Experimental and comparative theoretical study of thermal conductivity of MWCNTs-kapok seed oil-based nanofluid

International Communications in Heat and Mass Transfer 110 (2020) 104402

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Experimental and comparative theoretical study of thermal conductivity of MWCNTs-kapok seed oil-based nanofluid

T

Ahmad Mukhtara, Sidra Saqiba,b, Fatma Safdarc, Ayesha Hameedd, Sikander Rafiqe, ⁎ ⁎ Nurhayati Binti Mellona, , Rabia Amenf, Muhammad Saad Khang, Sami Ullahh, , Muhammed Ali Assirih, Muhammad Babara, Mohamad Azmi Bustama, Wajid Ur Rehmani, Z.M.A. Mericani a

Department of Chemical Engineering, Universiti Teknologi PETRONAS, Bandar, Seri Iskandar, 32610 Perak, Malaysia Department of Chemical Engineering, COMSATS University Islamabad, Lahore Campus, 54000 Lahore, Pakistan School of Environmental Sciences and Engineering, Government College University, 38000 Faisalabad, Pakistan d School of Chemical and Materials Engineering, National University of Science and Technology, Islamabad, Pakistan e Department of Chemical, Polymer & Composite Material Engineering, University of Engineering and Technology, Lahore, (KSK Campus), Pakistan f Institute of Soil and Environmental Sciences, University of Agriculture Faisalabad, Faisalabad 38040, Pakistan g Department of Petroleum Engineering, Texas A&M University, Qatar Campus, Doha, Qatar h Department of Chemistry, College of Science, King Khalid University, Abha 61413, P. O. Box 9004, Saudi Arabia i Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Bandar, Seri Iskandar, 32610 Perak, Malaysia b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Multi-walled carbon nanotubes Kapok seed oil Nanofluid Thermal conductivity Dimensionless group analysis Artificial neural network

Despite the significant potential of nanofluids in energy storage applications, the experimental determination of thermophysical properties is relatively costly and time-consuming. Therefore, the modeling techniques can be used for the accurate estimation of thermo-physical behavior. The predictive models are useful for the understanding of thermo-physical behavior. Due to restrictions on classical models, there is a need to develop more reliable models to simulate the thermophysical behavior of nanofluids. This paper deals with the synthesis and experimental thermal conductivity measurement of the multi-walled carbon nanotubes (MWCNTs)-Kapok seed oil nanofluid. Additionally, two new correlations based on multiple non-linear regression analysis along well as dimensionless analysis are proposed to estimate thermal conductivity with high precision compared to the classical models. Based on the statistical analysis, the prediction accuracy of the proposed model was ranked. Finally, the sensitivity analysis has been carried out in combination with the residual analysis to assure the accuracy of the model parameters of both proposed models and adequacy of estimated values of model parameters, respectively. The results revealed the global minimum values for all parameters at 0% perturbations indicating that the model parameters were estimated with high accuracy and adequacy.

1. Introduction The early inspiration of utilization of base fluid with enhanced thermal conductivity by the accumulation of highly thermally conductive nano-sized particles goes back to 100 years ago when Maxwell first time proposed this concept theoretically [1,2]. However, the commercialization of this concept realized many practical problems during the thermal conductivity enhancement by the accumulation of high thermally conductive nanoparticles such as the comparatively large size of the colloidal particles, their clogging, abrasion, and sedimentation [3]. To address these issues, Choi [4] pioneered in the

⁎

applications of nanofluids in 1995 used the nanofluids developed based on advanced synthesis methods for nano-metal particles exhibiting enhanced thermal conductivities. Since Lee et al. [5] first time measured the thermal conductivity of nanofluids, different types of nanofluids have been investigated to investigate their thermal conductivity [6–9]. Nanofluids are the colloidal suspension of certain kinds solid particles with nano-size (< 100 nm) having a higher thermal conductivity in base fluids such as thermal oils, ethylene glycol (EG), and/or water and, and consider as promising alternatives for heat transfer in numerous applications [10,11]. These applications include air

Corresponding authors. E-mail addresses: [email protected] (N.B. Mellon), [email protected] (S. Ullah).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104402

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International Communications in Heat and Mass Transfer 110 (2020) 104402

A. Mukhtar, et al.

Nomenclature A d K L m N n OD T V w x y

MWCNTs R R2 RMSE SSE SWCNTs

Surface area (m2) Diameter (nm) Thermal conductivity (W/m.K) Length (nm) Mass (gm) Number of data points Empirical shape factor Outer diameter (nm) Temperature (°C) Volume (cm3) Weight of neurons Inputs Outputs

Multi-walled carbon nanotubes Enhancement factor (Eq. 21) The correlation coefficient (Eq. 7) Single-walled carbon nanotubes Sum of squared error (Eq. 5) Single-walled carbon nanotubes

Greek Letters: γ ρ φ ψ

Sphericity (Eq. 16) Density (Kg/m3) Nanoparticle weight fraction (Eq. 1 & 2) Curve fit parameter

Subscripts: bf exp. f np o p pred.

Abbreviations a, b, c, d Curve fit parameters AAE Average absolute error (Eq. 3) CNTs Carbon nanotubes EG Ethylene glycol EMT Effective medium theory MSE Mean squared error (Eq. 4)

Base fluid Experimental Fluid Nanoparticles Reference temperature Particle Predicted

transfer through nano-convection governed by the Brownian movement of nanoparticles is one of the most commonly adopted mechanisms for improvement in the thermal conductivity of nanofluids. Brownian movement of nanoparticles is often cased with nanofluids provided that the suspended nanoparticles in base fluid are very small [22–27]. Jang and Choi [22] firstly reported a hypothesis that the convective heat transfer at the nanoscale is induced by the random motion of nanoparticles in base fluids and this nano-convection is prominent parameter in the thermal conductivity enhancement of nanofluids and further they have reported a Brownian movement-based model considering nano-convection. Koo and Kleinstreuer [23] reported the further extension of the idea of nano-convection proposed by Jang and Choi and they have proposed that Brownian movement of the nanoparticles in different base fluids could produce a mixing at microscale which should be a key parameter in the improvement of thermal conductivity of nanofluids. Several experimental and modeling studies have been reported for the modeling of thermal conductivity enhancement using various techniques. The systems for these studies are based on MWCNTs-CuO/ water [28], copper ferrite nanoparticles coated with the silica/water [29], Al2O3/deionized water [30], MWCNTs-Fe3O4/ethylene glycol [31], CuFe2O4/SiO2 nanocomposite water/ethylene glycol [32], FeCuO/ethylene glycol-water [33], and SWCNTs-nano-antifreeze [34]. In our case, we have selected the Kapok seed oil for the nanofluid synthesis due to its various advantages over traditional base fluids. These advantages include abundant availability of kapok seed oil in Malaysia, cheaper than traditional base fluids, and eco-friendly compared to other base fluids. Additionally, the kapok seed oil is found to be not suitable in the food industry; therefore, its utilization in heat transfer does not influence the food industry. In view of the above literature review, this work has the following objectives:

conditioning, transportation, power generation, heat pipes, solar collectors, defense, electronic cooling, space, biomedicines, lubricating agent, corrosion inhibitors, and nuclear systems cooling [12,13]. In the last decades, nanofluids attracted the researchers because of their improved thermal conductivity characteristics. According to Eastman et al. [6], the dispersion of 0.3 volumetric fraction percent of copper (Cu) nanoparticles (< 10 nm) in the ethylene glycol resulted in a 40% improvement in the thermal conductivity of pure ethylene glycol. According to the heat transfer theory, the convection-mechanism based heat transfer is directly influenced by the thermal conductivity. With this observation, many researchers paid significant attention to the precise measurement of the thermal conductivity of nanofluids [14]. In the past few decades, the thermal conductivity of various nanofluids as a significant heat transfer characteristic has been extensively reported, and many theoretical and empirical models have been proposed based on different heat transfer mechanisms as reported in the literature [15–19]. However, in many cases, because of measurement conditions and experimental deviations, the experimental thermal conductivity of nanofluids from different researchers could not agree properly. Thus, further research on the thermal conductivity of nanofluids is still required for a deep understanding of the mechanisms for heat transfer for nanofluids. Traditional classical models for the thermal conductivity prediction of nanofluids are based on classical effective medium theory (EMT), such as Hamilton-Crosse Model [20], Maxwell Model [1], and Bruggeman Model [21], which are failing to provide a deep understanding in the improvement of nanofluids thermal conductivity. This is because of the intrinsic nature of these classical models as they only consider the effect of nanoparticle weight/volume fractions in the base fluids for enhancement of thermal conductivity. In fact, the thermal conductivity improvement of nanofluids was found to be influenced by different factors not only nanoparticles concentration in base fluids, other factors including type of nanoparticle material, base fluids, temperature, chemical additives, nanoparticle size and shape, volume fraction, state of dispersion, and the presence of clusters [17]. Enhanced theoretical thermal conductivity models have also been developed by accounting the influence of various mechanisms of heat transfer via conduction such as the nature of heat transfer in the nanoparticles, nanolayers, clustering, and the Brownian motion of the nanoparticles [18]. Heat

• Synthesis of MWCNTs and Kapok seed oil-based nanofluid using one-step synthesis method. • Experimental measurement of thermal conductivity at various temperatures and nanoparticle weight fractions. • Performance evaluations of classical models for thermal conductivity prediction.

2

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• Development of new correlations based on regression analysis and •

indication of the closeness of correlation between experimental and predicted data if its value touches one, and proposed model has been considered as the model with great accuracy. These indicators can be formulated as follows (Eqs. 3–7).

dimensionless groups analysis approach and their performance evaluation for thermal conductivity prediction. Comparison of all proposed modeling results based on statistical indicators.

AAE = 2. Materials and methods 2.1. Materials

1 N

1 MSE = N

The Kapok seeds were bought from Bota Kinnin, Perak, Malaysia. The analytical grade n-Hexane and MWCNTs (O.D. × L = 6–13 nm × 2.5–20 μm, purity > 99%) were taken from Merck (Malaysia).

n

∑

(Kpred . − K exp .) Kpred .

i=1

(3) 2

n

∑ i=1

Knf ⎞ ⎤ ⎡ ⎛ Knf ⎞ − ⎜⎛ ⎟ ⎟ ⎥ ⎢⎜ K K bf bf ⎢ ⎠exp . ⎝ ⎠ pred .⎥ ⎦ ⎣⎝

(4)

n

SSE =

∑ (K exp . − Kpred.)2

(5)

i=1

n

2.2. Extraction of Kapok seed oil

RMSE =

Soxhlet extractor was used for the extraction of oil from Kapok seeds using n-Hexane as a solvent at 68 °C for 6 h. After the extraction of kapok seed oil, a laboratory-scale evaporator was used to evaporate the n-hexane in vacuum conditions using water bath for 60 min at 70 °C. After that the airtight glass was used to store kapok seed oil.

R2 = 1 −

wρbf

(1 − ) ρ

np

−

( )ρ w 100

bf

(1)

m

∅=

⎛ np ⎞ ⎝ ρnp ⎠ m

m

⎛ np ⎞ + ⎛ bf ⎞ ⎝ ρnp ⎠ ⎝ ρbf ⎠

Kpred .)2 K exp .)2

(7)

The difficulties came into existence during the development of correlation or the dimensionless model for the precise estimation of the nanofluids thermal conductivity. These difficulties can be directed to the several factors including the diversity of the experimental data, complication of the developed models, the number of output parameters to be predicted with high accuracy, the method of solution employed to solve the set of differential equation, and also the process of optimization to find the model parameters values. To assures that the values of parameters of models developed in this work are optimized and are the best set of predicted values, a sensitivity-based analysis based on a combination of sensitivity analysis and residual analysis (Fig. 2) is performed [37]. The first step of this analysis involved the initialization of the model parameters as the optimal solution of the model mostly depends on the initial guesses-based values of parameters [38]. However, if the values of the model parameters are previously reported in the literature by someone else, these values can be employed as the initial guesses-based values of the model parameters. In case if there are no reported values of the model parameters into the literature, the simulation techniques can be applied to estimate the values of the objective function i.e. SSE [39]. This calculation is subjected to repetition several times until the minimum possible value of the objective function i.e. SSE is obtained. Once the initial values of the model parameters have been estimated with minimum possible value of objective function i.e. SSE, the second step is to apply the non-linear regression analysis to search out the optimal solution that minimizes the values of objective function [40–42]. After the estimation of the

The one-step method was used to prepare kapok seed oil and MWCNTs based nanofluids [35,36] In this method the MWCNTs were spread in kapok seed oil in a different proportion ranging from 0.2–0.6 wt% and the temperature were in the range of 25 °C to 65 °C. The mechanical mixing technique was used to scatter the MWCNTs evenly in the kapok seed oil at power (70%) and pulse (30%) for 6 h. For the stable dispersion of MWCNTs in kapok seed oil, the ultrasonic probe was preferred instead of bath-type ultrasonication. But, the ultrasonic waves disturb the clusters due to heat transfer, as a result significant increase in temperature was observed throughout the synthesis process. To regulate the temperature, the nanofluid samples were cooled down by sonication on water bath, Hashnin, HS 3005 N. A diagrammatic explanation of the nanofluid synthesis used in numerous experiments is shown in (Fig. 1) [35]. The anticipated volume and weight proportions of nanoparticles could be measured by means of the subsequent equations (Eq. 1–2). w 100

(6)

N n ∑i = 1 (K exp . − n ∑i = 1 (K exp . −

2.5. Sensitivity analysis

2.3. Preparation of MWCNTs-Kapok seed oil-based nanofluid

∅=

∑i = 1 (K exp . − Kpred .)2

(2)

Where, w and φ denote the nanoparticles volume and weight proportion, respectively. The ρbf and ρnp represent the nanoparticles and the base fluids densities, respectively. While mnp and mbf stand for nanoparticles and base fluids weight, respectively. The nanoparticle characterization is given in our preceding published research work. [35].

Ultrasonic Disruptor

Cooling Water Bath

MWCNTs Cluster Kapok Seed Oil

2.4. Statistical analysis To evaluate the precision and the performance of the proposed models, statistical parameters including the average absolute error (AAE), mean squared error (MSE), the sum of squared error (SSE), root mean squared error (RMSE), and the correlation coefficient (R2) are utilized. If the values of these above-mentioned indicators are approaching to zero, it indicates the models have the highest accuracy. The regression coefficient or the correlation coefficient (R2) is an

Stable Nanofluid

Cooling Coils (Refrigerant)

Fig. 1. The schematic diagram for the synthesis of MWCNTs and kapok seed oilbased nanofluid using one-step synthesis method. 3

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Initialization of Parameters

Experimental Data Developed Models

Non-Linear Process Optimization

Estimated Models Parameters

Residual Analysis New Initial Guess based on Sensitivity Analysis

Not

Sensitivity Analysis

Same Global Minimum of the Objective Function for all the Parameters?

Yes Successful Non-Linear Optimization Process Fig. 2. The proposed methodology for sensitivity analysis.

within the nanofluid system and the kinetic energy of nanoparticles present in base fluids for the formation of stable nanofluid was also intensified because of collisions between the excited nanoparticles causing heat and energy transfer. The thorough detail of nanoparticle systems temperature and weight proportion impact on intensification of the thermal conductivity is illustrated in the subsequent sections. The main mechanism behind intensification in nanofluid thermal conductivity was the Brownian motion of nanofluid particles which are strongly influenced due to altering temperature of nanofluid system.

optimal values of the model parameters, a sensitivity analysis is implicated to each model parameters by using the concept of perturbations i.e. modifying one model parameter as time and by keeping all other model parameters constant without any variation. The objective function is re-estimated for each perturbation and then a graph is plotted between the percentage of the perturbation of the respective model parameter and the value of its objective function. If the perturbations of the all model parameters yield the minimum value of the objective functions with their original values i.e. 0% perturbation, then the global minimum has been obtained. On the other hand, if at least of the model parameter does not give the global minimum at the 0% perturbation, that will be an indication of the poor non-linear regression analysis-based estimation of the values of the model parameters. Finally, residual analysis has been performed by plotting a graph against the experimental observation and the predicted data with regular distribution and without any specific pattern or tendency should be observed, thus proving the accuracy of the values of the model parameters of the proposed model [43]. Another useful method for the residual analysis is to plot the graph against the experimental data and predicted data to assure the quality of the best-fitted model. However, the residual analysis is useful to some extent to assure the accuracy of the values of the model parameters of the proposed model graphically and it cannot guarantee by itself the achievement of the global minimum of the objective function i.e. SSE. Therefore, it is recommended to use both sensitivity analysis as well as residual analysis to assure the accuracy of the values of the model parameters of the proposed model.

3.1. Stability of MWCNTs-Kapok seed oil-based nanofluid The vast commercial and industrial applications of nanofluids are greatly influenced by means of stable scattering of nanoparticles in the base fluids. The amalgamation of nanoparticles could be avoided by using nanoparticles having smaller sizes and large specific surface areas such as use of nano-sized particles to improve the intermolecular forces on the surface of the nanoparticles. The aforesaid mechanism is proficiently described by the DLVO theory and colloidal particle theory. [44]. The conception of DLVO theory is essential to recognize the magnetic, electrostatic, and van der wall forces. Occasionally, the formation of microchannels occurs due to amalgamation of nanoparticles because of uneven scattering of the nanoparticles due to gravity which overall decreases the heat transfer phenomena of nanofluids in the heat transfer apparatus. Hence, the knowledge and clarification of intermolecular forces among base fluids and nanoparticles are of substantial importance in order to understand the even dispersion of nanofluids in the base fluids [45]. In the present case, the ultrasonic disrupter was used for the even distribution of MWCNTs in base fluid (kapok seed oil). The ultrasonic disrupter was subjected to samples for 6 h, resulting in the formation of extremely stable nanofluid having longer duration stability (minimum a month as conveyed in our preceding work) [35]. The fallouts tremendously favored the use of mechanical mixing (ultrasonic disruptor) instead of chemical mixing for the formation of extremely stable nanofluids. The synthesis of stable nanofluids could be used in range of fields comprising solar collectors, space, air conditioning, heat pipes, power generation, transportation, cooling of

3. Results and discussions One of the key aims concerning the applications of heat transfer in nanofluids is intensification in thermal conductivity of thermal oils via addition and formation of evenly dispersed nanoparticles suspension such as MWCNTs. In most cases, the thermal conductivity intensification strongly influenced the mobility of free electrons, molecular vibrations, nanoparticle aggregation, nanoparticle's weight proportion, weight and size of nanoparticles. The rise in temperature was observed 4

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of thermal conductivity at high weight proportions of nanoparticles plays an important part in the conduction enhancement because of intensified aggregation of vastly conductive nanoparticles. [55]. Thus, maximum researchers recommended that there must be linearity in association among the weight proportion of nanoparticles and thermal conductivity at lesser nanoparticles weight proportion typically lower than 10 wt%. [13]. In our research study, the distinguishing performance of intensified thermal conductivity of nanofluids based on MWCNTs-kapok seed oil with upsurge in the weigh fraction of nanoparticles is presented in (Fig. 3b). The fallouts revealed that thermal conductivity intensifies with an increase in nanoparticles weight proportion. The utmost enhancement in thermal conductivity is detected from 0.35 to 1.485 W/m.K (Table 1) with the escalation in nanoparticles weight proportion from 0.2 to 0.8 wt% with the temperature range of 25–65 °C. This trend of intensification in thermal conductivity with escalating weight proportion of nanoparticles could directed towards the improvement in the nanoparticles Brownian motion and MWCNTs Soret effect which is supported by intensifying temperature [46,56]. This could also be attributed to the MWCNTs thermal conductivity which in turn effect the base fluids thermal conductivity. The upsurge in volume of MWCNTs results in enhancement of surface to volume ratio for the purpose of heat conduction which is also identical to the conclusions of other scientists. [57–60]. The enhancement in MWCNTs thermal conductivity with increase in its weight proportion could be credited to its nature of high thermal conductivity in contrast to the base fluid (kapok seed oil). It could also be focused on the fact that there is presence of improved three-dimensional dispersal of MWCNTs possibly linked to their direction and ratio characteristic where the links between nanoparticles and the ratio of associations is more crucial.

electronics and nuclear system. 3.2. Influence of temperature The intensification in nanofluids thermal conductivity based on MWCNTs-Kapok seed oil is testified in this study (Fig. 3a) showed that intensification in thermal conductivity is directly proportional to the temperature (25–65 °C) and weight proportion (0.2 wt% to 0.8 wt%) of nanoparticles in the nanofluid system. The fallouts are identical to the published study on nanofluids based on MWCNTs-thermal heating oil [46]. Specifically, at the higher weight proportion of nanoparticles, a straight link among the temperature and intensification in thermal conductivity is observed. This direct link could be credited to the intensified Brownian movement, interfacial thermal resistance, and nanoparticles arbitrary motion to promote higher energy and the bombardment of nanoparticles substantial to the heat transmission process through conduction [47]. Two types of noteworthy mechanisms have been described concerning the intensification of thermal conductivity; one is nanoparticle's inner thermal carriers and other is the Brownian motion [48]. The Brownian motion of nanoparticles is recognized through estimation of mean velocity in the favorable route and the mean of nanoparticles free path. Occasionally, the enhancement in nanoparticles Brownian motion causes the agglomeration of nanoparticles which is not suitable for Brownian motion. Hence, the elevation in temperature could be proved as unsuitable mechanism for the improvement of nanofluids thermal conductivity and frequently the fallouts are opposite. The varied outcomes of escalated temperature impact on the nanofluids thermal conductivity are enlightened in this way: primarily the variation rule for diverse types of base fluid comprising varied thermal conductivity with temperature escalation is different. In general, the thermal conductivity of aqueous solutions declines with intensifying temperature excluding the water and glycerol where opposite trend is observed (increase in temperature intensifies the thermal conductivity). Secondly, in various forms of nanofluids, the impact of temperature on the nanoparticles Brownian motion is significantly estimated via composition, size, agglomeration, crystal formation and surface charge of nanoparticles base fluids, hence, persuading varying impacts of temperature on the intensification of thermal conductivity. Thirdly, despite the fact that similar variation tendencies of thermal conductivity have been observed with varying temperatures for both water and base fluids, the comparative escalation in the nanoparticle and base fluids thermal conductivity will also change the fallouts affected by temperature [49].

3.4. Classical models In history, Maxwell finely reported the matter of thermal

The rmal Conductivity (W/m.K)

1.6

3.3. Influence of nanoparticle weight fraction

(a) 1.4

25 °C 35 °C 45 °C 55 °C 65 °C

1.2

1 0.8 0.6 0.4

0.2 0.1

Most of the research studies has been exposed that if the nanoparticles are well isolated in base fluids, the increase in nanoparticles proportion by weight in base fluids cause escalation in the nanofluids overall thermal conductivity and viscosity, however, decreasing its distribution stability entirely signifying that there must be a linear association among the nanoparticles thermal conductivity and weight proportion [50]. It focused on a thorough assessment of the nanoparticle's thermal exchange transport process. This might be directed to various other parameters influencing the nanofluid's thermal conductivity comprising the vital significance of nanoparticles, nanoparticle clustering and the base or layer fluids on the nanoparticles colloidal surface [51]. Though few researchers stated that the nonlinear link could cause clustering of nanoparticles in base fluids. [52]. The increase in the nanoparticle's weight proportion beyond a certain level results in the formation of clusters which is unsuitable and consequently executes non-linearity behavior of intensification in thermal conductivity with the weight proportion of nanoparticles. [53]. But, the proportion of nanoparticle weight constantly has a positive impact on the thermal conductivity [54]. The non-linear link in the intensification

0.3 0.5 0.7 Nanoparticle Weight Fraction (wt. %)

0.9

The rmal Conductivity (W/m.K)

2.5 (b)

Kapok Seed Oil (Base Fluid) 0.2 wt. % 0.4 wt. % 0.6 wt. % 0.8 wt. %

2

1.5 1

0.5 0 20

30

40 50 Temperature (°C)

60

70

Fig. 3. Thermal conductivity of the MWCNTs and Kapok Seed oil-based nanofluid; (a): effect of temperature, (b): effect of nanoparticle weight fraction. 5

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Table 1 Thermal conductivity of MWCNTs-Kapok seed oil-based nanofluid at different temperatures and nanoparticles weight fractions. Nanoparticle weight fraction (wt%)

Thermal conductivity (W/ m.K)

25

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

0.35 0.55 0.75 0.95 0.459 0.6325 0.864 1.152 0.567 0.7458 0.954 1.2647 0.684 0.895 1.0987 1.394 0.784 0.954 1.1235 1.485

35

45

55

The rmal Conductivity (W/m.K)

65

2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

The rmal Conductivity (W/m.K)

Temperature (°C)

Table 2 Curve-fit parameters for a dimensionless model for thermal conductivity prediction of the MWCNTs and kapok seed oil-based nanofluid.

Experimental (25 °C) Experimental (35 °C) Experimental (45 °C) Experimental (55 °C) Experimental (65 °C) Hamilton & Crosser Model Maxwell Model

0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Nanoparticle Weight Fraction (wt. %)

The rmal Conductivity (W/m.K)

2.5

Experimental (25 °C) Experimental (35 °C) Experimental (45 °C) Experimental (55 °C) Experimental (65 °C) Dimensionless Model

2 1.5 1 0.5 0

0.2 0.4 0.6 0.8 Nanoparticle Weight Fraction (wt. %)

1

Models

AAE

MSE

SSE

RMSE

R2

Correlation Dimensionless model

0.0664 0.0674

0.1211 0.1182

0.0846 0.0830

0.0650 0.0644

0.9781 0.9784

0.9

Knf Kf

=

Knp + 2Kf + 2φ (Kf − Knp ) Knp + 2Kf − φ (Kf − Knp )

(8)

Where, φ represents the thermal conductivity of nanoparticles weight fractions, while Knf, Knp and Kf symbolize the nanofluids, base fluids and nanoparticle's thermal conductivities, respectively. The Maxwell model was used for the estimation of experimental data and the fallouts are given in (Fig. 4). It has been observed from the results that the Maxwell model is unable to evaluate the thermal conductivity of MWCNTs and kapok oil seed-nanofluid at different proportions of nanoparticle weight and temperature range. There might be following reasons behind these outcomes. Firstly, the Maxwell model was projected on the basis of two hypotheses comprising that the nanofluid's thermal conductivity is dependent on the ratio of nanoparticles in base fluids, nanoparticle's thermal conductivity (specifically round-shaped) and base fluids thermal conductivity. Therefore, the second hypothesis is that for the precise evaluation of nanoparticle's thermal conductivity by implicating Maxwell model, the irregular phases should be of round shaped [61]. The unsuitable estimation of MWCNTs and kapok seed oil-based nanofluids was might be due to the fact that the Maxwell model doesn't deal with other factors that significantly influence the thermal conductivity of nanofluids i.e. interfacial strain among the base fluids and nanoparticles, temperature, shape, and size of nanoparticles and Brownian movement. Thus, the Maxwell theory miscalculates the accurate evaluation of MWCNTs and kapok seed oil-based nanofluid's thermal conductivity and the error intensifies with the increase in ratio of nanoparticles volume because of neglecting the part of substantial parameters in the Maxwell theory.

0.5

0 0.2 0.4 0.6 0.8 Nanoparticle Weight Fraction (wt. %)

1.0071 2.5009 0.5605 3.1155 0.8424

Table 3 Statistical analysis of the developed correlation and dimensionless model.

1

0

γ a b c d

Fig. 6. Performance evaluation of the dimensionless model.

Experimental (25 °C) Experimental (35 °C) Experimental (45 °C) Experimental (55 °C) Experimental (65 °C) Correlation

1.5

Value

0

Fig. 4. Performance evaluation of Maxwell and Hamilton & Crosser Model

2

Parameter

1

Fig. 5. Performance evaluation of developed correlation.

conductivity approximation of standardized suspensions of liquids/solids in late 19th century. A number of the thermal conductivity prediction models given by the scientists in the past were Maxwell modelbased. In order to forecast the thermal conductivity of huge spherical and less concentrated particles-based liquid/solid mixtures he anticipated a model (Eq. 8) based on the effective medium theory explained as follows:

6

International Communications in Heat and Mass Transfer 110 (2020) 104402

SSE

10

1.5

0.9

1.3

0.8

1.1

0.7

0.9

0.6

0.7

0.5 1

0.4

0.1

0.01 -20

-10

0 % Perturbation

10

20

(c) γ a b d c

19.8

14.8

0.1

0.2

-0.1

0.1

-0.3

0

-0.5

SSE

0.2 0 -0.2 0

1.25

1E+28

0.75

5

10 15 20 Experimental Observations

25

5.00E+08

0.00E+00

0.25

1E+22 -0.25

1E+19

1E+10

-5.00E+08

-0.75

-1.25

-1.00E+09

-1.75

10000000

γ a b d c

-2.25

10000 -2.75

10

-0.2

-3.25

0.01 -30

-20

-10

0 % Perturbation

10

20

30

(d)

1E+25

1E+13

4.8

0.4

1E+31

1E+16 9.8

0.6

0.3

30

24.8

0.8

0.5

0.3

SSE

-30

1

a b d e c

(b)

0

30

5

10 15 20 Experimental Observations

25

Residual

100

1

Re s idual

a b d e c

Same Global Minimum of the Objective Function for All Parameters

Re s idual

(a)

SSE

1000

Residual

A. Mukhtar, et al.

-1.50E+09

-2.00E+09 30

Pre dicte d The rmal Conductivity (W/m.K)

Fig. 7. Sensitivity and residual analysis of proposed models; (a) sensitivity analysis of proposed correlation, (b): residual analysis of proposed correlation, (c) sensitivity analysis of the proposed dimensionless model, and (d): residual analysis of proposed dimensionless model.

0.6

The nonconformity in Maxwell model evaluations from the experimental statistics might be due to the fact that the Maxwell model only deals with the huge sized [62] and this error is similar to the published research studies [63–67]. Though in few studies, for example, SiO2 nanoparticles-ethylene glycol-based nanofluid, the Maxwell model accurately estimated the thermal conductivity in precise proportions of nanoparticle volumes studied in that experiment. This is a thoughtprovoking opinion because most of the published research studies have been reported that the Maxwell model does not support the nanofluids in which ethylene glycol was used as base fluid. The illogicality of Maxwell model with the experimental data of thermal conductivity has been conveyed in many nanofluids based on ethylene glycols such as Y3Al5O12 suspension [68], ZnO [69], SiC [70], Co3O4 [71], AIN [72], SnO2 [73] and BN [74]. The application of Maxwell model on nanofluids tends to miscalculate the approximation of thermal conductivity rate according to low or high temperatures and ratio of nanoparticles volume, respectively as underestimated in the present study and flops. The Hamilton and Crosser (HeC) projected a novel model (Eq. 9) for the enhanced approximation of thermal conductivity and to modify the theory of the Maxwell model by adding a new parameter into Maxwell model recognized as shape factor. The main objective was the modification of Maxwell model for the constant phase along with the irregular phase.

0.4

Knf

1.6 (a)

1.4 1.2 1.0 0.8 0.6 0.4 0.2

Pre dicte d The rmal Conductivity (W/m.K)

0.2

0.4 0.6 0.8 1 1.2 1.4 1.6 Experimental Thermal Conductivity (W/m.K)

1.6 (b)

1.4 1.2 1 0.8

Kf

0.2 0.2

0.4 0.6 0.8 1 1.2 1.4 1.6 Experimental Thermal Conductivity (W/m.K)

=

Knp + (n − 1) Kf + (n − 1) φ (Kf − Knp ) Knp + (n − 1) Kf − φ (Kf − Knp )

(9)

Here n denotes the experiential shape factor for nanoparticles Knp/ Kf > 100, n factor is illustrated further as follows:

Fig. 8. An alternative way for residual analysis; (a): proposed correlation and (b): proposed dimensionless model.

n=

7

3 ψ

(10)

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A. Mukhtar, et al.

Here ψ denotes the round-shaped nanoparticles (Eq. 11) and stated as the proportion of the surface area of flawless sphere to particles surface area (Ap), by signifying that the sphere volume is alike the nanoparticles volume (Vp), which is illustrated as given behind: 1

ψ=

could be formulated as:

2

π 3 (6Vp ) 3 Ap

0.10

) φ0.48T 0.33

(15)

π3 =

T To

(16)

π4 =

L dp

(17) (18)

It is assessed from the dimensionless groups given above that the first dimensionless group (π1) is a function (Eq. 19) of other dimensionless groups as given:

π1 =

Knf Kbf

Knp T L , , , φ ⎞⎟ = f (π2 , π3 , π4 , π5) = f ⎛⎜ ⎝ Kbf To dp ⎠

(19)

The nanofluids thermal conductivity is greater as compared to the base fluid's thermal conductivity. Resultantly, we gain (Eq. 20):

Knf ⎞ ⎛ Knf ⎞ = 1 + R Knf > Kbf ≫ ⎛⎜ ⎟ > 1 ≫ ⎜ ⎟ K bf ⎝ ⎠ ⎝ Kbf ⎠

(20)

Here R denotes the enhancement factor (Eq. 21) and can be stated as: a

c

b

⎡ Knp ⎞ ⎛ T ⎞ ⎛ L ⎞ d⎤ R = γ ⎢ ⎛⎜ ⎜ ⎟ (φ) ⎟ ⎥ K T d ⎦ ⎣ ⎝ bf ⎠ ⎝ o ⎠ ⎝ p ⎠ ⎜

⎟

(21)

In the end, the dimensionless model (Eq. 22) can be formalized as:

Knf

a

b

Kbf

c

⎡ Knp ⎞ ⎛ T ⎞ ⎛ L ⎞ d⎤ = 1 + γ ⎢ ⎜⎛ ⎜ ⎟ (φ) ⎟ ⎥ K T d ⎦ ⎣ ⎝ bf ⎠ ⎝ o ⎠ ⎝ p ⎠ ⎜

⎟

(22)

Where, γ, a, b, c, and d denote the curve-fit parameters measured though the least-square method and tabularized in (Table 2). The efficiency of the dimensionless model is given in (Fig. 6). An effective link was obtained among the predicted and experimental data evaluated by using dimensionless models in temperature ranging from 25 to 65 °C, MWCNTs aspect fraction (O.D. × L = 6–13 nm × 2.5–20 μm) and weight proportion of nanoparticles in the range of 0.2–0.8 wt% presented in (Fig. 6). Moreover, the projected dimensionless models also provide knowledge about the non-linearity in the association among the nanofluid thermal conductivity and nanoparticle's weight proportion however both classical models (Hamilton-Crosser (HeC) and Maxwell) discussed in the earlier section were failed to find non-linear linkage. Fundamentally, these classical models have been anticipated for evaluation of thermal conductivity of large particles and these models are incapable of evaluation at noncontinuous level or molecular level linkages which is significantly vital for nanoparticles. Therefore, both classical models (Hamilton-Crosser (HeC) and Maxwell) resulted in almost similar miscalculation in forecast for this system because these models failed to deal with dimension characteristics of MWCNTs and interfacial shell. Though, the impact of MWCNTs shape and size couldn't be ignored as exposed in the experimental outcomes.

(12)

Here T and φ denote the temperature and nanoparticle's weight proportion, respectively. The Kbf and Knf represent the thermal conductivity of base fluids and thermal conductivity of nanofluids, respectively. The process of the established correlation is presented in (Fig. 5). The statistical evaluation has been done to evaluate the precision of correlation for the forecast of thermal conductivity and the fallouts are tabularized in (Table 3). 3.6. Dimensionless groups-based model development With the support of suitable dimensionless groups based on the features that have a strong impact on the nanofluids thermal conductivity, a model has been projected in this part for the precise estimate of nanofluids based on MWCNTs and kapok seed oil. The assumption was made about this model that the thermal conductivity influenced significantly due to the thermal conductivity of base fluids (Kbf), nanoparticles (Knp), reference temperature (To), nanofluid temperature (T), nanoparticles diameter (dp) and length (L) (for example MWCNTs and nanoparticles proportion in the base fluid). Consequently, the subsequent eq. (Eq. 13) is obtained:

Knf = f (Knp , Kbf , To , T , L, dp , φ)

Knp

π5 = φ

A novel regression based-model has been developed on the basis of experimental data to ensure the need for an appropriate model for accurate approximation of thermal conductivity and to improved considerate and assessment of temperature and nanoparticle's weight proportion influence on the MWCNTs and kapok seed oil-based nanofluids thermal conductivity. At first, the correlation is established and then least-square method was used for the regression analysis by applying the attained experimental data to assess the best curve-fit parameters. The developed correlation is a function of nanoparticle's weight proportion and temperature and effective in the nanoparticles volume fraction ranging from 0.2–0.8 wt% and temperature in range of 25–65 °C. The established correlation (Eq. 12) could predict the thermal conductivity with high precision (R2 = 0.9781) as similar to the published correlation, i.e. nanofluids based on MWCNTs-thermal oil [46].

= 0.38(e1.78φ

(14)

Kbf

(11)

3.5. Correlation development

Kbf

Kbf

π2 =

The roundness is 0.5 and 1 for cylinder-shaped such as MWCNTs and round-shaped nanoparticles, respectively. It has been detected from the fallouts (Fig. 4), that HeC model was also not suitable for the precise prediction of thermal conductivity and miscalculate the rate of nanofluid thermal conductivity. Though in other similar reported studies it has declared that HeC models could precisely predict the pure water-based nanofluids in contrast with the Maxwell model, however it couldn't predict the thermal conductivity of nanofluids based on ethylene glycol or oil accurately. The reason behind might be the interfacial resistance among the base fluids and nanoparticles surface. The HeC model accurately predicts the thermal conductivity when base fluid is pure water, due to the low interfacial resistance however in the present case where kapok seed oil is used as base fluid the HeC model couldn't precisely predict the thermal conductivity due to high interfacial resistance as compared to pure water [75].

Knf

Knf

π1 =

3.7. Sensitivity analysis The fallouts of the sensitivity and residual assessment of the anticipated dimensionless and correlation models are shown in (Fig. 7) and a different method of residual analysis outcome illustrated in (Fig. 8). The outcomes of sensitivity analysis of both models (dimensionless and correlation models) have confirmed that all features of

(13)

The dimensionless groups (Eq. 14–18) for further model derivation 8

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models have global least solution of 0% miscalculation. This suggests that features of both models (dimensionless and correlation models) are optimized and the predicted values are the finest values with lowest value of the objective function (SSE). The correlation model demonstrated the least value of the objective function (SSE) also recognized as global lowest which is 0.0846 however the projected dimensionless model demonstrated the lowest value of 0.0830 which is lowest amid all apprehension. Moreover, the residual analysis has confirmed the dissemination of values repeatedly deprived of any definite tendency or pattern (Fig. 7) which presented the sign of suitability of the assessed features of both projected models (dimensionless and correlation model. Moreover, few investigators did the residual analysis by plotting graph among the model projected values and experimental values by means of the model feature values measured by the sensitivity analysis. The fallouts of residual analysis are presented in (Fig. 8) signifying the precision among the predicted data and experimental data for both projected models (dimensionless model and correlation model).

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4. Conclusion By considering the significance of the thermal and physical characteristics of the nanofluids for their applications in energy storage and heat transfer equipments, in this work, new type of nanofluid based on the MWCNTs as a nanoparticles and Kapok seed oil as a base fluid has been synthesized and subjected to the experimental measurement of thermal conductivity over various temperatures (25–65 °C) and nanoparticles weight fractions (0.2–0.8 wt%). The experimental data has been used to predict the thermal conductivity using classical models such as the Maxwell model and Hamilton-Crosser model which unfortunately failed in the precise estimation of thermal conductivity of the MWCNTs and kapok seed oil-based nanofluid. Furthermore, two new correlations have been developed based on regression analysis and dimensionless group analysis approach which were able to predict the thermal conductivity of the MWCNTs and kapok seed oil-based nanofluid comparatively with higher accuracy. In the view of modeling results by all these proposed approaches, the models can be ranked based on statistical indicators in the following order: Dimensionless Group Analysis Model > Correlation. Finally, the sensitivity analysis has been performed to assure the accuracy of the model parameters of both proposed models i.e. proposed correlation and dimensionless model. The results revealed the global minimum values for all parameters at 0% perturbations indicating that the model parameters were estimated with high accuracy and their adequacy is confirmed by residual analysis. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgment The authors would like to acknowledge the research grant YayasanUTP (YUTP-0153AA-H01) and research grant and the Department of Chemical Engineering at Universiti Teknologi PETRONAS (UTP), Malaysia for providing state of the art research facilities. Among the authors, Sami Ullah also extends his appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia for support through Research Groups Project under research grant number (R.G.P.2/59/40). Furthermore, the authors would like to acknowledge the funding support from the grant of Z. M. A. Merican (Grant No. FRGS-0153AB-K26), and Universiti Teknologi PETRONAS for the use of facilities and technical assistance. References [1] J.C. Maxwell, A treatise on electricity and magnetism, 1 Clarendon Press, 1881.

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