input parameter [45].
Based on experimental data of aqueous Al2O3 and TiO2 nanofluids published by Pak and Cho [46], Buongiorno [47] proposed two correlations to calculate the TC of these nanofluids as followsfor Al2O3 hydrosol, andfor TiO2 hydrosol.
An empirical equation was developed by Godson et al. [48] for estimating TC of deionized aqueous silver nanofluids. The correlations were developed as function of particle volume fraction using experimental data for temperatures in the range of 50–90 °C and volume fractions ranging from 0.3 to 0.9.
Sundar et al. [49] used 189 experimental TC points; including their experimental results for Fe3O4 in a mixture of water and ethylene glycol as well as those for Vajjha and Das [50] for CuO nanofluid, to propose an empirical correlation for predicting TC. Their correlation is given below [49]
Based on experimental data for Fe3O4-water nanofluids in a temperature range from 20 °C – 60 °C, Sundar et al. [51] proposed the following equation:
Mintsa et al. proposed two correlations for predicting the TC of CuO and Al2O3 hydrosols by considering particle volume fraction as input parameter [52]:for CuO hydrosol, andfor Al2O3 hydrosol.
Abdolbaqi et al. [53] developed the following correlation from experimental data of thermal conductivity of bioglycol/water-based SiO2 nanofluids over a range of 30–80 °C and NP volume fraction of 0.5–2.0%
The same group proposed the following correlation for prediction of the TC of bioglycol/water-based TiO2 nanofluids [54]:
Lastly, for correlating the TC of bioglycol/water-based Al2O3 nanofluid in a range of 30–80 °C and volume fraction range of 0.5–2.0%, Abdolbaqi et al. proposed the following correlation from their experimental results [55]:
Based on experimentally measured data of TiO2 nanofluid in water and ethylene glycol mixture, Abdul Hamid et al. [56] developed a new correlation to describe TC for hydrosols in a range of 30–80 °C and particle volume fraction in the range of 0.5–1.5% as follows:
By regression analysis of experimental data of SiO2 NPs dispersed in polyalkylene glyco lubricant, Redhwan et al. [57] introduced a TC correlation with temperature and volume fraction as inputs:
A correlation for prediction TC ratio between Fe3O4 nanofluids and water was obtained by curve fitting experimental data [58]:
Sundar et al. [59] presented an equation for prediction of TC ratio of nanodiamond nanofluids and propylene glycol and water-base mixtures as a function of NP volume fraction and temperature,
Hojjat et al. [60] developed artificial neural network (ANN) models with three-layer feed forward to predict the TC of various non-Newtonian nanofluids including, CuO, Al2O3 and TiO2 dispersed in 0.5 wt% of carboxymethyl cellulose (CMC) hydrosol. They developed an ANN for predicting TC of single type of NPs by considering temperature and NP volume fraction as inputs. They also proposed an ANN with more general applicability to predict the TC of all investigated NPs as a function of temperature, NP volume fraction, and the TC of a given NP. Their experimental data covered a range of 5–45 °C and NP volume fraction of 0.1–4.0%. The model fit agreed well with the experimental data. Papari et al. [61] employed multilayer perceptron diffusion neural network (MLPDNN) to predict TC ratio of multi wall carbon nanotubes (MWCNTs) in different base fluids including oil, decane, ethylene glycol, distilled water and also single wall carbon nanotubes (SWCNTs) in base fluids of epoxy and poly methylmethacrylate. They used 43 experimental data in the model development while varying TC of the fluid and the carbon nanotubes volume fraction. The MLPDNN model fit agreed well with the experimental data and other literature models [61]. Longo et al. [62] developed a three-input and a four-input feed-forward ANNs to predict the TC ratio of metal oxide-water nanofluids. They used 30 experimental data for Al2O3 hydrosol and 40 experimental data TiO2 hydrosol. The three-input and the four-input models use temperature, volume fraction and TC of the nanofluids as input parameters. The four-input model consider NP cluster average size as an additional input parameter. Although both models provided good fit to the experimental TC of metal-oxide nanofluids, taking the cluster average size (four-input model) increased the accuracy of the model. Mehrabi et al. [63] used fuzzy C-means clustering (FCM) neuro fuzzy interference system (FCM-ANFIS) approach and genetic algorithm-polynomial neural network (GA-PNN) approach to model the TC ratio of Al2O3-water nanofluids as a function of NP volume fraction, temperature and Al2O3 NP size. They employed 125 experimental data during the model development. The model results matched very well the experimental data. Nevertheless, FCM-ANFIS model provided better fit compared. Mechiri et al. [64] represented an ANN to fit TC ratio of the nanofluid relative to base fluid as a function of temperature, NP volume fraction, NP diameter and ratio of TC of NP to the base fluid. Their model was based on experimentally measured TC for CuZn hybrid NPs dispersed in vegetable oils. The temperature range was 30–60 °C and volume fraction range was 0–0.5%. The model fit showed better performance than theoretical models and agreed well with the experimental results. Ariana et al. [65] presented a multilayer perceptron ANN with temperature, NP volume fraction and diameter as input variables to fit TC data of Al2O3-water nanofluids. Experimental data of 285 Al2O3-water nanofluids from different literature sources spanning temperatures ranging from 1 °C to 133.8 °C, Al2O3 NP sizes from 8 to 283 nm, NP volume fraction from 0.13–16% and TC ratio, relative to water, of 0.99–1.2902. The model fit very closely matched the experimental TC ratio of Al2O3 hydrosols. Moreover, the model outperformed other literature correlations. Ahmadloo and Azizi [66] proposed a multilayer perceptron (MLP) ANN to predict the TC ratio of various nanofluids based on 778 experimental data belonging to 15 different nanofluids. Their data set was obtained from literature pertaining to TC of nanofluids having TiO2, Al, Al2O3, Cu, CuO, Mg(OH)2 NPs in three different base fluids; including a type of transformer oil, ethylene glycol and water. Ahmadloo and Azizi's model fit experimental data very well. By considering 1273 experimental data belonging to 26 different nanofluids, Aminian [67] represented a cascade-forward neural network for predicting the TC ratio of nanofluids. Aminian [67] took into account the effect of temperature, NP size, NP volume fraction, TC of the NP and TC of the base fluid as input parameters. The data set consisted of different nanofluids; including Al2O3, TiO2, CuO, Cu, Ag, Al, SiO2, ZnO and MWCNT NPs with diameter ranging from 10 nm to 150 nm dispersed in ethylene glycol, water, mono ethylene glycol, refrigerants, transformer oil and other oils [67]. The average absolute deviation of the model was 3.06% and the coefficient of determination (R2) was 0.9309. This model was more accurate and reliable than the existing correlations. Afrand et al. [58] experimentally measured the TC of 30 different Fe3O4 magnetic nanofluids for volume fraction ranging from 0.1–3% and temperature ranging from 20 to 50 °C and modeled the data with feed forward ANN trained by Levenberg-Marquardt algorithm. The temperature and NP volume fraction were used as two input parameters to the ANN model and the TC as its output. The model fit strongly correlated with the experimental data. Khosrojerdi et al. [68] introduced a multilayer perceptron ANN to estimate the TC of graphene nanofluids. They developed their model based on experimentally measured data of graphene nanoplatelets suspended in deionized water between 25 and 50 °C and 0.00025–0.005 wt% graphene. The two input parameters were the temperature and NP wt% and TC was the model output. Khosrojerdi et al.'s model outperformed theoretical models and showed high accuracy fit.
In the present study, we establish accurate, widespread and inexpensive to use smart models to predict relative TC (RTC) of nanofluids. To this end, several intelligence models, including multilayer perceptron (MLP), radial basis function neural network (RBFNN) and least square support vector machine (LSSVM) were applied separately, and then combined under a single committee machine intelligence system (CMIS). An extensive database collected from published literature including 3211 experimental points were considered in the development of the models. Several statistical and graphical error analyses were utilized to assess the robustness of the introduced models and compare between them with literature theoretical and empirical models. Finally, trend analysis was conducted to assess whether the best proposed paradigm is capable of accurately predicting the trend of the independent variables.
Section snippets
Multilayer perceptron
ANNs are considered as one of the most promising machine learning techniques. ANNs are known to be very efficient in recognizing relationships describing complex systems mainly by identifying the links between the inputs and outputs of these systems [69]. The inspiration of this soft computing method and its conception come from the human brain and its method of learning [70]. Several classes of ANNs exist. MLP and RBF neural networks are the most widely applied [[71], [72], [73]].
An MLP model
Data gathering and preparation
To develop reliable predictive paradigms with a wide domain of applicability for predicting the TC of nanofluids, a large databank with high variety of nanofluids should be included. For this purpose, 3211 experimental data points were gathered from published literature [[48], [49], [50], [51],[53], [54], [55], [56], [57],59,[84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110],
Implementation of the models
Before highlighting the main outcomes of this study, it is constructive to detail the computational procedure followed in each model. As previously indicated, four different algorithms, viz. LM, BR, SCG and RB, were implemented in the learning phase of MLP. Accordingly, the following notations are considered: MLP-LM, MLP-BR, MLP-SCG and MLP-RB. Trial and error method was applied to determine the proper structure and the activation functions in each case. Two hidden layers with 20 and 10
Conclusions
This work presented a detailed overview of literature models; including theoretical, empirical and smart models, pertaining to predicting the TC of nanofluids. In addition, several intelligence techniques; including MLP, RBFNN and LSSVM, were developed in this work to establish accurate predictive models of TC of nanofluids. More than 3200 experimental points were used in order to capture the variation of TC with respect to the main influential variables. In contrast to many modeling studied
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.
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