Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 2, pp 1451–1464 | Cite as

Experimental investigation of the effect of an external magnetic field on the thermal conductivity and viscosity of Fe3O4–glycerol

  • Mohammadhossein Hajiyan
  • Soroush Ebadi
  • Shohel MahmudEmail author
  • Mohammad Biglarbegian
  • Hussein Abdullah


Thermophysical properties, such as thermal conductivity and viscosity, of magnetic nanofluids (MNFs) can be enhanced by applying external magnetic fields. Such property enhancement can be beneficial for having a non-contact control of heat transfer rates in many applications such as cooling of electronic devices, heating mediator for targeted cancer treatment, drug delivery, and heat transfer medium in energy conversion systems. In this study, a detailed experimental investigation has been carried out to measure the thermal conductivity and viscosity of a magnetic nanofluid under the influence of a uniform external magnetic field. The MNF (i.e., glycerol–Fe3O4) is prepared by dispersing Fe3O4 magnetic nanoparticles in glycerol at different volume fractions of nanoparticles (i.e., φ = 0.5, 1.0, 1.5, 2.0, and 3.0%). The experimental results showed that the viscosity linearly increased with the increase in volume fractions while significantly decreased with the increase in temperature. With respect to the viscosity measurement, the maximum ratio revealed a value of 7.2 for 3.0% volume fraction and 50 °C subjected to 543 [G] magnetic field. Also, a 16.9% thermal conductivity enhancement was achieved when φ = 3.0% at 40 °C under 543 [G] magnetic field. Using the experimental results, a nonlinear model was developed as a function of temperature (T), magnetic field (B), and volume fractions of nanoparticles (φ) to predict the thermal conductivity of glycerol–Fe3O4. The proposed model provided satisfactory performance with an R2 value of 0.961, MSE value of 0.00015, and MAE value of 0.00932.


Fe3O4 nanoparticle Glycerol Magnetic nanofluid Thermal conductivity enhancement Symbolic regression 

List of symbols


Magnetic field (G)


Hexadecyltrimethylammonium bromide


Characteristic size of the nanoparticle (nm)


Ethylene glycol


International Centre for Diffraction Data


Joint Committee on Powder Diffraction Standards


Thermal conductivity (W m−1 K)


Boltzmann’s constant = 1.38066 × 10−23 J K−1


Mean absolute error


Magnetic nanofluid


Mean square error


Poly-glutamic acid


Prandtl number


Root-mean-square error


Reynolds number


Sodium dodecyl sulfate


Temperature (°C)

Greek symbols


Density (g mL−1)


Volume fraction of nanoparticles (%)


Dynamic viscosity (mPa s)


Sphericity of nanoparticles











The effective heat removal in heating/cooling system and thermal management of mechanical/electrical components are key factors of an optimum operation and durability of such systems. To reach this aim, different types of heat transfer fluids (HTF) such as oil, water, and air have been implemented as the working fluid in systems such as heat exchangers [1, 2, 3]. From an economic point of view, it is critical to design highly compact and effective heat exchangers. However, this fact is restricted by the low thermal conductivity of HTFs. To address this issue, various methods such as the implementation of highly conductive metal fins, porous medium, and nanoparticles have been proposed. Recently, nanotechnology, particularly employment of nanofluids, has attracted a great deal of attention of researchers to develop efficient HTFs that can be used as working fluids in many applications especially in heat exchangers for heating and cooling purposes [4]. For instance, nanofluids, being efficient HTFs, are commonly used in condensers and evaporators for heating/cooling purposes to enhance the thermal performance of such systems [5].

Moreover, a combination of a nanofluid and inserts as a passive technique for enhancing the performance of energy conversion systems has been used by many researchers [6].

Additionally, nanofluids have shown considerable potential for other applications including cutting fluids in machining processes, biomedical devices, electronic cooling systems, and oil recovery enhancement [7, 8, 9, 10, 11, 12]. Nanofluids, which were firstly coined by Choi [13], can be produced by dispersing highly conductive nanoparticles such as CuO, ZnO, Al2O3, TiO2 (i.e., size of 1–100 nm) into the base fluids including water, oil, kerosene, and glycerol which can lead to an improvement in the thermal conductivity of such fluids. In addition to the thermal conductivity, other thermophysical properties of base fluids such as viscosity (μ) and density (ρ) would be varied by dispersion of nanoparticles at different volume/mass fractions [14]. Moreover, the performance of nanofluids depends on the size of the nanoparticles and their stability, which is their ability to stay suspended. Although having all suitable features of using nanoparticles makes them a proper solution for enhancing the thermal conductivity of a base fluid, several drawbacks remain such as aggregation, sedimentation, and clogging in microchannels which may restrict the potential of using high volume/mass fraction of nanoparticles [15]. Magnetic nanofluid (MNF) is a special type of nanofluid in which the thermophysical properties (e.g., thermal conductivity, viscosity) can be varied in a non-contact way by applying an external magnetic field. MNFs can be produced by dispersing magnetic nanoparticles including but not limited to CO, Fe2O3, and Fe3O4 into base fluids such as water, ethylene glycol, kerosene, glycerine, and thermal oils. In MNFs, the application of a magnetic field causes the nanoparticles to aggregate in the carrier fluid, creating highly conductive paths for the heat flow inside the convective heat transfer environment. Therefore, the thermal conductivity and viscosity of MNFs can be tuned by varying the intensity of magnetic field. Such unique behaviors of MNFs in the presence of a magnetic field make it possible to induce and control the heat transfer process as well as fluid flow. A review on the thermophysical properties, heat transfer characteristics, and the practical applications of MNFs is available in Bahiraei and Hangi’s review article [16]. Recently, MNFs have been under considerable investigation for their medical and biological applications. For instance, the ease with which the properties of MNFs can be controlled and their compatibility with human biology provide them with the potential to be used in drug delivery and in hyperthermia method of cancer treatment. A comprehensive review regarding the use of MNFs in hyperthermia applications can be found in [17]. Bio-magnetic fluids are suitable for biomedical applications, since their flow and properties can be tuned using an external magnetic field. Magnetohydrodynamics, which refers to the flow of magnetic nanofluid in presence of magnetic field, is governed by the Navier–Stokes and Maxwell’s equations that can characterize the fluid flow and magnetic field. A review on the mathematical simulation of magnetohydrodynamics is provided in [18].

Over the years, considerable experimental and numerical studies have been carried out to identify the behaviors of nanoparticles suspended in commonly used base fluids with the aim of improving heat transfer performance. Water has been the most common base fluid for nanofluids and magnetic nanofluids [19, 20, 21, 22, 23, 24]. Magneto-hydrodynamic (MHD) analysis and convective heat transfer of CuO–water was studied in [25]. The effect of non-uniform magnetic field on water-based nanofluid was numerically studied and reported in [26]. Kannadasan et al. [27] compared the pressure drop and heat transfer of CuO–water experimentally under turbulent region. Thermal performance of Al2O3–water in a mini-channel heat sink was studied experimentally by [28]. Significant enhancement in thermal conductivity was reported when compared with previous studies having different mass fractions of nanoparticles (ωnf = 0–14%). Since long-term stability of nanofluids is the main goal in almost all nanofluids applications, viscous fluids like kerosene, thermal oil, and glycerol have been used since their higher viscosity slows sedimentation of the nanoparticles. In addition to stability of nanofluids, using surfactant [e.g., sodium dodecyl sulfate (SDS), hexadecyltrimethylammonium bromide (CTAB), oleylamine, and oleic acid] is an efficient way to enhance the stability of suspending nanoparticles in the nanofluids [19]. In fact, by adding the surfactant (organic or non-organic) to base fluids, the adhesion behaviors of nanofluids change and help to control the growing particles size. However, adding surfactant affects the pH of the mixture and may react with either nanoparticles or base fluid at high temperatures. Magneto-hydrodynamic analysis of Cu–kerosene over a cone was implemented by Raju et al. [29] and was shown that kerosene’s temperature-dependent viscosity enhanced the heat transfer process. In [30], the laminar heat transfer of kerosene–nanofluid carbon nanotubes in microchannel heat sink was studied. It was reported that enhancing the slip velocity coefficient on the solid surfaces significantly affected the minimum temperature. In an experimental investigation by Huang and Luo [31], the buoyancy-induced free convection of kerosene-based magnetic nanofluid was observed on a horizontal temperature gradient. Goshayeshi et al. [32] examined the effect of magnetic field on kerosene–Fe2O3 inside an oscillating heat pipe, and it was reported that the heat transfer performance improved by 16% using kerosene-based magnetic nanofluids. Agarwal et al. [33] investigated the thermal performance of kerosene–graphene at different mass percentages (0.005, 0.02, 0.05, 0.1, 0.2 mass%). Oleylamine was used as a surfactant, and the highest thermal conductivity was measured at 1.3 (which was 23–29% enhancement) when 0.2 mass% of nanoparticle was used. Recently, MNFs are used in some diverse applications including (1) HTF in energy conversion systems or thermal management applications, (2) heat mediator in targeted cancer treatment, and (3) biomedical applications such as bioseparation, drug testing, and magnetic resonance imaging (MRI) [34, 35]. Since the heat transfer via MNFs is usually convection dominated, investigation of the thermophysical properties especially thermal conductivity and viscosity will be necessary for having an efficient HTF. These properties will be tuned by using different volume/mass fractions of nanoparticles and various strengths of the applied magnetic field. Also, using a base fluid with a low freezing point temperature is advantageous for heating/cooling systems in cold environments. Researchers try to not only increase the thermal conductivity of liquids, but also to expand the range of working temperatures of nanofluids. Thus, liquids such as ethylene glycol (EG) and glycerol (or their mixtures with water) have been used as base fluids in nanofluids applications. Glycerol has many applications in the pharmaceutical and personal care industries, as well as in the food industry and as a chemical intermediate [36]. In engineering applications, glycerol has been used in many experimental and theoretical studies [37, 38, 39]. Since the viscosity of glycerol and EG are relatively high, they are great candidates to be used for preparation of nanofluids with long-term stability. As discussed earlier, magnetic nanoparticles are added to a base fluid to have more control over some thermophysical properties, namely thermal conductivity and viscosity in the presence of a magnetic field. Table 1 provides a summary of recent studies using glycerol- and EG-based fluids with different magnetic nanoparticles.
Table 1

Summary of experimental studies on glycerol- and EG-based magnetic nanofluids



Volume fraction/concentration

Base fluid


Abareshi et al. [40]




Studied rheological properties of nanofluids

Viscosity increased by a factor of 1.3 by increasing φ at 40 °C

Linear relation between shear stress and shear rate

Tshimanga et al. [41]




Thermal conductivity enhanced by 19% using MgO-glycerol

New correlations developed to predict thermal conductivity (R2 = 0.99)

Hemmat Esfe et al. [42]




Provided a correlation using ANN successful to predict knf

Maximum improvement ratio was 1.48 at T = 55 °C and dp = 20 nm

Tsai et al. [43]




Experimentally showed that kcond. = kMaxwell

Brownian motion is a factor to improve thermal conductivity

The highest enhancement ratio was observed at 1.08 using φ = 2.0% Fe3O4

Atashrouz et al. [44]




The reason of increase in viscosity is formation of aggregates

Proposed two models with less than 5% MARD

Sundar et al. [45]




Oleic acid surfactant

60:40% EG/W + Fe3O4 was 2.94 times more viscous than base fluid

The effect of surfactant and pH was not studied

Ishiki et al. [46]



Glycerol and serum

Increasing the glycerol concentration from 0 to 60% increased the viscosity by a factor of 10

Brownian motion, aggregation, and hysteresis played key role in magnetic nanofluids

Xu et al. [47]


2–22 mg


Magnetic solid phase extraction of proteins via Fe3O4NH2GO-DES

Wan et al. [48]


0.28 g


Stable MNF (coated iron oxide particles with PGA or PGMA) was seen in 10% NaCl, 10%CaCl2, or pH range of 2–14

Organic surfactant was added to increase the stability

Pastoriza-Gallego et al. [49]


Mass fraction up to 25%


Enhanced knf is temperature independent

Enhanced k by 16.8 and 13.4%

For Fe2O3 and Fe3O4, respectively

Sonawane and Juwar [50]


0.2, 0.5, and 0.8%


Optimized conditions for φ, sonication time, and temperature in MNF

Maximum thermal conductivity was measured at 0.694 Wm−1 C−1

Harandi et al. [51]




Maximum thermal conductivity ratio was

30% at 50 °C when φ = 2.3%

Nonlinear correlation for thermal conductivity with R2 = 0.9904

Afrand et al. [52]




Rheological behavior of MNF was examined for range 25–50 °C

Non-Newtonian behavior for high values of φ

Most recently, there has been a notable effort to measure the viscosity of nanofluids, particularly those using glycerol or EG as a base fluid. In much of the relevant literature, many precise correlations and models were proposed to predict the behaviors of MNFs, namely thermal conductivity and dynamic viscosity [53, 54, 55, 56]. To the best of the authors’ knowledge, there has been no experimental investigation on the effect of magnetic field on the thermal conductivity and viscosity of pure glycerol and Fe3O4 without surfactant. Pure glycerol is chosen as the base fluid due to its high viscosity which enables the prepared MNF to carry higher volume fractions of nanoparticles without sedimentation. Additionally, in some applications it is necessary to have liquid with low freezing point or higher boiling temperature than water. Thus, the motivation behind this study is to experimentally investigate the effect of magnetic field with various intensities on MNF and establish the thermal conductivity model as a function of volume fraction of nanoparticles (φ), and magnetic field (B) for viscous magnetic nanofluids.

The rest of the paper is organized as follows: Sect. 2 presents the characterization and preparation of magnetic nanofluid. Sections 3 and 4 provide the detailed information on thermal conductivity and viscosity measurements, respectively. Section 5 comprises the results and discussion. Finally, Sect. 6 concludes the paper.

Characterization and preparation of magnetic nanofluid

To investigate the thermal conductivity and viscosity of glycerol + Fe3O4, the apparatus used in this study are: ultrasonic liquid processors (Qsonica), variable gap magnet (EM8618-PASCO), KD2 Pro thermal properties analyser single needle sensor, SV-10 viscometer, and Bell 5170 single-axis digital magnetometer. The two-step method is commonly used to prepare nanofluids because the process is simple and economical [57, 58]. This method entails adding the nanoparticles, in the form of a dry powder, directly into the liquid. To avoid agglomeration and to ensure the uniform distribution of nanoparticles in the fluid, a dispersion method should be used. The iron (II, III) oxide nanopowder of 50–100 nm and density of 4.8–5.1 g mL−1 at 25 °C was purchased from Sigma-Aldrich, Missouri, USA [59]. Figure 1 shows the SEM image of the magnetic nanoparticles Fe3O4 used in this study captured by the Quanta TM 250 FEG at University of Guelph. Comparing the images of nanoparticles in Fig. 1 and SEM images in [60], it is apparent that the shapes of nanoparticles in our study are combination of both spherical and hexagonal.
Fig. 1

SEM image of Fe3O4 nanoparticles—a ×100,000, b ×50,000

The average particle size distribution is calculated based on SEM image analysis, and the results are provided in Fig. 2. From eight different SEM images, about 200 particles were counted and their diameters were estimated. As can be seen in Fig. 2, the Gaussian profile is fitted to the histogram (R2 = 0.98) and the average diameter of the Fe3O4 particle is estimated to be about 80 nm. Figure 3 shows the X-ray diffraction pattern for crystal information of Fe3O4 and the magnetic nanoparticle powder. From Fig. 3, the XRD pattern of the as-received sample shows only the Bragg diffraction peaks of Fe3O4. The peaks of Fe3O4 were identified using ICDD (JCPDS) card # 19-629. As it is shown in Fig. 3, the most two intense peaks of the Fe3O4 X-ray diffraction are at 2ϴ = 35.4° and 2ϴ = 62.5° from the lattice planes of (311) and (440), respectively.
Fig. 2

Diameter distribution of Fe3O4 nanoparticle

Fig. 3

XRD pattern of Fe3O4 [54] and magnetic nanoparticles powder

Preparing homogeneous nanofluid with a long-term stability is a crucial factor for thermal conductivity enhancement. As stated previously, stable nanofluids can be obtained by adding surfactants to base fluids; however, this method has some drawbacks such as undesired increase in viscosity. Since glycerol as a base fluid is highly viscous, the long-term stability and minimum sedimentation can be achieved. In addition, due to the high resistance between layers of glycerol, it can carry higher volume fractions of nanoparticles. In this study, the sonication method was used to disperse the nanosized particles within the glycerol. Magnetic nanofluids (glycerol + Fe3O4) with different volume fractions of nanoparticles (0.5–3%) were prepared in a 50-mL beaker in a room temperature (20–25 °C) without any surfactant. Then the nanofluids were ultrasonicated with a sonicator operated at 40 kHz and 500 W for 1 h to ensure uniform dispersion of nanoparticles within the base fluid.

Thermal conductivity measurement

In this study, thermal conductivity of nanofluids was measured with a Decagon Devices KD2 Pro thermal properties analyser, which measures the thermal conductivity using the transient hot wire technique [61]. This technique is more accurate and reliable for samples with low temperature (below 50 °C), since, as the temperature of fluid increases, the convective heat transfer between the sensor surface and the surrounding fluid may affect the measurement. The duration of measurement was set to 1 min to ensure the accuracy of the readings and to reduce convective heat transfer between the needle and the fluid. Also, it is strongly recommended to eliminate any vibrations during the measurement and the sensor and fluid/nanofluids should be absolutely still during the measurement [62]. Two samples of pure glycerine and distilled water were used to calibrate the KD2 Pro thermal property analyser, and each measurement was repeated five times to reduce the measurement uncertainties. Then the average of the five data was compared with the standard sample provided by manufacturer [63], and in both cases the device provided less than 5% error which was also reported by the manufacturer. The average measured thermal conductivities for glycerine and water were 0.282 W m−1 K at 21.1 °C and 0.598 W m−1 K at 21.2 °C, respectively. Comparing the measured thermal conductivities with reference data [64, 65], it was concluded that the KD2 Pro was successful in providing accurate measurement for the liquid in the room temperature range. Figures 4 and 5 are a schematic of the apparatus and the experimental setup, respectively, used to measure the thermal conductivity of nanofluids by subjecting them to a magnetic field. The prepared nanofluid samples were transferred into to a cylindrical enclosures (27 × 72 mm), shown in Fig. 4b, and placed between the magnetic poles. The magnetic field could be changed using two screws connected to the both poles on right and left sides of the device (Fig. 4a). The KS-1 sensor (with built-in heater and temperature sensor) was inserted vertically into the enclosure filled with the magnetic nanofluid. The enclosure was secured between two magnetic poles which could be moved further from or closer to the enclosure using the adjustable screws.
Fig. 4

a Schematic view of the experimental setup for thermal conductivity measurement, b pure glycerol versus MNF

Fig. 5

Experimental setup for thermal conductivity measurement under magnetic field

The strength of the magnetic field between the poles was measured at different gap distances, and the results are shown in Fig. 6. This was done before measuring the thermal conductivity, as taking both measurements at the same time could affect their accuracy. Then the results were fitted, and curve is provided in Fig. 6 to be used later, with coefficient of determination R2 = 0.99.
Fig. 6

Characterization of the variable gap magnet (EM8618-PASCO)

Viscosity measurement

In this study, the viscosities of MNFs were measured using a sine wave SV-10 vibro viscometer with a range of 0.3–10,000 mPa s. According to the manual, the device is accurate to within 1% (repeatability), for the aforementioned viscosity range. Figure 7 shows the viscometer used in this study to measure the viscosity of the samples at different temperatures. The viscometer includes two viscosity detection sensor plates and one temperature sensor. The temperature sensor was located between the two plates which were submerged in the sample liquid and vibrating at a frequency of 30 Hz. The device was tested and calibrated with pure glycerol (99.9%) at room temperature (20 °C) with a given viscosity of 1410 mPa s. Then, magnetic nanofluid samples with different volume fractions of nanoparticles (0.5, 1.0, 1.5, 2, and 3%) temperatures between 27 and 50 °C were transferred into the cup of the apparatus which was secured between magnetic poles. The device constantly monitored and recorded the viscosity and temperature of the samples using its display unit.
Fig. 7

SV-10 sine wave vibro viscometer and display unit

Regarding the accuracy of measurements in this experimental study, Table 2 presents the uncertainty analysis and measurement range of all devices used in this work.
Table 2

Range and accuracy of different measuring devices


Measurement range


Bell 5170

1 G–20 kG

± 1 G from 1 G–20 kG

SV-10 vibro

0.3 mPa s–10 Pa s

±1% of repeatability

KD2 Pro

0.1–4.0 (W m−1 K)

± 0.02 W m−1 K from 0.1 to 0.2 (W m−1 K)

± 5% from 0.2 to 4.0 (W m−1 K)

Results and discussions

The effects of temperature and magnetic field on the viscosity and thermal conductivity of glycerol–Fe3O4 are studied, and results followed by discussions are provided in this section.

Viscosity analysis

The nanofluid’s viscosity was first measured under no magnetic field, since in highly viscous fluids such as glycerol, the viscosity can vary significantly with temperature. The accuracy of these results was then verified and validated against the following classical models:

The Einstein model is given by [66]:
$$\mu_{\text{nf}} = \mu_{\text{f}} (1 + 2.5\varphi )$$
The Batchelor model to predict the viscosity of nanofluids is as follows [40]:
$$\mu_{\text{nf}} = \mu_{\text{f}} (1 + 2.5\varphi + 6.5\varphi^{2} )$$
The Brinkman model which is the modified version of Einstein model is [1]:
$$\mu_{\text{nf}} = \frac{{\mu_{\text{f}} }}{{(1 - \varphi )^{2.5} }}$$
Amani et al. [66] provided the model for viscosity as follows:
$$\mu_{\text{nf}} = \mu_{\text{f}} 1.101(1 + \varphi )^{3.495} T^{ - 0.023}$$
where T is the temperature, and finally, the Sundar model is as follows [45]:
$$\mu_{\text{nf}} = \mu_{\text{f}} \left( {1 + \frac{\varphi }{12.5}} \right)^{6.356}$$
Figure 8 compares the predicted viscosity for different volume fractions of nanoparticles using different models against the experimental results obtained in this study. It is apparent that the previous models were not successful in computing the viscosity of magnetic nanofluid. Note that the viscosity models were obtained for nanofluids that had base fluids with low Prandtl numbers (e.g., water, kerosene, EG). Based on our experimental data in Fig. 8, in order to predict the viscosity of MNF with highly viscous base fluid under no magnetic field, the exponential function is provided as follows:
$$\mu_{\text{nf}} = \mu_{\text{f}} {\text{e}} ^{0.2055 \varphi }$$
Equation 6 was obtained via curve fitting through the experimental data with R2 = 0.98. Note that this equation is valid at 30 °C under no magnetic field for MNF with viscous base fluids.
Fig. 8

Comparison between classical models and experimental results for viscosity at 30 °C

Figure 9 shows the viscosity of magnetic nanofluids with different volume fractions at different temperatures. From Fig. 9, it can be observed that increasing the volume fraction of nanoparticles results in increasing the viscosity. However, the viscosity decreases significantly with increasing temperature. Regardless of the temperature’s effect, the increase in viscosity with volume fraction of nanoparticles can be explained by the increased shear between layers of the nanofluids in the presence of nanoparticles. Also, by increasing the volume fraction of nanoparticles, the amount of liquid trapped between particles increases. This increases the internal friction of the nanofluid, thus increasing its viscosity.
Fig. 9

Viscosity of pure glycerol and magnetic nanofluids (glycerol + Fe3O4) with different volume fractions versus temperature under no magnetic field

In order to further elaborate on the effect that magnetic field and volume fraction of nanoparticles have on the viscosities of MNFs, Figs. 1012 depict the influence of magnetic fields on viscosity for different volume fractions (1.0, 2.0, and 3.0%) at various temperatures (25–55 °C). A decreasing trend in viscosity was noted when temperature was increased for all three volume fractions. The viscosities of the samples increased proportionally with the volume fraction of nanoparticles, given a uniform magnetic field. Also, the faster rate of change in viscosity with temperature for samples with a higher volume fraction of nanoparticles was observed. It was found that the magnetic field had a more pronounced effect on high-volume fraction MNFs than on low-volume fraction MNFs. This is because high-volume fraction MNFs have a higher concentration of nanoparticles that construct the chain-like structure along magnetic fields which resist fluid flow. From Fig. 12, it can be seen that the influences of the magnetic field on viscosity are more notable at high temperatures than at low temperatures. This is mainly because of the strong magnetic forces which are more dominant than internal forces of the MNF at high temperatures. Also, the non-Newtonian behavior of MNFs is observed at high temperatures in the presence of the strong magnetic field when φ = 3.0%.
Fig. 10

Variation of viscosity against temperature under different magnetic field intensities (φ = 1.0%)

Fig. 11

Variation of viscosity against temperature under different magnetic field intensities (φ = 2.0%)

Fig. 12

Variation of viscosity against temperature under different magnetic field intensities (φ = 3.0%)

Thermal conductivity analysis

Thermal conductivities of magnetic nanofluid (glycerol + Fe3O4) for five different volume fractions of nanoparticles (0.5, 1.0, 1.5, 2.0, and 3.0%) were measured experimentally. Since the thermal conductivity depends strongly on temperature, the samples were kept at a fixed temperature of approximately 300 K during the measurements. The results obtained without the application of a magnetic field were compared to the following classical models:

The Maxwell–Garnett model is given by [11]:
$$\frac{{k_{\text{nf}} }}{{k_{\text{f}} }} = \frac{{k_{\text{s}} + 2 k_{\text{f}} - 2\varphi (k_{\text{f}} - k_{\text{s}} )}}{{k_{\text{s}} + 2 k_{\text{f}} + \varphi (k_{\text{f}} - k_{\text{s}} )}}$$
The Bruggeman model is as follows [15]:
$$\frac{{k_{\text{nf}} }}{{k_{\text{f}} }} = \frac{1}{4} \left[ {\left( {3\varphi - 1} \right)\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + (2 - 3\varphi )} \right] + \frac{{k_{\text{f}} }}{4} \sqrt \Delta$$
where Δ = \(\left( {3\varphi - 1} \right)^{2} \left( {\frac{{k_{\text{s}} }}{{k_{\text{nf}} }}} \right)^{2} + 2(2 + 9\varphi - 9\varphi^{2} )\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }}} \right)\)The Hamilton–Crosser which is given by [67]:
$$\frac{{k_{\text{nf}} }}{{k_{\text{f}} }} = \left[ {\frac{{k_{\text{s}} + \left( {n - 1} \right)k_{\text{f}} - (n - 1)\varphi (k_{\text{f}} - k_{\text{s}} )}}{{k_{\text{s}} + \left( {n - 1} \right)k_{\text{f}} + \varphi (k_{\text{f}} - k_{\text{s}} )}}} \right]$$
where \(n = \frac{3}{\psi }\) and ψ is sphericity of nanoparticles (1 for spherical shape and 0.5 for cylindrical shape).
The following approach is the Rayleigh model [11]:
$$k_{\text{nf}} = k_{\text{f}} + 3\varphi \left[ {\frac{{k_{\text{s}} k_{\text{f}} }}{{2k_{\text{f}} + k_{\text{s}} \varphi \left( {1 + \frac{{3.939\varphi^{2} \left( {k_{\text{s}} k_{\text{f}} } \right)}}{{4k_{\text{f}} }} + 3k_{\text{s}} } \right)k_{\text{f}} k_{\text{s}} }}} \right]$$
The Jeffrey model for thermal conductivity is given by [68]:
$$k_{\text{nf}} = k_{\text{f}} + 3\varphi k_{\text{f}} + k_{\text{f}} \varphi^{2} \left[ {3\chi^{2} + \frac{{3\chi^{2} }}{4} + \frac{{9\chi^{3} }}{16}\left( {\frac{{\alpha_{t} + 2}}{{2\alpha_{t} + 3}}} \right) + \frac{{3\chi^{3} }}{64}} \right]$$
The Lu–Li model is given by [69]:
$$k_{\text{nf}} = k_{\text{f}} + k_{\text{f}} \alpha_{t} \varphi + \chi \varphi^{2} k_{\text{f}}$$
where \(\chi = \frac{{\alpha_{t} - 1}}{{\alpha_{t} + 2}}\) and \(\alpha_{t} = \frac{{k_{\text{s}} }}{{k_{\text{f}} }}.\)
Table 3 shows the difference between the experimental investigation versus classical models for thermal conductivity under no magnetic field in which the metrics for comparison were mean absolute error (MAE) and root-mean-square error (RMSE). According to Table 3, all theoretical models provided higher values than our experimental results. The discrepancy can be explained by the fact that the temperature plays a key role in thermal conductivity of high-viscous fluids (e.g., glycerol) than low-viscous fluids (e.g., water, kerosene). The best model which predicted the thermal conductivity of MNF was Lu–Lin model with MAE and RMSE values of 0.003645 and 0.000014, respectively.
Table 3

Summary of measured and calculated thermal conductivity using classical models































































Figure 3 shows the variation of thermal conductivity ratio versus volume fraction at different temperatures (20–45 °C). The linear incremental trend and direct temperature dependency can be observed for all MNF samples. Figure 13 also shows that this upward linear trend of thermal conductivity with volume fraction of nanoparticles is further exaggerated at higher temperatures.
Fig. 13

Effect of volume fraction of nanoparticles on thermal conductivity enhancement at different temperatures under no magnetic field

As stated previously in Table 2, the uncertainty measurement of the KD2 Pro is 5% over the temperature range of 0–50 °C. To include this uncertainty in our measurements, we have added this value to the thermal conductivity readings. Figures 1416 show the thermal conductivity versus magnetic field strength at different temperatures for three different volume fractions, 1.0, 2.0, and 3.0%, respectively, considering 5% uncertainty appeared as error bars. It should be mentioned that applied magnetic fields were parallel to temperature gradient during the thermal conductivity measurements. The effect of magnetic field on thermal conductivity of MNF directly depends on the direction of magnetic field. As stated in the similar study by Parekh and Lee [70], if the direction of magnetic field is parallel to the temperature gradient, it helps to improve the thermal conductivity by creating chain-like structure and consequently accelerate the energy transportation within the magnetic nanofluids. This is the main reason behind the thermal conductivity enhancement in our experimental study.
Fig. 14

Effect of magnetic field on thermal conductivity of glycerol + Fe3O4 at different temperatures when φ = 1.0%

Fig. 15

Effect of magnetic field on thermal conductivity of glycerol + Fe3O4 at different temperatures when φ = 2.0%

Fig. 16

Effect of magnetic field on thermal conductivity of glycerol + Fe3O4 at different temperatures when φ = 3.0%

In all cases, effective thermal conductivity increments under magnetic field. Note that the variation of thermal conductivity for 1.0% volume fraction was small, and slight improvement was observed. This can be explained by the fact that the amount of particles was not enough to create the chain-like structure along the direction of magnetic field. In Fig. 16, significant improvement in thermal conductivity was observed due to the applied magnetic field when temperature was 40 °C. It can be explained by the attendance of a magnetic field; magnetic forces between particles become dominant compared to other internal forces (such as van der Waals forces). This makes particles align with the direction of the magnetic field into a zipper-like formation which is the main reason for the improved thermal conductivity. Since highly viscous fluids are more mobile when subjected to high temperature than to low temperature, the effect of temperature on thermal conductivity in these fluids cannot be neglected. In order to quantify the improvement in the thermal conductivity in the presence of magnetic field, the improvement ratio is defined as follows:
$$I_{\phi ,{\text{H}}} = \frac{{k_{\text{H}} - k_{0} }}{{k_{0} }} \times 100$$
where I is the percent improvement for a particular volume fraction of nanoparticles and applied magnetic field, and where kH and k0 are the thermal conductivities with and without magnetic field, respectively. Using Eq. 12, the maximum improvements in thermal conductivity of MNFs under the magnetic field at 40 °C are calculated to be 6.1 and 16.9% for φ = 2.0% and φ = 3.0%, respectively.

Proposed correlation for thermal conductivity

As mentioned before, it is always desired to have an accurate model to predict the thermophysical properties of MNFs. Since the thermal conductivity depends strongly on temperature, volume fraction of nanoparticles, and magnetic field intensity, an accurate model is provided as a function of temperature (T), magnetic field (B), and volume fraction of nanoparticle (φ). Thus, after regression analysis using on our experimental data for each parameters versus thermal conductivity, the following nonlinear expression was developed as a single objective symbolic regression via genetic programming:
$$k\left( {\varphi ,B,T} \right) = \left( {a_{1} + a_{2} T + \frac{1}{{a_{3} e^{{a_{4} \varphi }} + a_{5} }} + a_{6} e^{{a_{7} \varphi }} } \right) \left( {a_{8} + a_{9} T + a_{10} B} \right) + a_{11}$$
where T is temperature in °C, B is the magnetic field intensity in Gauss, and φ is the volume fraction of nanoparticles in percentage (%). Note that a1,…, a11 are the coefficients of this nonlinear function which are provided in Table 4.
Table 4

Coefficient and accuracy of the proposed model for thermal conductivity


R2 (training)

R2 (test)

a 1

a 2

a 3

a 4

a 5

a 6

a 7

a 8

a 9

a 10

a 11




− 4.588



− 0.591

− 2.69



− 6.8e−6

− 1.27e−7


Based on Table 4, the coefficient of determination for training and testing of the model is 0.974 and 0.961, respectively. In addition, Table 5 provides the performance of the proposed correlation (Eq. 13) using mean square error (MSE), mean absolute error (MAE), and root-mean-square error (RMSE). Figure 17 shows the predicted values using the proposed model in comparison with the corresponding observed values. As seen in this figure, most of the data are close to the exact line and the error does not exceed the error lines (± 5%). Thus, it can be seen that the proposed nonlinear function provided acceptable accuracy for the thermal conductivity as a function of temperature, volume fraction, and magnetic field intensity. The model can be used to optimize the required magnetic field intensity and to control the heat transfer rate in thermal systems or cooling applications. To further investigate the accuracy of the developed model in Eq. 14, it was compared with the empirical correlation for effective thermal conductivity provided by Corcione [71] as follows:
$$\frac{{k_{\text{eff}} }}{{k_{\text{f}} }} = 1 + 4.4{\text{Re}}^{0.4} { \Pr }^{0.66} \left( {\frac{T}{{T_{\text{fr}} }}} \right)\left( {\frac{{k_{\text{p}} }}{{k_{\text{f}} }}} \right)^{0.03} \varphi^{0.66}$$
Where Tfr is the freezing point of the base fluid, \({\text{Re}} = \frac{{2 \rho_{\text{f}} k_{\text{b}} }}{{\pi \mu_{\text{f}}^{2} d_{\text{p}} }}\), and Pr = \(\frac{{c_{\text{f}} \mu_{\text{f}} }}{{k_{\text{f}} }}\)
Table 5

Performance analysis of the proposed correlation using MAE, MSE, and RMSE
















Fig. 17

The predicted thermal conductivity using the proposed model versus actual experimental results

Note that Eq. 15 is the thermal conductivity model as a function of volume fraction, temperature, Reynolds, and Prandtl numbers. Table 6 shows the comparison between Corcione’s model in [71] and our proposed model under no magnetic field.
Table 6

Comparison between the proposed model for thermal conductivity and Corcione [70] under no magnetic field at different temperatures


\(\varphi\) = 1%

\(\varphi\) = 2%

\(\varphi\) = 3%

Proposed model

Corcione [70]

Proposed model

Corcione [70]

Proposed model

Corcione [70]






















As can be seen in Table 6, both models provided satisfactory results and showed that thermal conductivity is an increasing function of both temperature and volume fraction of nanoparticles. Thus, it can be concluded that the proposed model in this study is a robust and accurate model for thermal conductivity since it includes the magnetic field term alongside temperature and volume fraction of nanoparticles.


In this work, the characterization and preparation of glycerol–Fe3O4 was carried out. The viscosity and thermal conductivity were experimentally investigated for five different volume fractions of nanoparticles (0.5, 1.0, 1.5, 2, and 3%) at different temperatures (20–55 °C) under uniform magnetic fields. The results showed that applying magnetic field significantly increased the viscosity and thermal conductivity of MNFs. Also, it was found that the effect of magnetic field was more pronounced on MNFs with high volume fractions of nanoparticles. With respect to the thermal conductivity enhancement, the maximum improvement of 16.9% was observed at 3.0% volume fraction and 40 °C. With respect to the viscosity, the maximum ratio revealed a value of 7.2 for 3.0% volume fraction and 50 °C subjected to 543 [G] magnetic field. Based on the experimental results, a new nonlinear correlation was developed using genetic programming to predict the thermal conductivity of glycerol–Fe3O4 as a function of temperature (T), magnetic field intensity (B), and volume fraction of nanoparticles (φ).



The authors would like to thank Dr. Ashutosh Singh for providing laboratory apparatuses for the present experimental study. Also, the first author is grateful to Dr. Amirreza Shirani Bidabadi and Peter J. Krupp for their supports and recommendations.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • Mohammadhossein Hajiyan
    • 1
  • Soroush Ebadi
    • 1
  • Shohel Mahmud
    • 1
    Email author
  • Mohammad Biglarbegian
    • 1
  • Hussein Abdullah
    • 1
  1. 1.School of EngineeringUniversity of GuelphGuelphCanada

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