Development of a neural architecture to predict the thermal conductivity of nanofluids

doi:10.21203/rs.3.rs-2627942/v1

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Development of a neural architecture to predict the thermal conductivity of nanofluids

https://doi.org/10.21203/rs.3.rs-2627942/v1

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published 24 Nov, 2023

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The present study proposes an artificial neural network (ANN) model for correctly estimating the thermal conductivity property of nanofluids. The ANN model was designed based on using 800 existing experimental data containing spherical nanoparticles of TiO₂, ZnO, CuO, Al₂O₃, ZrO₂, Fe₂O₃, Fe₃O₄, SiO₂, CeO₂, MgO, Fe, Al, Cu, Ag, Sic and diamond in various fluids of oil, ethylene glycol, water, and radiator cooling. Here, effective parameters of thermal conductivity of the base fluid and dispersed nanoparticle, nanoparticles volume fraction (0.4 − 0.4%), temperature (10 − 80 ℃), and particle diameter (4 − 150 nm) were considered as input variables, while the thermal conductivity of nanofluid was defined as the target variable. The Levenberg-Marquardt (L-M) back-propagation algorithm was used to design this model. According to the results, the best R and lowest MSE using 5-13-1 topology were founded to be about 0.9975 and 0.000238, respectively, indicating good fitting between predicted results and target points. Also, the results of the comparison between the ANN model and experimental points indicated successful validation of the presented model for estimating the thermal conductivity of nanofluids.

Thermal Conductivity

Nanofluids

Nanoparticles

Artificial Neural Network

Heat transfer.

Benefiting from a high-efficiency of heat transfer medium is one of the basic needs of today’s industries [1]. Heat transfer media is usually composed of fluids such as water, ethylene glycol, or oil. As a result, it is important to use more efficient heat transfer fluids with higher thermal conductivity to provide proper heat delivery in industries [2]. Several methods can be used to promote the heat delivery efficiencies, such as the use of microchannels and microtubes, creating more turbulence and reducing the thermal resistance of the circulating fluid [3, 4]. In this regard, the use of efficient nanofluids, as a new generation of fluids with proper potential in industrial applications, has been recommended to improve the heat exchange process [5, 6]. Nanofluids are colloidal suspensions consisting of nanometer-scale solid particles suspended in a base fluid [7]. Many researchers have also found that dispersing nanoparticles into a base fluid can improve thermal conductivity and enhance the heat delivery rate [8]. These particles are conventionally carbon nanotubes, metal particles, metal oxides, and non-metallic solid particles. Whereas water, engine oils, and ethylene glycol are used as common main fluids [9–11].

Thermal conductivity is considered the most important parameter to investigate the potential of nanofluid to increase heat transfer so that even low concentrations of nanoparticles are quite noticeable in heat transfer performance. Therefore, knowing the thermal conductivity coefficient of nanofluid is of immense importance to present a proper heat transfer model for a nanofluid-based thermal medium. However, the experimental calculation of this thermophysical property in different operating conditions may be faced many limitations [12]. Hence, presenting the proper models to predict and determine the thermal conductivity coefficient seems to be necessary [13, 14]. In this regard, the theoretical, analytical, and experimental models for the prediction of the thermal conductivity of nanofluids were widely developed in recent years [15, 16]. Maxwell initially presented a model based on the classical theories of two-phase mixtures which took the size, shape, and distribution of particles into consideration [17, 18]. This model suffered from a high error for non-spherical particles [19]. Considering the empirical shape factor and sphericity of nanoparticles, another experimental-based model was then proposed by Hamilton and Crosser [18]. The modified Maxwell model was then suggested which takes into consideration the Brownian motion and particle-fluid system thermal diffusivity [20]. However, the theoretical developed models were not able to estimate and predict this thermophysical property of nanofluids properly.

Thereafter, analytical models were developed to estimate the nanofluid’s thermal conductivity, which was based on the effects of some parameters such as the type, size, and shape of nanoparticles [21, 22]. Some of these proposed models have been derived based on macro models and did not able to show good agreement with laboratory data [14]. To overcome these problems, theories and mechanisms were proposed that showed more suitable adaptation to laboratory results. These mechanisms often took into account the effect of Brownian motion.

This work intends to present a proper artificial neural network model to estimate the thermal conductivity coefficient of nanofluids over 800 experimental data based on considering effective parameters of thermal conductivity of the main fluid and nanoparticles, the volume fraction of the nanoparticles, temperature, and particle size. The Levenberg-Marquardt (L-M) back-propagation algorithm, available in the neural network toolbox of MATLAB software, was used to design this model. The mean square error and the correlation coefficient were used to compare the predicted values and experimental data and confirm the accuracy of the presented ANN model.

2.1. Description of Artificial neural network

Computational systems based on artificial neural networks are modern systems that can easily mimic the complex relationships between input and output and utilize this computational ability for parallel data processing. One of the most basic neural models is the Multi-Layer Perceptron (MLP) model, which simulates the transfer function of the human brain to solve the regression problem [23]. The basic topology of ANNs has been constructed from three sequent connected layers, namely, input, hidden, and output, as well as many intralayer processing elements called neurons. A random number generator is responsible for assigning the weight value in a typical neural network [24]. Each neural in the MLP network is determined through a nonlinear activation function, as indicated in Eq. 1. Except for the input neuron, each neuron receives an input vector (Pi) from each node in the previous layer, multiplies it by the corresponding weight between the neurons j and node i (W_ij) in the previous layer, and then passes the sum multiplies (P_iW_ij) plus the bias of the neurons j (b_j) through the activation function to the other neurons [25].

While neurons in the hidden layer will convert into target value in the output layer through a linear activation function. The commonly transfer functions for the hidden layers in the MLP network are sigmoid logarithmic and hyperbolic sigmoid tangent [26]. ANN is able to analyze and detect the accurate relationship among variables and target values in complex systems through a proper algorithm using a training process, validation and test of data. "Training process in the neural network is performed with the aim of finding the best values of weight and bias for learning relationship between inputs to outputs, according to optimizing the network performance [27]. The Levenberg-Marquardt (L-M) back-propagation algorithm is conventionally used as a training function due to its suitable ability to quickly converge to an optimal solution and minimize a function [25, 28].

2.2. Data collection

To present a suitable semi-experimental model for accurately estimating the nanofluid's thermal conductivity, we need a high number of experimental data, with different nanoparticles and base fluids in various conditions such as temperature ranges and volume fractions. In this research, 800 experimental points and corresponded variables have been extracted from reported experimental results in previous studies. Several nanofluids including different spherical nanoparticles of Al₂O₃, SiO₂, ZnO, TiO₂, ZrO₂, CuO, CeO₂, MgO, Fe₂O₃, Fe₃O₄ Cu, Al, Ag, Sic, Fe, and diamond in various base fluids of ethylene glycol, water, and radiator cooling (RC, 50% water and 50% ethylene glycol), engine oil, kerosene, transmission oil, gear oil, and TH-66 were used to design the presented ANN model. The thermal conductivity of the used nanoparticles (K_p), the thermal conductivity of the main fluid (K_bf), temperature (T), the volumetric concentration of used nanoparticles (φ), and nanoparticle diameter (d_np) were also considered as effective parameters on enhancing the thermal conductivity property in the nanofluids. The temperature range, nanoparticle volume fractions and nanoparticles diameter in the nanofliuds used to architect the current ANN model are 10–80 ℃,0.4–0.4%, and 4.150 nm, respectively. Table 1 shows the database of used nanofluids and their experimental conditions. The thermal conductivity of the base fluids and nanofluids was collected and reported at different temperatures due to changes in the thermal conductivity with temperature.

Table 1

used database to architect the current ANN model [29]*.
nanoparticle	Base fluid	Kp (w/mk)	d_np (nm)	Φ (%)	T (°C)	reference
TiO₂	EG	8.04	15	1–5	21	[30]
Al	EO	204	80	1, 3	21–60
Al₂O₃	EG	27	80	1, 5	21–60
Al₂O₃	Water	25	47	1, 4	21–61	[31]
Al₂O₃	Water	27	38.4	1, 4	21–51	[32]
CuO	Water	18	28.6	1–4	21–51	[32]
CuO	D. water	18	29	2–14	22–40	[33]
Al₂O₃	D. water	27	36	3	22–35	[33]
Al₂O₃	RC	27	13	0.2 -1	10–50	[34]
ZnO	EG	29	18	0.5–5	24–50	[35]
SiO₂	Water	1.2	14	2.2	20, 45	[36]
SiO₂	EG	1.2	14	1.2, 2.5	24–80	[36]
ZnO	EG	29	15	0.4 -5	10–60	[37]
TiO₂	Water	8.04	21	1, 3	20–80	[38]
Al₂O₃	Water	40	25	2.5–7.5	25	[38]
TiO₂	Water	8.04	21	0.2–2	15–35	[39]
Al₂O₃	Water	27	11, 45	1–3	20–50	[40]
Al₂O₃	EG	27	11, 45	1–3	20–50
Cu	Water	383	80	1–3	20–50
Cu	EG	383	80	1–3	20–50
Al	Water	204	80	1–3	20–50
Al	EG	204	80	1–3	20–50
Fe	EG	80	10	2.5–5	25	[41]
CuO	EG	20	23.6	1–4	25	[42]
CuO	Water	20	23.6	1–4	25
Al₂O₃	EG	27	38.4	1–5	25
Al₂O₃	Water	27	38.4	1–4	25
ZnO	D. water	29	10,30,60	1–3	25	[43]
ZnO	D. water	29	30, 60	1–3	25
TiO₂	D. water	8.04	10, 34, 70	1–3	25
TiO₂	EG	8.04	10, 34, 70	1–3	25
SiO₂	Water	2	12	0.5–3.6	30	[11]
TiO₂	Water	10	35	0.5–5	30
Al₂O₃	Water	36	30	0.5–5	30
Al	Water	280	25	0.75–3.6	30
SiO₂	EG	2	12	0.5–3.6	30
CuO	Water	18	23	1–3.5	30
Diamond	Water	1000	11.8	0.2 -1	20–60	[44]
Diamond	Water	1000	8	0.1-1	30	[45]
Diamond	Water	1000	4	0.8–1.5	30, 40	[46]
MgO	Water	30	40	0.14–1.13	25–60	[47]
Ag	EG	430	12	0.1- 2	30–60	[48]
Fe₃O₄	Water	4.5	25	0.25–3	20–60	[49]
Fe₂O₃	Water	20	30	0.5-2	20–50	[49]
Cu	TH-66	380	62	0.5-2	27	[50]
SiO₂	TH-66	1	15	1.2 -7	25	[51]
Fe₃O₄	Water	4.5	6.7	0.5 -3	10–40	[52]
ZrO₂	Water	3.5	20	1.8–10	27	[53]
TiO₂	Water	8.04	40	0.6–2.5	10–40	[53]
Al₂O₃	Bio glycol	36	13	0.1-1	30–80	[54]
Fe	EG	80	40	0.12 -3	25,35	[55]
MgO	EG	48	20	0.5 -5	25, 35	[55]
CeO₂	Water	12	30	0.5	25–50	[56]
CeO₂	Water	12	30	0.5- 3	40	[57]
SiO₂	Water	1.2	10	0.5- 3	40	[57]
Al₂O₃	Water	36	150	1–6	25	[58]
TiO₂	Water	8.04	150	1–6	25
ZrO₂	Water	2.2	44	2–8	25
CeO₂	Water	12	25	1.6–6.6	20	[59]
SiC	EG	490	30	0.2 -	20–50	[60]
ZnO	Water	13	20	1–3	27–52	[61]
SiO₂	Water	1.2	20	1–3	27–52	[61]
SiC	Water	490	30	0.04- 1	10–50	[62]
Cu	Water	400	48	0.06–2.5	25	[63]
Fe₃O₄	Water	4.5	8	0.8–7.8	25–50	[64]
Cu	Gear oil	380	40	0.11- 2	80	[65]
Fe₃O₄	Cerosen	4.5	10	1.7–4.7	25–75	[66]
* the same database was used in our previous study [29]

2.3. ANN model architecture

A 5-input ANN model was designed using 800 experimental data points for accurately estimating the thermal conductivity of nanofluids. The neural network toolbox of MATLAB R2019 was used to architecture this ANN model. The Levenberg-Marquardt (L-M) algorithm as a fast backpropagation tool was used for the training process and converge the network output and the target points [26]. Here, variables of K_bf, K_p, φ, T and d_np were presented as first input to fifth input, respectively. The thermal conductivity of nanofluid was also introduced as the target variable. As schematically exhibited in Fig. 1, an input layer, a hidden layer, and an output layer formed the main construction of the developed ANN model, so that the tangent sigmoid activation function and linear activation function were utilized to determine the output of network in the hidden and output layers. Network training and validation were performed using 70% (560 experimental points) and 15% (120 experimental points) of the entire data points, and the remaining 120 unused data (15% of the entire data points) were also chosen for network testing. It should be mentioned that the data points in the sub-sets of training, validation, and testing were randomly selected and divided by the network itself. Due to the notable effect of the number of neurons in the hidden layer on the performance and accuracy of created network, the training process was performed using various neurons from 1 to 17. The optimal number of neurons is conventionally obtained using the trial-and-error technique.

Here, the statistical indexes of correlation coefficient (R) and mean square error (MSE) between estimated values and related experimental values were applied to address the accuracy and performance of the designed ANN model [25], as follows:

\(MSE=\frac{1}{N}{\sum }_{m=1}^{N}({{K}_{Exp,m}-{K}_{Pre,m})}^{2}\)	Eq. 2
\(\text{R}=\frac{\sum _{m=1}^{N}({K}_{Pre,m}-{K̅}_{Pre,m})({K}_{Exp,m}-{K̅}_{Exp,m})}{\sqrt{\sum _{m=1}^{N}{({K}_{Pre,m}-{K̅}_{Pre,m})}^{2} \sum _{m=1}^{N}{({K}_{Exp,m}-{K̅}_{Exp,m})}^{2}}}\)	Eq. 3

Here, K_exp,m and K_pre,m demonstrate the m-th element in the laboratory values and corresponded m-th element in predicted values by ANN model, respectively. N represents the number of entire data. Also, \({\stackrel{-}{K}}_{exp}\) and \({\stackrel{-}{K}}_{pre}\) are the average of the laboratory data, and mean of the values obtained from the ANN model, respectively.

3.1. ANN model using variables of K_bf, K_p, φ, T, and d_np

To address the accuracy and performance of the presented ANN model, the results of experimental data were compared with the results of the predicted thermal conductivity using the ANN model. The variables of K_p, K_bf, φ, T, and d_np were respectively defined as input variables, and thermal conductivity obtained from laboratory data was applied as the objective function in this modeling. As reported in Table 2, the number of neurons gradually increased from 1 to 17 in each step to obtain a proper and acceptable result. Here, increasing the number of neurons up to 13 led to a higher correlation coefficient and lower Mean squared error for all data.

Table 2

The results of neural networks with different numbers of neurons for total data
Number of Neurons	Mean Square Error (MSE)	Correlation Coefficient (R)
1	0.9914	0.000811
2	0.9933	0.000627
3	0.9946	0.000513
4	0.9951	0.000471
5	0.9955	0.000422
6	0.9962	0.000358
7	0.9962	0.000359
8	0.9963	0.000349
9	0.9968	0.000295
10	0.9969	0.000296
11	0.9969	0.000281
12	0.9972	0.000262
13	0.9975	0.000238
14	0.9975	0.000270
15	0.9974	0.000289
16	0.9974	0.000293
17	0.9974	0.000292

However, it was observed that using a greater number of neurons, up to 17 caused in increasing the MSE, as well no positive change was achieved in R. So, the results of the prediction with 13 neurons were satisfactory, and this neurons number was chosen as an optimum number of neurons. According to the results, the best R and lowest MSE using 5-13-1 topology were founded to be about 0.9975 and 0.000238, respectively, indicating good fitting between predicted results and target points.

Also, the results of the presented network using 13 neurons for all data sets and sub-sets including training, validation, and test data are given in Table 3. As is clear, the best correlation coefficient and the lowest mean square error were related to the network training sub-set, about 0.9976 and 0.000229, respectively.

Table 3

The results of developed ANN model using 13 neurons
Data Set	Correlation Coefficient (R)	Mean Square Error (MSE)	Data Points Number
training	0.9976	0.000229	560
Validation	0.9977	0.000240	120
Testing	0.9971	0.000275	120
Total data	0.9975	0.000238	800

Figure 2 shows the comparison between the estimated nanofluid's thermal conductivity using the presented ANN model (output) and actual data (target), for all data and sub-sets. Accordingly, a good fifing was observed for all data and each sub-sets, indicating proper accuracy and performance of the presented ANN model.

Also, the results of the ANN model include MSE and ± maximum of the error (max error) from laboratory data (K_exp,m − K_ANN,m) for water, oil, ethylene glycol, and RC base nanofluids were reported in Table 4. As is clear, the ANN has a more appropriate ability for predicting the thermal conductivity of nanofluids based on radiator cooling, oil and ethylene glycol. However, the most positive and negative errors from laboratory data and the highest MSE are + 0.077, -0.063, and 0.000359, respectively, which are related to water-based nanofluids.

Table 4

The results of the developed ANN model using 13 neurons for various base fluids
Base fluid	MSE	+ Max Error	− Max Error	Data points No.
Water	0.000359	+ 0.077	−0.063	472
EG	0.000072	+ 0.027	−0.018	238
Oil	0.000037	+ 0.009	−0.027	65
RC	0.000041	+ 0.003	−0.010	25

The error histogram graph was used to further evaluate the amount of error between values predicted by ANN model and experimental data. The results in Fig. 3, illustrated an approximately narrow distribution of error so that most of the error between values predicted by 5- parameter ANN model and used experimental data is less than ± 0.02, confirming the good performane and proper accuracy of the designed 5-input ANN model. According to the results presented in Fig. 3, the values of error maximum are approximately + 0.077 and − 0.063 and include a limited amount of data.

The details of developed 5-13-1 topology in the current ANN model including generated weights and biases in hidden and output layers are given in Table 5. These reported datails is need to predict the thermal conductivity of the nanofluid in similar conditions and systems. Here, parameters of i, j and z represent the number of input variables equal to 5, the number of optimal neurons equal to 13, and the number of elements of the output layer and equals 1.

Table 5

details of developed 5-13-1 topology including generated weights and biases in hidden and output layers
Neurons	Hidden layer						Output layer
	Weights (j,i)					Bias (j)	Weights (z, j)	Bias (z)
	w_j,1	w_j,2	w_j,3	w_j,4	w_j,5	b_j	W_z,j	b₂
1	-2.373	-0.750	-2.868	-2.1324	− .63109	3.4503	-1.3477	-0.0666
2	2.6033	-2.6548	1.0633	-0.6431	-0.00685	-5.8058	1.1825
3	5.3041	6.2036	− .8895	11.5857	-3.1982	6.6723	-0.1936
4	-0.07695	-1.9939	0.11738	-1.0823	-1.8938	0.2621	-1.6987
5	-1.7939	0.5535	0.39141	0.3496	-1.4243	0.3815	-2.3237
6	3.9334	1.2779	2.0192	-8.7285	-0.12049	-5.4235	0.04595
7	4.2268	-4.819	-0.33342	2.703	-1.722	-1.9611	-0.11795
8	1.5564	-0.6616	-0.46827	-0.49512	1.3717	-0.6102	-1.8786
9	5.399	0.6448	-0.07069	11.1908	-1.6698	2.7623	0.2034
10	1.5471	4.5512	-0.6625	-0.93448	4.9159	-3.4479	-0.7357
11	3.5399	0.0980	-0.37568	1.2641	0.71327	4.2202	0.3323
12	1.6815	-9.2135	1.2068	0.012136	-0.29878	-11.479	-1.4107
13	-0.7935	01.9723	-0.3867	2.5227	0.034926	-1.6416	-1.8418

3.2. Validation of Presented ANN model

To evaluate the accuracy of the ANN model to estimate the thermal conductivity of nanofluids, the validation of presented model was done with the test of 104 other experimental data points, which were collected from some previous research. The details of the used database for validation of the 5-input ANN model were reported in Table 6.

Table 6

details of the used database to validat the 5-input ANN model
nanoparticle	Base fluid	d_np (nm)	Φ (%)	T (°C)	Kp (w/mk)	reference
ZnO	water	30	1–3	22–75	29	[67]
Al₂O₃		75	1.5-6	20–50	27	[68]
CuO		31	1–3	20–50	18	[40]
Fe		25	1–5	20	80	[69]
CuO	EG	31	1–3	20–50	18	[40]
Al₂O₃		12	2–3	25	27	[70]
ZnO		50	0.7, 1	15–55	18	[37]
Cu	Oil	40	0.11-2	60	383	[65]
CuO		31	1–3	20–50	18	[40]
Al₂O₃		45	1–3	20–50	27	[40]
Ag		20	0.012–0.180	40–70	430	[71]

The results of the comparison between predicted values of thermal conductivity (output) versus corresponding actual laboratory data (target) were represented in Fig. 4. As is clear, the results of the developed ANN model indicated a proper agreement with the experimental data, confirming the proper ability of the presented model. The maximum deviation between values predicted by ANN model and corresponded experimental data for nanofluids based on oils and ethylene glycol is less than + 1.2% and − 0.5%, respectively. However, the maximum error for nanofluids based on water is about + 3.7%, which shows a acceptable and satisfactory amount. The outcomes proved the accuracy and proper strength of the developed 5-input ANN model to predict the thermal conductivity of nanofluids that are in the limitation of reported operating conditions in the current study.

Also, the results of the semi-experimental model, presented by Ahmadlou et al. were compared with the proposed 5-input ANN model [27]. They developed an ANN model to predict the thermal conductivity of nanofluids using 800 different experimental data points related to different nanoparticles and main fluids of water, ethylene glycol, and oil in the range of 10–71°C, volume fractions of 0.1-1%, and nanoparticles diameter of 5-150 nm. According to their results, the R and MSE were found to be 0.993 and 0.000527. Ali Aminian [23] also investigated and predicted the thermal conductivity of nanofluids using an ANN model over various nanofluids. They defined the parameters of T, ϕ, d_np, K_bf and K_p as input variables of ANN. The presented model by them was performed in the range of 0-200°C and a nanoparticle diameter of 9-150 nm and a nanoparticle volume fraction of 0.005-14%. The calculated correlation coefficient was about 0.9309 and the mean squared error was 3.06% for all data. While the 5-input ANN model presented in the current study, showed a better mean squared error of 0.000238 and a further correlation coefficient of 0.9975, confirming the accuracy and appropriate performance of the proposed 5-input ANN model in predicting the thermal conductivity of nanofluids.

An artificial neural network model using 800 laboratory data was presented to estimate the thermal conductivity of nanofluids in this study. Here, the variables of K_bf, K_p, φ, T, and d_np were defined as input variables, and the experimental thermal conductivity coefficient was determined as target values. tTe nanofluids were containing different spherical nanoparticles of ZnO, ZrO₂, SiO₂, Al₂O₃, TiO₂, CuO, MgO, CeO₂, Fe₂O₃, Fe₃O₄, Al, Cu, Ag, Fe, Sic and diamond suspended in fluids of water, ethylene glycol, oil and radiator coolant in the temperature range of 10–80°C and a volume fraction of 0.14–0.4% and a nanoparticle diameter of 4-150 nm. The 5-13-1 topology was found to be the best construction with proper performance for the ANN model. The results indicate a proper performance for the 5-input ANN model, so the mean squared error and correlation coefficient were obtained at about 0.000238 and 0.9975, confirming the ability of the present model in predicting the thermal conductivity of nanofluids.

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An artificial neural network model using 5-13-1 topology for predicting thermal conductivity of nanofluids

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published 24 Nov, 2023

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28 Feb, 2023
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Development of a neural architecture to predict the thermal conductivity of nanofluids

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Method

2.1. Description of Artificial neural network

2.2. Data collection

2.3. ANN model architecture

3. Results And Discussion

3.1. ANN model using variables of K_bf, K_p, φ, T, and d_np

3.2. Validation of Presented ANN model

4. Conclusion

References

Supplementary Files

Status:

Journal Publication

Version 1

Development of a neural architecture to predict the thermal conductivity of nanofluids

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Method

2.1. Description of Artificial neural network

2.2. Data collection

2.3. ANN model architecture

3. Results And Discussion

3.1. ANN model using variables of Kbf, Kp, φ, T, and dnp

3.2. Validation of Presented ANN model

4. Conclusion

References

Supplementary Files

Status:

Journal Publication

Version 1

3.1. ANN model using variables of K_bf, K_p, φ, T, and d_np