Measurement of the thermal conductivity of SiO2

Thermochimica Acta 661 (2018) 84–97

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Thermochimica Acta journal homepage: www.elsevier.com/locate/tca

Measurement of the thermal conductivity of SiO2 nanofluids with an optimized transient hot wire method

T

⁎

Wenwen Guo, Guoneng Li, Youqu Zheng , Cong Dong Department of Energy and Environment System Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Thermal conductivity SiO2 nanofluids Transient hot wire method Measurement error Particle size

Thermal conductivity enhancement of nanofluids can be affected by measurement deviation and particle size distribution. In the present work, deionized water (DW) and ethylene glycol (EG) based nanofluids were made and characterized. A nanofluid thermal conductivity measurement system founded on transient hot wire method was built up and a comprehensive analysis regarding measurement error was made for the two base fluids. The proper hot wire working current and the measurement time were acquired to decrease the measurement error caused by non-constant heating power and natural convection. Then, the nanofluid thermal conductivity was measured with optimized measuring parameters. It was found that the thermal conductivity increases 3.2% and 9.6% for 0.5 vol.% SiO2-EG and 1.0 vol.% SiO2-EG, higher than the result for DW based nanofluids (1.0% and 3.4%). This may be attributed to the higher value of λBrownian and the difference in particle shape distribution.

1. Introduction As a new heat transfer medium with high thermal conductivity, nanofluid has aroused increasing attention since the concept was firstly proposed in 1995 by Choi et al. [1]. Then nanoparticles of metals or oxides began to be used as additives to traditional heat transfer fluids (water, alcohol, heat conduction oil, etc.) to form uniform and stable nanofluids. Nanofluid has much higher heat-conducting capability than conventional fluid. It’s available for improving the rate of heat transfer and reducing the dimensions of heat exchanger. And thus nanofluid has a promising prospect in many industrial applications. Nanofluid thermal conductivity enhancement is considered to be closely associated with the characteristics of nano-particles and the type of base fluid [2,3]. Much attention has been paid to enhance the nanofluid thermal conductivity in previous research. Different nanoparticles were applied in making nanofluids, including metal oxides (CuO, Al2O3, SiO2, TiO2, ZrO2, etc.), new carbon materials with high thermal conductivity (carbon nanotube and graphene, etc.) and hybrid nanoparticles [4]. Masuda et al. [5] used ultrasonic dispersion method to prepare water based nanofluids and found that the base fluid thermal conductivity would rise by 30% and 10% with the addition of 4.0 vol.% Al2O3 and TiO2 respectively. Sundar et al. [6] studied the magnetic Fe3O4 nanofluids and reported that 2.0 vol.% Fe3O4 nanofluid prepared by precipitation method can increase 25% thermal conductivity at 20 °C. Carbon based nano-materials were also used to prepare nanofluids and showed extraordinary enhancement effect on thermal ⁎

conductivity in literatures [7–9]. Wusiman et al. [8] prepared water based MWCNT nanofluids with addition of SDBS as a dispersant. They found that the thermal conductivity enhanced 2.8% when 0.5 wt.% CNTs and 0.25 wt.% SDBS were added to water. Chen and Xie [9] prepared double and single walled carbon nanofluids without dispersant addition and claimed that the nanotubes with smaller diameter can contribute to higher thermal conductivity. Base fluid is also of vital importance to the thermal conductivity increment. Compared with deionized water (DW), ethylene glycol (EG) has a longer temperature range of liquid state, and EG-based nanofluids have attracted the attention of researchers due to its potential in refrigeration and heat transfer engineering [10]. Lee et al. [11] prepared CuO-EG nanofluids and found that with the addition of 4.0 vol.% CuO (particle size of 23.6 nm) the thermal conductivity presented an 20% increase. Agarwal et al. [12] investigated the influence factors on the thermal conductivity of Al2O3 nanofluid. It was reported that the EG base fluid had a great impact on thermal conductivity increase when nanoparticle concentration was kept unchanged. When 0.5 vol.% Al2O3 nanoparticles were added in, the thermal conductivity was increased 3.5% and 5.0% for water base fluid and EG base fluid respectively. There have been many published data on nanofluid thermal conductivity measurement. These results, however, differ from one another even for one type of nanofluid with similar volume concentration and primary particle size. Table 1 lists the thermal conductivity enhancement results from literatures (water based Al2O3 nanofluids). Perhaps part of the reason for the deviation was that while the primary particle

Corresponding author. E-mail address: [email protected] (Y. Zheng).

https://doi.org/10.1016/j.tca.2018.01.008 Received 2 August 2017; Received in revised form 4 January 2018; Accepted 11 January 2018 Available online 12 January 2018 0040-6031/ © 2018 Elsevier B.V. All rights reserved.


W. Guo et al.

Nomenclature

Greek symbols

DW EG THW T t F0 Ra Lw Cp q

a λ ρ γ δ β ν ξ μ θ

Q U U0 r0 I g vol.% wt.%

Deionized water Ethylene glycol Transient hot wire method Temperature (°C) Time (s) Fourier number Rayleigh number Hot wire length (m) Specific heat (J/kg K) Heating power of the platinum hot wire per unit length (W/m) Heating power of the platinum hot wire (W) Voltage (V) DC supply voltage (V) Wire radius (m) Hot wire working current (A) Gravitational acceleration (m2/s) Volume percent Weight percent

Thermal diffusivity (m2/s) Thermal conductivity W/(mK) Density (kg/m3) Euler constant, 0.5772 Thermal-boundary-layer thickness (m) Thermal expansion coefficient (1/K) Kinematic viscosity (m2/s) Ratio of hot wire length to diameter Ratio of hot wire conductivity to fluid conductivity Ratio of fluid specific heat capacity to hot wire specific heat capacity

Subscripts bf nf w 0

Base fluid Nanofluid Hot wire Initial state

Table 1 Literature data on the thermal conductivity enhancement of water based Al2O3 nanofluids. Primary particle size (nm)

Temperature (°C)

Volume concentration (%)

Thermal conductivity enhancement (%)

Test method

References

45 45 30 53 53 47 47 47 47 11 40 40 40

25 25 21 20 20 20 20 20 20 20 20 20 20

3.7 7.4 0.3 0.5 1.0 5.0 8.0 1.0 4.0 1.0 1.0 1.5 2.0

6.0 12.5 2.9 3.5 5.8 4.8 10.0 2.5 8.0 5.0 8.0 13.0 15.0

SSCCM

Barbes et al. [16]

THW THW

Lee et al. [29] Agarwal et al. [12]

THW

Mintsa et al. [30]

THW

Chon et al. [31]

THW

Heyhat et al. [32]

curve in a relatively large time range. Shi et al. [25] investigated the effect of radiation on the measure accuracy of liquid propane, and demonstrated the radiation effect could not be neglected at 372 K. De Castro et al. [26] studied the influence of hot wire heat capacity on the thermal diffusivity measurement accuracy of liquid and gas. The proper tstart time for gas and liquid were discussed and obtained. Recently, several papers reported the measurement accuracy about nanofluids thermal conductivity by THW method. Hong et al. [27] studied the influence of convection on EG-based 1.06 vol.% ZnO thermal conductivity measurement accuracy by THW method, and found that the thermal conductivity results calculated from different temperature ranges were remarkably different. While Lee et al. [28] found the thermal conductivity measurement accuracy for EG was not affected by natural convection during the test time range (tstart = 0 ∼ 15s, tend = 0 ∼ 20s) at a selected voltage value of 1.0 V. Then they applied the optimized measurement system with the voltage to measure the thermal conductivity of Al2O3-EG nanofluids accurately. To improve the measurement accuracy of nanofluid thermal conductivity, detailed error analysis is needed. But as yet few papers have made detailed error analysis regarding nanofluid thermal conductivity measurement based on THW method. In the present work, a comprehensive analysis of measurement error for THW method has been made to determine the key parameters (hot wire temperature rise, measurement time range, etc.) that influence the measurement accuracy of nanofluid thermal conductivity. Then the thermal conductivity and particle size distribution measurement of two types of SiO2 nanofluid

sizes are similar, the real particle sizes in nanofluids are distinct from each other due to the particle agglomeration and the difference in preparation method (ultrasonic time, dispersant type, etc.). In fact, the real particle size of nanofluids should be measured for the better understanding of the thermal conductivity enhancement effect. Another influence factor lies in the measurement deviation [13]. How to precisely obtain the nanofluid thermal conductivity is where the shoe pinches. Various approaches are used to measure nanofluid thermal conductivity: transient hot wire method (THW) [14], parallel-plate technique (STPPH) [15], coaxial cylinders method (SSCCM) [16] and temperature oscillation technique (TOT) [17], among which THW is one of the most widely used methods due to its short measurement time and convenient operation. The general idea of THW method is to generate heat in a thin metallic wire (often called hot wire). The hot wire is immersed in the fluid where it acts as heat source and thermometer. The temperature rise rate of the hot wire can reflect the fluid thermal conductivity [18]. The THW method is widely used in the thermal conductivity measurement of liquid, gases, etc. Coated with a thin electrical insulation layer, hot wire can be applied to measure the thermal conductivity of electrically conducting liquids [19–21]. A few authors also did research on simultaneous measurement on liquid thermal conductivity and thermal diffusivity [22,23]. Some factors in the THW method, such as natural convection and hot wire heat capacity, can influence the measurement accuracy directly. Yu et al. [24] claimed that the thickness of hot wire insulation layer showed no significant effect on the slope of ΔT-lnt 85


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were implemented. And the effects of nanoparticle volume concentration, particle size and base fluid type on nanofluid thermal conductivity were further investigated.

λ=

2.1.2. Measurement system In the experiment an Isonel-insulated platinum wire was taken as the hot wire. Its diameter is 50 μm and length is 18.30 cm. During the measurement process, the platinum hot wire was immersed in the fluid and given a constant heat flow. The heat was then transferred to the surrounding fluid. By recording the changing trend of the hot wire electrical resistance, the temperature change of the fluid with time was obtained. According to Eq. (9), the thermal conductivity of the fluid is acquired. As shown in Fig. 1, the Wheatstone bridge was used in the thermal conductivity measurement system for the purpose of obtaining the electrical resistance variation of the hot wire. R1 and R2 are two precision resistors of 20 Ω, with an accuracy of 0.01%. RPt represents the electrical resistance of the platinum hot wire. Rx and Rd are two resistance boxes. A DC power with constant voltage supplied power for the Wheatstone bridge circuit. The unbalanced DC voltage acquisition was carried out by Agilent 34970A, with a sampling frequency of 50 Hz and measuring range of 100 mV. At first, the bridge was balanced under a current of 1.0 mA, and the voltage change ΔU acquired by data acquisition system (DAQ) should approach to zero if possible. Then the hot wire was immersed in the fluid and the relay switch was switched to Rd circuit, followed by starting the stabilized voltage supply as well as the DAQ. After the voltage was stable, the relay was turned to the bridge to collect the voltage variation with time (ΔU-t). The relationship of ΔU with U0 is given by:

2.1. Thermal conductivity measurement 2.1.1. Principle of THW method The ideal model of THW method is built grounded on the following assumptions: (1) the hot wire is infinitely long and vertically placed in isotropic fluid. (2) The thermal conductivity of the hot wire is infinite and the heat capacity is zero. Initially, the system is thermodynamically balanced at T0. Then a constant heat flow is applied to the hot wire. The temperature rise at a radial position r conforms to the following equation: (1)

According to Fourier’s heat conduction equation, the differential equation of heat conduction in the process is:

∂ΔT 1 ∂ ⎛ ∂ΔT ⎞ =a r ∂r r ∂r ⎝ ∂r ⎠

(2)

The initial condition and the boundary condition are as follows:

T (r , t ) = T0 t = 0 ∂T

(3)

q

lim r ∂r = − 2πλ t > 0

(4)

r→0

lim T (r , t ) = T0 t ≥ 0

(5)

r →∞

Assuming the thermophysical properties of the fluid are constant, the solution of Eqs. (1) and (2) can be expressed as follows [33]:

R2 R1 ⎞ ΔU = ⎛ − U0 R + R R + RPt ⎠ 2 x 1 ⎝ ⎜

r2

q ⎞ E1 ⎛ ΔT (r , t ) = − 4πλ ⎝ 4at ⎠ ⎜

(10)

The partial differential of ΔU with respect to lnt is:

(6)

d (ΔU ) d (ln t )

λ=

r02

q ⎡ ⎛ 4at ⎞ ln ⎜ 2 ⎟ + + ⋅⋅⋅⎤ ⎥ 4πλ ⎢ 4at ⎦ ⎣ ⎝ r0 C ⎠

q 4at ⎞ ln ⎜⎛ ⎟ 4πλ ⎝ r02 C ⎠

= R1 U0

d (RPt ) d (ΔT ) 1 (R1 + RPt )2 d (ΔT ) d (ln t )

Q d (ln t ) 4πLW d (ΔT )

=

Q 1 d (RPt ) d (ln t ) RU 4πLW 1 0 (R1 + RPt )2 d (ΔT ) d (ΔU )

(11)

2

U0 ⎞ RPt Q=⎛ ⎝ R1 + RPt ⎠

(7)

⎜

Considering that r02 4at → 0 and the high order term can be ignored, Eq. (7) can be written as:

ΔT (r0, t ) =

⎟

⎟

Where E1 (x ) indicates the exponential integral function. Supposing that the temperature distribution of hot wire is uniform, its temperature rise is:

T (r0, t ) − T0 =

(9)

From Eq. (9) the thermal conductivity can be calculated by measuring ΔT in a certain time range.

2. Experimental

ΔT (r , t ) = T (r , t ) − T0

q q d ln t d ln t = 4π dΔT (r0, t ) 4π dΔT (t )

⎟

(12)

The relationship between temperature and hot-wire resistance is:

RW = R 0 (1 + αΔTW ) = R 0 + R 0 αΔTW

(13)

where Rw, R0, α, and ΔTW represent the measured and the initial resistance of hot-wire, the temperature coefficient of resistance (TCR) of hot-wire, and the temperature rise from the initial temperature

(8)

And the fluid thermal conductivity is:

Fig. 1. Schematic of the thermal conductivity measurement system.

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Table 3 Physical properties of materials. Material

Thermal conductivity (W/m/K)

Density (Kg/m3)

Specific heat (J/kg K)

References

SiO2 (27 °C) Ethylene glycol (20 °C) Water (20 °C) Platinum (20 °C)

1.4 0.249

2220 1110

745 2356

[35] [36,37]

0.597 71.6

998 21450

4182 135

[38] [39]

The volume concentration (ϕ ) of nano-SiO2 particle was calculated by Eq. (14): Fig. 2. The temperature-resistance curve of hot wire.

ϕ=

respectively. TCR describes the relative change in the resistance of a substance per unit change in temperature. The TCR of the hot wire was obtained from a calibration experiment. First, the temperature was set to the desired value. Then the hot wire was put into an incubator until it reached a steady temperature with a multimeter (Pro'skit MT-1210-C) measuring the electric resistance of the hot wire. The process was repeated at different temperatures to obtain the hot wire resistance temperature dependence. The temperature-resistance curve is shown in Fig. 2. It was found to be liner in the range of 0–30 °C (Tw − T0) with a slope of 0.033 Ω/°C (d (RPt ) d (ΔT ) = R 0 α = 0.033Ω/ °C ) and an intercept of 9.97 Ω (R0 = 9.97 Ω, Tw = T0 = 20 °C). The average TCR of the hot wire (in the temperature range from 20 to 50 °C) is 0.00331 K−1, which is close to the value 0.00362 K−1 reported by Hong et al. [27] and 0.0035241 K−1 reported by Hong et al. [34].

SiO2

XFNANO technology Inc. Wahaha

Deionized water Ethylene glycol

Sinopharm Chemical Reagent Co., Ltd

Purification method

Analysis method

99.0%

none

≤10 μS/cm*

none

99.0%

none

Certified purity Certified purity Certified purity

+

ω ρ 100 bf

(14)

3.1. Analysis of model error Some deviations in thermal conductivity results derive from the model error and the random error. The model error, i.e. systematic error, is mainly caused by the difference between theoretical modeling and practical testing condition. The random error includes the error produced from the observation of physical quantities (voltage, wire length, etc.). Two base fluids, DW and EG, were applied in the error analysis of the measurement system in order to optimize the measurement parameters and improve the accuracy. The model errors induced in the THW method are discussed in the following sections in detail. 3.1.1. (δλ λ )1, the error caused by ignoringr02 4at In Eq. (7) the high order term is ignored as the diameter of hot wire is much smaller than the arithmetic product of thermal diffusivity and the measuring time. The temperature deviation caused by ignoring the high order term is: δT ΔT =

r02 4at

ln

( ). 4at r02 C

The diameter of the platinum hot wire is 50 μm. And the (δλ λ )1values for DW and EG in different measuring time are listed in Table 4. EG has relatively higher (δλ λ )1 due to its low thermal diffusivity. Both of the two base fluids show a gradual decline in (δλ λ )1value as time increases. When t ≥ 1s, the (δλ λ )1values for DW and EG are lower than 0.0175% and 0.0281% respectively. 3.1.2. (δλ λ )2 , the error caused by finite vessel boundary If b20 at > 5.783, (δλ λ )2 is lower than 10−4 [18]. b0, the inner diameter of the vessel, is 1 cm in the test. For DW at t = 10 s,

Table 2 Sample information. Initial mass Fraction purity

P

3. Results and discussions

The two-step preparation method was utilized in nanofluid preparation. The nano-SiO2 particles (99 wt.% purity, 20 nm particle size) used for nanofluid preparation were provided by XFNANO technology Inc. The sample information is listed in Table 2. The size distribution and average diameter of fresh nano-SiO2 particles were analyzed by transmission electron microscope (TEM, Philips-FEI, Tecnai G2 F30), while the size distribution of particles in nanofluids was detected by Malvern Nano Zetasizer (S type, Malvern Inc.) at 20 °C. The characterization and thermal conductivity measurement were both done within 3 h after sample preparation. In the preparation, nano-SiO2 particles were first added into the base fluids (DW or EG). Then the suspension was stirred for 30 min by magnetic stirrers and dispersed in an ultrasonic bath (100 W, 40 KHz) for 2 h to form a stable suspension. The DW based and EG based nanofluids with 0.5 vol.% and 1.0 vol.% nano-SiO2 particle were denoted as: 0.5% SiO2-DW, 1.0% SiO2-DW, 0.5% SiO2-EG and 1.0% SiO2-EG.

Source

(1 − ) ρ

Where ω is the mass concentration. The physical properties of SiO2, DW and EG were listed in Table 3.

2.2. Nanofluid preparation and characterization

Chemical name

ωρbf ω 100

Table 4 The model error caused by ignoring the high order term for DW and EG in different measuring time.

* The electrical conductivity of deionized water is less than 10 μS/cm (25 ± 1 °C).

87

Base fluid type

Thermal diffusivity a (m2/s)

Measuring time (t s)

(δλ λ )1 (%)

Water

1.43 × 10−7

0.5 1 5 10

0.0396 0.0175 0.0027 0.0018

EG

9.52 × 10−8

0.75 1 5 10

0.0394 0.0281 0.0044 0.0020


W. Guo et al.

Fig. 3. The ΔT-t curve of (a) DW and (b) EG during the thermal conductivity measurement process. (c) The relationship between the temperature rise and the error caused by nonconstant heating power.

b20 at = 69.9 > 5.783 and (δλ λ )2 < 0.01%.

The error caused by non-constant heating power of the hot wire can be calculated in the following equation:

3.1.3. (δλ λ )3 , the error caused by heat dissipation from the hot wire end In fact, there is heat dissipating from the platinum hot wire end in the axial direction because of its finite length, which furtherly leads to smaller temperature decrease of the hot wire compared with theoretical value. Blackwell [40] proposed that when the ratio of length to diameter is higher than 250 the error generated from axial heat transfer would be kept less than 0.2%, and that the error can be calculated by the following equation: −ζ 2

1/2

⎛ δλ ⎞ = π 1/2e 4F0 ⎡ (4F0) ⎢ ⎝ λ ⎠3 ⎣ ζ

( )

δλ λ 4

= =

= [t

1

t

∫ 2 Q (t ) dt − Q0]/Q0 ≤

2 − t1 t1

Q(t 2) − Q0 Q0

[U0 / (R1(t 2) + RPt (t 2) )]2 RPt (t 2)−[U0 / (R1(0) + RPt (0) )]2 RPt (0) [U0 / (R1(0) + RPt (0) )]2 RPt (0) 1 R1(t 2) 2/RPt (t 2)+2R1(t 2) + RPt (t 2)

RPt (t 2) = RPt (0) + αRPt (0) ΔT

×

(R1(0) + RPt (0) )2 RPt (0)

−1

(16) (17)

The TCR of the precision resistors (R1 and R2) is 25 ppm/°C. During the test time (10s), the actual temperature rise of the precision resistors was very low and assumed to be 2 °C, the resistance change was 0.005% (R1(t2) ≅ R1(0)). According to Eq. (16), the measurement error derived from the precision resistors temperature is 0.006% at T = 20 °C and Istart = 140 mA. This impact is relatively small and can be neglected. According to Eqs. (16) and (17), the relation between the error caused by non-constant heating power (δλ λ )4 and the hot wire temperature rise ΔT can be obtained. The results are displayed in Fig. 3(c). As illustrated in the figure, higher initial fluid temperature can contribute to a lower error. Eqs. (16) and (17) indicate that the error (δλ λ )4 enhances with temperature rise increasing (at t < 50 °C, RPt(t2) < R1(t2) ≅ 20 Ω). When the temperature rise reaches 1 °C with an initial fluid temperature of 20 °C, the error caused by non-constant heating power is 0.11%. When the temperature rise approaches 10 °C, the error climbs to 1.1%. As a result, the temperature rise should be controlled in a reasonable range during the measuring process. The ΔT-t curves of DW and EG during the thermal conductivity measurement process are illustrated in Fig. 3(a) and (b). During the

+ 2ζ(μ − θ−1)(ln 4F0 − γ)(4F0)−3/2⎤ × 100% ⎥ ⎦ (15)

In the present study, Lw = 18.30 cm, r0 = 25 μm, and ζ = 3660. The calculated value of (δλ λ )3 from the equation above is less than 10−5 and can be ignored. 3.1.4. (δλ λ )4 , the error caused by non-constant heating power of the hot wire The hot wire temperature will experience a growth during the measurement, resulting in the change of its electrical resistance and heating power. The error caused by the change of hot wire heating power is dependent on the hot wire initial electrical resistance (decided by the initial fluid temperature) as well as its temperature rise. After the electric power is supplied to the Wheatstone bridge, heat is generated in the precision resistors, leading to a resistance change which would further cause non-constant heating power of the hot wire. The higher the TCR is, the larger the resistance change would be. 88


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connecting lines of the inflection point separate the test area to three zones. The error caused by hot wire heat capacity can affect the test accuracy especially in the nonlinear part (stage 1–2). For DW, the hot wire heat capacity shows a significant influence on the measurement results before t = 0.20 s at Istart = 220 mA. The error caused by hot wire heat capacity (δλ λ )6 dropped down as the test time increases. The actual ΔT of the hot wire considering the hot wire heat capacity, the finite thermal conductivity and the finite radius is reported by Healy et al. [18]:

measurement, a constant voltage was supplied to the Wheatstone bridge. The start working current of RPt is equal to the voltage divided by the original resistance (Istart = U0/(R1(0) + RPt(0))). For the same U0 value, the working current changes with R1. Start working current (Istart) is an accurate parameter and is chosen as the comparison base. On one hand, a stronger current usually results in higher hot wire temperature rise for the same type of fluid, and further leads to a larger model error. On the other hand, a lower working current leads to a smaller value of ΔU, and also means the measuring signal is more likely to be interfered by noise signal. In the measurement, the proper start current is 140 mA for DW and 100 mA for EG. In this condition, (δλ λ )4 is lower than 0.22% and the temperature rise would be controlled under less than 2 °C in the measuring time (10s).

ΔT =

((ρCP ) w − (ρCP )f ⎤ r2 r2 q ⎧⎡ 4at λ ⎫ 1 − r02 × ln ⎜⎛ 2 ⎟⎞ + 0 − 0 + ⎢ ⎥ ⎨ 4πλ ⎩ ⎣ 2λt 2at 2a w t 2λ w ⎬ ⎦ ⎝ r0 C ⎠ ⎭

(19) 3.1.5. (δλ λ )5 , the error caused by radiation The energy dissipation through radiation directly depends on the temperature rise. Assuming the platinum wire is a black body, the heat loss caused by radiation for per unit length of the hot wire can be estimated by the following equation: [18]

qrad = 2πrσ (TW4 − T04 ) ≅ 8πrσT03 ΔTw

Where aw and λw is the thermal diffusivity and thermal conductivity of the hot wire respectively. The physical properties of platinum are shown in Table 3. In most applications the factor [1 − r02 ((ρCP ) w − (ρCP )f ) 2λt ] constitutes the major correction and is due to the finite heat capacity of the hot wire [18]. Thus, the temperature deviation due to the hot wire heat capacity can be expressed as:

(18) −3

For ΔTw = 2 °C and T0 = 20 °C, qrad = 1.79 × 10 W/m. Here the working current in the measurement for DW and EG is 140 mA and 100 mA respectively. The initial heating power is 10.68 W/m and 5.45 W/m respectively. So the heat loss caused by radiation is 0.017% for DW and 0.033% for EG.

(ρCP ) w − (ρCP )f δT1 ≈ r02 ΔT 2λt

(20)

For water and ethylene glycol, the time dependence of δT1/ΔT at 20 °C is displayed in Fig. 5. |δT1/ΔT| drops sharply with time increasing from 0 s to 0.5s. When t > 0.5s, the decrease trend slows down. |δT1/ ΔT| of DW is higher than that of EG, which is due to the fact that (ρCP ) w − (ρCP )DW is higher than (ρCP ) w − (ρCP )EG . At 0.5s, |δT1/ ΔT| = 0.13% for water and |δT1/ΔT| = 0.067% for EG. De Castro et al. [26] studied the influence of hot wire heat capacity on the thermal diffusion measurement accuracy of liquid (λ= 0.129 Wm−lK−l, −1 −1 ρ = 855 kg m−3, Cp = 1720 J kg K ) and gas −l −l −3 (λ = 0.01438 W m K , ρ = 92.86 kg m , Cp = 664.7 J kg −1 K−1). They found δT1 initially fell rapidly with increasing time to zero at 0.3 s for liquid. While for gas, the decline trend of δT1 with time lasted for 0.5s. |δT1/ΔT| is closely related with fluid properties, hot wire properties and test time. Prolonging test time can reduce the value of |δT1/ ΔT|, which may also cause higher measurement error derived from natural convection. In the following study, the start time for DW and EG is set to later than 0.5s.

3.1.6. (δλ λ )6 , the error caused by the finite heat capacity of the hot wire Hot wire heat capacity is not an infinitesimal amount. At the initial stage of charging, the heat generated spread in the hot wire, and then diffused to the surrounding fluid. The heat capacity difference between the hot wire and the surrounding fluid influences the measurement accuracy in the initial stage. According to Eq. (11), the ideal ΔU-lnt curves are linear, and their slopes can be used to calculate liquid thermal conductivity. As shown in Fig. 4, two inflection points divide the curve into 3 stages: (1) hot wire heating dominant stage (stage 1–2, with a large slope); (2) heat conduction dominant stage (stage 2–3, with a high linearity), in which the heat conduction from hot wire to fluid plays a key role; (3) nonlinear part caused by natural convection (stage 3–4). For each curve in Fig. 4, a fourth order polynomial fitting function f(x) is established to fit the ΔU-lnt scatter plot (R2 > 0.999). For x∈[−4, 4.5], f ′ (x ) , f ′ ′ (x ) and f ′ ′′ (x ) are the corresponding first, second and third order differential equations. If f ′ ′′ (x ) = 0 and f ′ ′ (x ) reaches its maximum value at xm. Define k1 = 0.95 f ′ (x m) , and k2 = 1.05 f ′ (x m) . Then let f ′ (x1) = k1 and f ′ (x2) = k2 . The inflection points (x1,f (x1)), (x2,f(x2)) and the slope of the corresponding straight line f ′ (x m) can be obtained. The inflection point (marked as dotted circles) in Fig. 4 can be used to roughly predict the influence time range. And the

3.1.7. (δλ λ )7 , the error caused by natural convection Natural convection driven by fluid temperature gradient can disturb the measurement results of nanofluid thermal conductivity. To reduce the deviation from the ideal model, the influence of natural convection should be considered. Natural convection is in favor of heat dissipation

Fig. 4. The ΔU-lnt curve of (a) DW and (b) EG with different working current at T = 20 °C.

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caused by natural convection. It is worth noting that higher current is easier to induce natural convection, and the nonlinear part of EG is posterior to that of DW due to its high viscosity. Some researchers [41] proposed a judgment rule for natural convection:

Ra = Gr⋅Pr = gβΔTδ 3/(νa) ≤ 105

(21)

Where δ = (28at)1/2 is thermal-boundary-layer thickness. As seen in Fig. 4(a), the ΔU-lnt curve of DW seriously deviates from the linear part in the condition of Istart = 220 mA and t > 9.3 s. At Istart = 220 mA and t = 9.3 s, ΔT is 4.4 °C, and Ra = 1.1 × 105 > 105. The results are consistent with Equation (21), indicating the reliability of the measurement system. To obtain more accurate measurement results, stage 2–3 was selected as the measuring range and the natural convection effect was analyzed for DW and EG by linear fitting method. Fig. 6 displays the influence of natural convection on the measurement accuracy of DW and EG at 20 °C. The hot wire start working current is 140 mA for DW and 100 mA for EG. The plot of locally estimated values for water was calculated based on 50 adjacent data points with the measuring time range of 0–10 s. R2 stands for the goodness of fit. The locally estimated values of DW thermal conductivity show a trend of gentle increase. For DW, the data of initial stage are lower than reference value due to the

Fig. 5. Time dependence of δT1/ΔT at 20 °C for DW and EG thermal conductivity measurement.

from heat source to surroundings, resulting in the slow-down of temperature rise rate and the measurement deviation from the linear part of ΔU-lnt curve. Reasonable measurement parameters (such as measurement time range) can delay the starting of natural convection and reduce the measurement error. The ΔU-lnt curves of DW and EG at T = 20 °C are shown in Fig. 4. The nonlinear part of stage 3–4 was

Fig. 6. The influence of natural convection on the thermal conductivity measurement accuracy of DW and EG (T = 20 °C).

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Fig. 7. Deviations of measured ΔU from fitted straight line for a typical run.

effect of hot wire heat capacity; while those of later stage are higher on account of the influence of natural convection. As time goes on, the influence of natural convection continuously increases and the goodness of fit goes down. It’s evident that the fluid has seen considerable fluctuation in the locally estimated values of thermal conductivity. Although the locally estimated data couldn’t be directly utilized to the measurement for its instability, the goodness of fit of the locally estimated values is a good indicator of natural convection. In Fig. 6, the cumulatively estimated values of thermal conductivity were calculated by using a series of cumulative data. When the measuring end time is later than 4.5 s for DW, the error is larger than 1%. The proper end time for DW is from 1.7 s to 4.5 s (tstart = 0.5 s). The linearity of the results within this range is higher than 0.999. For EG the proper end time is from 3.4 to 9.6 s (tstart = 0.75 s), in which the linearity also reaches 0.999. The total error of thermal conductivity measurement in this range is lower than ± 1% as displayed in Fig. 6.

δQ Q is the error caused by non-constant heating power of the hot wire. It’s already discussed in Section 3.1.4 and classified as a model error. The uncertainty of the hot wire length measuring with a vernier caliper is δLw Lw = 0.03%. The uncertainty of the DC stabilized power supply is δUO UO = 0.1%. δR1 R1 and δ (R1 + RPt ) (R1 + RPt ) are resistance errors. The uncertainty of d (RPt ) d (ΔT ) is estimated to be less than 0.1%. Fig. 7 displays the deviation of measured ΔU (ΔUexp) from the fitted straight line (ΔUfitted). The maximum deviation is lower than 0.25% during the measurement time range (0.5–4.5s). The average deviation is about 0.04%. Thus, the uncertainty of d (ln t ) d (ΔU ) is estimated to be 0.25%. The error analysis results in the thermal conductivity measurement by THW are listed in Table 5, including all the model errors and random errors mentioned above. Among the model errors, the error caused by non-constant heating power of the hot wire (δλ λ )4 and the error caused by natural convection (δλ λ )7 account for the main part. Some of the errors are dependent on hot wire temperature rise, such as(δλ λ )4 , (δλ λ )5 , (δλ λ )7 . So it’s necessary to keep an appropriate temperature rise to measure the fluids thermal conductivity accurately. The hot wire temperature rise is associated with hot wire working current, measuring time and fluid type. For DW it is 1.86 °C (at tend = 4.5 s and Istart = 140 mA) and for EG is 1.69 °C (at tend = 9.6 s and Istart = 100 mA). The value of (δλ λ )6 is closely related with test time. For DW, it is actually much lower than 0.13% (at tstart = 0.5s). At tend = 1.7s, it decreased to 0.04%. And the average value (0.085%) is

3.2. Analysis of random error Random error (E2) is unavoidable and can be expressed as the mean square error based on Eq. (22):

⎧ ⎪ E2 = ⎨ ⎪ ⎩

δLW 2 LW

δU0 2 U0

δR1 2 R1

( ) + ( ) + ( ) + (2 ( ) +( ) δ (dRPt / dΔT ) 2 dRPt / dΔT

1/2 δ (R1 + RPt ) 2 ⎫ +⎪ R1 + RPt

)

δ (d (ln t ) / d (ΔU )) 2 d (ln t ) / d (ΔU )

⎬ ⎪ ⎭

(22)

Table 5 The error analysis results in the thermal conductivity measurement by transient hot wire method at T0 = 20 °C, P0 = 0.1 MPa.a Item Model error

Random errorb

DW (%)

EG (%)

< 0.040 (t ≥ 0.5s)

< 0.039(t ≥ 0.75s)

< 0.01 ignored < 0.20 < 0.016 0.085 0.20

< 0.01 ignored < 0.19 < 0.027 0.028 0.20

δLw/Lw δUO UO

0.03 0.10

0.03 0.10

δR1 R1 δ (R1 + RPt ) (R1 + RPt ) δ(d (RPt ) d (ΔT ) )/(d (RPt ) d (ΔT ) ) δ(d (ln t ) d (ΔU ) )/(d (ln t ) d (ΔU ) )

0.01 0.20 0.10 0.25 < 1.24

0.01 0.19 0.10 0.25 < 1.17

Error Error Error Error Error Error Error

caused caused caused caused caused caused caused

by by by by by by by

ignoring r02 4αt (δλ λ )1 finite vessel boundary (δλ λ )2 heat dissipation from the hot wire end (δλ λ )3 non-constant heating power of the hot wire (δλ λ )4 radiation heat transfer (δλ λ )5 finite wire heat capacity (δλ λ )6 natural convection (δλ λ )7

Maximum total error

Conventional designations: Lw – the hot wire length, U0–the DC supply voltage, R1–the resistance of precision resistors, RPt –the resistance of the hot wire, and ΔT – the hot wire temperature rise. a Standard uncertainties u are u(T) = 0.1 °C, and u(P0) = 10 kPa. b Random errors are the relative expanded uncertainties Ur (0.95 level of confidence).

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Fig. 8. The reliability test results of the measurement under optimized measuring conditions (T = 20 °C).

Fig. 10. SiO2-DW and SiO2-EG nanofluids.

adopted in the error analysis. The value of model error is less than 0.55% and 0.49% for DW and EG, while the value of total error is less than 1.24% for DW and 1.17% for EG.

distribution of SiO2-DW and SiO2-EG nanofluids is displayed in Fig. 11. In all cases, the size of measured nanoparticle exceeds the primary particle size (20 nm). It may be because that the nanoparticles tend to agglomerate by van der Waals force as soon as they are added to the solution. For the SiO2-DW nanofluids, the size distribution is rather dispersed with a wide peak. The SiO2-EG nanofluids, however, have a narrow particle size distribution between 200 ∼ 300 nm. For the two base fluids, the peak widths of the size distribution are not the same. The narrower distribution peak in the case of EG was also observed by other researchers [43]. As the particle concentration grows, the particle size of the nanofluids increases as well.

3.3. Reliability test According to the analysis above, the temperature rise of the hot wire and measurement time range are two key factors that affect the measurement accuracy. In the reliability test, the temperature rise was kept below 2 °C. The thermal conductivity measurement results for DW and EG under optimized measuring conditions are shown in Fig. 8 (Istart = 140 mA, tstart = 0.5 s and tend = 1.7–4.5 s for DW; Istart = 100 mA, tstart = 0.75 s and tend = 3.4 ∼ 9.6 s for EG), from which the total measurement error is 1.2% and 1.3% respectively, close to the error analysis results.

3.4.2. Measurement of SiO2 nanofluids 3.4.2.1. Analysis of measurement error. (a) Model error analysis

3.4. Application to SiO2 nanofluids The nanofluid measurement error also consists of model error and random error. From analysis results in Section 3.1, it can be concluded that the error caused by non-constant heating power of the hot wire (δλ λ )4 , the finite wire heat capacity (δλ λ )6 and natural convection (δλ λ )7 account for the main part. The hot wire temperature rise can influence the measurement accuracy. As displayed in Fig. 12, during the same measurement time range, the hot wire temperature rise in nanofluids is lower than that in the base fluids. The nanoparticle concentration enhancement can facilitate the reduction of hot wire temperature rise. Within the same test time, the error caused by non-constant heating power of the hot wire in

3.4.1. Particle size analysis of SiO2 nanofluids Fig. 9 shows the TEM image of fresh SiO2 nanoparticle. It could be seen that the nano-SiO2 particles are mutually aggregated and spherical in shape with a particle diameter of about 20 nm. The prepared nanofluids are shown in Fig. 10. With the same SiO2 volume concentration, the SiO2-DW nanofluids present a milk-white color, while the SiO2-EG nanofluids are relatively transparent, in accordance with the report by Żyła et al. [42] Dynamic light scattering analysis provides further information on the nanoparticles in the prepared nanofluid. The particle size

Fig. 9. TEM images of primary nano-SiO2 particles.

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Fig. 11. Size distribution by intensity of the SiO2-DW and SiO2-EG nanofluids. (a)1.0 vol.% SiO2-DW, (b) 0.5 vol.% SiO2-DW, (c) 1.0 vol.% SiO2-EG, (d) 0.5 vol.% SiO2-EG.

• Analysis of random error

nanofluid is lower than in the base fluid, which can be calculated by Eqs. (16) and (17). According to Eq. (20), the error caused by hot wire heat capacity is related to fluid property, hot wire property and test time. Due to the small addition of SiO2 nanoparticles, the (ρCP )f of the nanofluids is slightly lower than that of the base fluids ((ρCP )DW = 4173636 J/(m3 K), (ρCP ) 1.0vol .% SiO2− DW = 4148439 J/(m3 K), (ρCP ) W = 2895750 J/(m3 K)) and (ρCP ) w − (ρCP )DW is slightly higher than (ρCP ) w − (ρCP )1.0vol .% SiO2− DW . At the same test time, |δT1/ΔT|nanofluid is also slightly lower than |δT1/ΔT|DW for λnanofluid > λDW. The same start time was chosen for nanofluid thermal conductivity measurement (tstart = 0.5 s for SiO2-DW nanofluid; tstart = 0.75 s for SiO2-EG nanofluid). Among the system errors, the error caused by natural convection (δλ λ )7 also plays an important role. Fig. 13 displays the influence of natural convection on the thermal conductivity measurement accuracy of DW and EG based SiO2 nanofluids. As seen in the figure, the tend-λ curves of the nanofluids and the base fluids both show a similar trend. As tend goes up, the measurement value of thermal conductivity show an increase trend. The influence of natural convection continuously grows with time. For the base fluids (DW and EG), in order to reduce the measurement error derived from natural convection, there are standard values of thermal conductivity in choosing the best test time range. For the nanofluids, however, there are no standard values of thermal conductivity for reference. Within the same test time, ΔTw during nanofluid thermal conductivity measurement is lower than that of the base fluid (displayed in Fig. 12). Therefore, the error caused by natural convection for nanofluid thermal conductivity measurement is lower than the error for the base fluid at the same tend. The curves of tend − λbf/λnf are also displayed in Fig. 13. λbf/λnf changes little with tend. For nanofluid thermal conductivity measurement, the same tend time ranges were selected (tend = 1.7–4.5 s for SiO2-DW nanofluid, tend = 3.4–9.6 s for SiO2-EG nanofluid).

The random error, which is also related to items including δLw Lw , δUO UO , δR1 R1 and etc, can be calculated by Eq. (22) for nanofluid thermal conductivity measurement. For 0.5 vol.%SiO2-DW and 1.0 vol. %SiO2-DW the hot wire temperature rise is 1.77 °C and 1.55 °C (at tend = 4.5 s and Istart = 140 mA), and for 0.5 vol.%SiO2-EG and 1.0 vol. %SiO2-EG, it is 1.63 °C and 1.47 °C (at tend = 9.6 s and Istart = 100 mA). Thus, the error dependent on hot wire temperature rise (δλ λ )4 , (δλ λ )7 can be calculated. Fig. 14 displays the error analysis results in the thermal conductivity measurement of nanofluids. As seen in the figure, (δλ λ )4 , (δλ λ )7 , δUO UO , δ (R1 + RPt ) (R1 + RPt ) and δ(d (ln t ) d (ΔU ) )/(d (ln t ) d (ΔU ) ) account for the main part of the total error. Among the errors, δUO UO and δ(d (ln t ) d (ΔU ) )/(d (ln t ) d (ΔU ) ) are determined by test equipment. Other errors such as (δλ λ )4 , (δλ λ )7 is up to the hot wire temperature rise and the test time. Higher thermal conductivity of the nanofluids can lead to a lower temperature rise in the same test time range compared with the base fluids, further decreasing the values of all the temperature rise dependent errors. The error caused by non-constant heating power of the hot wire (δλ λ )4 is 0.2% for DW. For 0.5 vol. %SiO2-DW and 1.0 vol.% SiO2-DW, it decreases to 0.19% and 0.16% respectively. In conclusion, in the same test time range the value of total error is less than 1.23% and 1.17% for 0.5 vol.%SiO2-DW and 1.0 vol.% SiO2-DW respectively, while for 0.5 vol.%SiO2-EG and 1.0 vol.% SiO2-EG, the value of total error is less than 1.15% and 1.12%.

3.4.2.2. Analysis of nanofluids thermal conductivity enhancement. The measurement was carried out on the system with the optimized measuring parameters. The results are displayed in Fig. 15. The thermal conductivity enhancement effect can be denoted as the relative magnitude of nanofluid thermal conductivity to base fluids Fig. 12. The ΔT-t curves of SiO2 nanofluids during the thermal conductivity measurement process (T = 20 °C).

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Fig. 13. The influence of natural convection on the thermal conductivity measurement accuracy of DW and EG based SiO2 nanofluids (T = 20 °C).

of λnf/λbf as a function of ϕ . Maxwell model is given as:

(λnf/λbf). As seen in Fig. 15, the addition of nanoparticles enables the thermal conductivity to increase as revealed from the value of λnf/λbf. That also means the hot wire temperature rise for the nanofluids is smaller than the base fluids, leading to a lower measurement error. For DW, the thermal conductivity increases about 1.0% and 3.4% respectively, with the addition of 0.5 vol.% and 1.0 vol.% nano-SiO2 particles. While for EG, the thermal conductivity increases about 3.2% and 9.6%, from 0.249 W/m/K to 0.257 and 0.273 W/m/K. Experimental values of thermal conductivity for the liquids are listed in Table 6. The enhancement effect varies with the type, volume fraction and shape of nanoparticles. Many models have been put forward in order to describe the relation of nanofluid thermal conductivity and nanoparticle properties, such as Maxwell Model, H&C Model and Yu&Choi Model. Hamilton and Crosser put forward a model for the calculation of nanofluid thermal conductivity in terms of nanoparticle sphericity [44]: λnf λbf

=

λnf λbf

2λP + λbf + ϕ (λP − λbf ) 2λP + λbf − 2ϕ (λP − λbf )

(24)

The data are from our measurement results (dashed line), model prediction (solid line) and other literatures [42,45–47] (dotted line). It is observed that the λnf/λbf distribution with the same vol.% are inconsistent, implying the complexity of enhancement mechanism. In the present work, the SiO2-DW and SiO2-EG nanofluids both behave higher thermal conductivity than H&C and Maxwell model prediction results. For 0.5 vol.% SiO2-DW and 1.0 vol.% SiO2-DW, the thermal conductivity increases about 1.0% and 3.4% (similar to the literature of Jahanshahi et al. [45]) respectively. While for 0.5 vol.% SiO2-EG and 1.0 vol.% SiO2-EG the thermal conductivity increases about 3.2% and 9.6% (similar to the literature of Xie et al. [46]). From the measurement results, the enhancement effect for EG is more remarkable than DW. Similar phenomenon was also observed by previous researchers [11,48], and the enhancement is supposed to be connected with

λP + (n − 1) λbf − (n − 1) ϕ (λbf − λP )

n = 3/ Ψ

=

λP + (n − 1) λbf + ϕ (λbf − λP )

(23)

Where n is shape factor, and Ψ is sphericity. Fig. 16 presents the curves

Fig. 14. The error analysis results in the thermal conductivity measurement of nanofluids.

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Fig. 15. The measurement results of SiO2 nanofluid thermal conductivity (T = 20 °C).

two key factors: the conventional static part λstatic and the Brownian motion part λBrownian. The Brownian motion in nanofluids can result in a lower temperature gradient and a higher thermal conductivity due to the microconvection effect [49]. The relations of λstatic and λBrownian is [50]:

Table 6 Experimental values of thermal conductivity for the liquids at T0 = 20 °C, P0 = 0.1 MPa.a Samples

Thermal conductivity (W/m/K)

Reference values

DW EG 1.11 wt% SiO2-DWb 2.20 wt% SiO2-DWb 1.00 wt%SiO2-EGb 1.98 wt%SiO2-EGb

0.597 0.250 0.603 0.617 0.257 0.273

0.597 0.249 – 0.615 0.264 0.274

± ± ± ± ± ±

0.010 0.004 0.010 0.016 0.008 0.010

λBrownian / λstatic = ρbf cbf / λbf

[28] [26]

For DW and EG:

(λBrownian / λstatic )DW : (λBrownian / λstatic )EG = 1: 1.5

[35] [36] [36]

Thus, compared with the SiO2-DW nanofluids, the SiO2-EG nanofluids have higher remarkable thermal conductivity enhancement. This may be also connected with the particle size distribution in nanofluids. The primary nanoparticles are all with a spherical shape. However, when added in the base fluid they tend to agglomerate into irregularshape agglomerates such as ellipse and fractal agglomerates [51]. SiO2 nanoparticles may have different shape distributions in the two base fluids, resulting in different particle size distributions. The particles with an elongated shape, for instance, can enhance transfer heat in the direction of the long axis [52]. This may be another reason for the

Conventional designations: DW – Deionized water, EG – Ethylene glycol. a Standard uncertainties u are u(T) = 0.1 °C, u(P0) = 10 kPa, u(ω(SiO2)) = 0.01 wt.%. Reported uncertainties correspond to the expanded uncertainties for 0.95 level of confidence (k ≈ 2). b 1.11 wt% SiO2-DW, 2.20 wt% SiO2-DW, 1.00 wt% SiO2-EG, 1.98 wt% SiO2-EG correspond to 0.5 vol% SiO2-DW, 1.0 vol% SiO2-DW, 0.5 vol% SiO2-EG and 1.0vol% SiO2-EG respectively.

Fig. 16. Comparison of λeff/λbf between the measured data and literature values.

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higher thermal conductivity enhancement in EG in comparison with DW. The proposed explanation still requires a further confirmation.

[10]

4. Conclusions [11]

In the present work, a nanofluid thermal conductivity measurement system based on THW method was built up. To find out the key parameters that influence the measurement accuracy, a comprehensive analysis of system error and random error was made and the thermal conductivity of two types of SiO2 nanofluids were measured with the optimized measuring parameters. The conclusions drawn from the experimental study are as follows:

[12] [13]

[14] [15] [16]

(1) Among all the model errors in the thermal conductivity measurement by THW, the error caused by non-constant heating power and by natural convection account for the main part. (2) Hot wire temperature rise and measurement time range are two key factors that influence the nanofluid thermal conductivity measurement accuracy. Under the optimized measuring conditions (Istart = 140 mA, tstart = 0.5 s and tend = 1.7–4.5 s for DW; Istart = 100 mA, tstart = 0.75 s and tend = 3.4–9.6 s for EG), the total measurement error of thermal conductivity measurement is 1.2% and 1.3% for DW and EG respectively, very close to the error analysis results. (3) At the optimized measurement conditions, the SiO2 nanofluid thermal conductivity measurement accuracy is higher than that of based fluids. The SiO2-DW and SiO2-EG nanofluids both show higher thermal conductivity than the H&C model prediction results. For 0.5 vol.% SiO2-DW and 1.0 vol.%SiO2-DW, the thermal conductivity increases about 1.0% and 3.4% respectively. While for 0.5 vol.%SiO2-EG and 1.0 vol.%SiO2-EG the thermal conductivity increases about 3.2% and 9.6%. (4) The thermal conductivity enhancement for EG is much higher than DW may be due to the higher value of λBrownian and the difference in particle shape distribution.

[17] [18] [19]

[20] [21]

[22]

[23]

[24] [25]

[26]

[27]

Acknowledgment

[28]

We acknowledge the financial support from the National Natural Science Foundation of China (No. 51476146, 51476145, 51606169).

[29]

Appendix A. Supplementary data [30]

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.tca.2018.01.008.

[31]

References

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