A novel semi-empirical model on predicting the thermal conductivity of diathermic oil-based nanofluid for solar thermal application

International Journal of Heat and Mass Transfer 138 (2019) 1002–1013

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A novel semi-empirical model on predicting the thermal conductivity of diathermic oil-based nanofluid for solar thermal application Meng-Jie Li, Ming-Jia Li ⇑, Ya-Ling He, Wen-Quan Tao Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

a r t i c l e

i n f o

Article history: Received 4 January 2019 Received in revised form 31 March 2019 Accepted 16 April 2019 Available online 28 April 2019 Keywords: Nanofluid Thermal conductivity Nanolayer Semi-empirical model Oil-based nanofluid Heat transfer fluid

a b s t r a c t In this paper, a novel semi-empirical model is developed for predicting the thermal conductivity of Diathermic Oil (DO)-based nanofluid for solar thermal application. First, a corrected thermal conductivity model of nanofluid is developed based on the widely used Yu-Choi model. In this corrected model, the thermal conductivity distribution of nanolayer is treated as a quadratic form instead of a constant value along with the distance from nanoparticle. The corrected model shows the predicted thermal conductivity of nanofluid is significantly influenced by nanolayer thickness. Then, the semi-analytical nanolayer thicknesses for different DO-based nanofluid are calculated utilizing the corrected model and the experimental data in literatures. It is found that the nanolayer thickness of DO-based nanofluid is not a constant value but varies with the volume fraction of nanoparticles, nanoparticle radius, and nanofluid temperature. According to the semi-analytical results, the empirical equations between the above variables and nanolayer thickness are summarized. Finally, by adopting the empirical equation of the nanolayer thickness in the corrected model, a novel semi-empirical thermal conductivity model is developed for predicting the thermal conductivity of DO-based nanofluid. The semi-empirical thermal conductivity model is validated by a large number of experimental data from literatures. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Diathermic oil (DO) is a traditional heat transfer fluid (HTF) which is widely used in concentrating solar power (CSP). In the parabolic trough concentrating solar power system, DO works as the HTF which absorbs the solar radiation energy in the solar thermal collector and transfers the thermal energy to the working fluid for the power cycle in the heat exchanger [1–7]. An efficient solar thermal collector and heat exchanger should be coupled with HTF that possesses excellent thermal properties [8,9]. Therefore, the enhancement of thermophysical properties for DO can be beneficial for the CSP performance. Nanofluid has been proven to possess good thermophysical properties, thus DO-based nanofluid could be the next-generation HTF for parabolic trough CSP [10,11]. Nanofluid is made by adding nano-diameter solid particles into the heat transfer fluid and dispersing nanoparticles uniformly, which was firstly reported in 1995 by Choi in Argonne National Laboratory [12]. Because of the higher thermal conductivity of nanofluid than that of base fluid, nanofluid can work as a novel HTF to enhance heat transfer. The thermal conductivity of nano-

⇑ Corresponding author. E-mail address: [email protected] (M.-J. Li). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.080 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

fluid is closely related to the thermal conductivity of the base fluid and nanoparticles. Besides, it is also influenced by the diameters, volume fraction, dispersion of nanoparticles, and the nanofluid temperature. Due to the importance and complexity of nanofluid thermal conductivity, how to predict the thermophysical properties accurately has attracted many investigations in recent years. The effective thermal conductivity model of solid particle suspension was first proposed by Maxwell at 1873 [13]. The model shows the thermal conductivity of the suspension is decided by the volume fraction of particles and the thermal conductivities of the base fluid and particles. The Maxwell model was expressed as Eq. (1).


 ksf kp þ 2kbf þ 2/ kp  kbf
 ¼ kbf kp þ 2kbf  / kp  kbf

ð1Þ

where ksf, kbf, and kp are the thermal conductivity of the suspension fluid, the base fluid, and the particles, respectively. And / represents the volume fraction of the particles. The Maxwell model can only predict the thermal conductivity of suspension fluid with a low /. The Bruggeman model was proposed in 1935 [14], which can better predict the thermal conductivity of suspension fluid with no limitations on /. The Bruggeman model was expressed as Eq. (2).

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Nomenclature a b c1 c2 k kl m n Q r T t

Greek symbols a thermal conductivity ratio of base fluid to nanoparticle b ratio of nanolayer thickness to the nanoparticle radius c thermal conductivity ratio of nanolayer to nanoparticle / volume fraction of nanoparticles

fitting parameter fitting parameter fitting parameter fitting parameter thermal conductivity (Wm1K1) equivalent thermal conductivity of nanolayer (Wm1K1) fitting parameter fitting parameter heat flow (W) radius (nm) temperature (°C) nanolayer thickness (nm)

Subscripts bf base fluid nf nanofluid l nanolayer p particle sf suspension fluid

h pffiffiffiffii k ¼ 14 ð3/  1Þ k p þ ð2  3/Þ þ D bf h i2
k k where : D ¼ ð3/  1Þ k p þ ð2  3/Þ2 þ 2 2 þ 9/  9/2 k p

ksf kbf

bf

bf

ð2Þ The Maxwell and Bruggeman models can predict the thermal conductivity of suspension fluid well when the particles are millimeter or micrometer-sized. However, when the diameter of particles reduces to nanometer-sized, the thermal conductivity of nanofluid increases enormously and cannot be predicted precisely by these classical models [15–17]. To better predict the thermal conductivity of nanofluid, researchers mainly proposed three possible mechanisms of enhancing thermal conductivity of nanofluid: (1) Brownian motion of the nanoparticles [18–20] (2) Nanoparticle clustering [21–25] (3) Surface adsorption [26–30]. Although there are many investigations focused on this area, researchers still failed to reach an agreement about the mechanism of enhancing thermal conductivity of nanofluid. The following studies in this paper were based on the surface adsorption mechanism. In the nanofluid, the liquid molecules which are close to a solid nanoparticle surface are attracted by the nanoparticle. These liquid molecules around the nanoparticle are more ordered than others in the base fluid. And the ordered liquid molecules form a solid-like structure between the nanoparticle surface and base fluid. This solid-like structure is named nanolayer. The nanolayer is just like a thermal bridge connecting the solid nanoparticle and the base fluid which reduces the contact thermal resistance [29,30]. Many researchers proposed that the nanolayer was one of the possible major reasons for the thermal conductivity enhancing of nanofluid. Many studies about the thermal conductivity of nanofluid are based on the nanolayer theory. In consideration of the effects of the nanolayer on thermal conductivity, Yu-Choi model is proposed to predict the thermal conductivity of nanofluid [27]. In the model, the nanolayer and the nanoparticle are treated as an equivalent particle to simplify the analysis, and the thermal conductivity of the equivalent particle is calculated by Eq. (3).

kpe ¼

½2ð1cÞþð1þbÞ3 ð1þ2cÞc ð1cÞþð1þbÞ3 ð1þ2cÞ

where : b ¼

t rp

;c ¼

kl kp

kp

ð3Þ

where kpe represents the thermal conductivity of the equivalent particle. b is the ratio of the nanolayer thickness (t) to the original

particle radius (rp). And c is the ratio of nanolayer thermal conductivity (kl) to particle thermal conductivity (kp). Considering the thermal conductivity of the equivalent particle, Yu-Choi model can be expressed as Eq. (4) by modifying the Maxwell equation.


 3 knf kpe þ 2kbf þ 2 kpe  kbf ð1 þ bÞ / ¼
 kbf kpe þ 2kbf  kpe  kbf ð1 þ bÞ3 /

ð4Þ

where knf represents the thermal conductivity of nanofluid. Due to the exact properties of the nanolayer can hardly be calculated by theoretical analysis or measured by experiments, the thermal conductivity of nanolayer (kl) is assumed to be the same with that of particle (kp) and the nanolayer thickness (t) is assumed to a constant in this model. It is worth noting that kl should be different with kp because the element type and molecule geometry of nanolayer are different with these of nanoparticles. And t should be a variable which could be influenced by the nanoparticle radius, the volume fraction of nanoparticles, and the nanofluid temperature. So it is obviously unreasonable to treat kl and t as constant. Since the nanolayer is made by the ordered base fluid molecules around the nanoparticle, it would have the intermediate properties between nanoparticle and the base fluid. Xie et al. [28] assumed that kl decreases linearly from kp at the nanoparticle surface to kbf at the nanolayer surface. The linear thermal conductivity profile in the nanolayer can be expressed as Eq. (5), and the equivalent thermal conductivity of nanolayer can be calculated by Eq. (6).


 kp r p þ t  kbf r p kp  kbf kl ðrÞ ¼ rþ t t kl kbf

¼ a1

ð5Þ

ð1þbaÞ2

ð1þbaabÞlnð1þb a Þþbð1þbaÞ

where : a ¼ kkbfp ; b ¼ rtp

ð6Þ

Assuming the thermal conductivity of the nanolayer as linear distribution, the Xie model can be built for predicting the thermal conductivity of nanofluid by solving Eqs. (3)–(6). This model was used by Ren et al. at 2005 [31], Rizvi et al. at 2013 [32], Tso et al. at 2014 [33]. It is known that the nanolayer is made by the base fluid molecules, so the properties of nanolayer should be close to the base fluid at the nanolayer surface. This means the thermal conductivity and its slope should be contiguous at the nanolayer surface. The thermal conductivity is contiguous but its slope is discrete of the linear profile proposed by Xie model. Jiang et al. proposed a nonlinear thermal conductivity distribution in the nanolayer [34]. In the

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nonlinear profile, the thermal conductivity and its slope are contiguous not only at the nanolayer surface but also at the nanoparticle surface. However, due to the elements and molecule geometry of nanolayer are different with these of nanoparticle, we think the thermal conductivity slope does not have to be contiguous at the nanoparticle surface. From the literature review, it can be concluded that when the nanolayer theory is utilized to predict the thermal conductivity of nanofluid, there are three major drawbacks in studies so far. (1) The thermal conductivity profile of nanolayer is may not be assumed reasonably in above models. (2) The nanolayer thickness is influenced by many parameters and has direct influence on the calculated thermal conductivity of nanofluid. But the nanolayer thickness is treated as a constant in above models. (3) The effect of temperature on thermal conductivity is not considered in above models. The objective of this work is to develop a novel semi-empirical model for predicting the thermal conductivity of oil-based nanofluid for solar thermal applications. In this paper, first, a nonlinear thermal conductivity distribution in the nanolayer is proposed. In this nanolayer, the thermal conductivity and its slope are contiguous at the outer nanolayer surface, and the thermal conductivity at the inner nanolayer surface (the nanoparticle surface) equals that of nanoparticle. Second, based on Yu-Choi model, a corrected thermal conductivity model of nanofluid is given, which employs the equivalent thermal conductivity of nanolayer with the nonlinear distribution instead of a constant value. After analyzing the effect of the volume fraction of nanoparticles, nanoparticle radius, and the temperature on nanolayer thickness in the oil-based nanofluid, an empirical correlation of the nanolayer thickness is proposed. Finally, by employing the nonlinear distribution of nanolayer thermal conductivity and the empirical correlation of nanolayer thickness, a novel semi-empirical model for predicting the thermal conductivity of oil-based nanofluid is developed, and the model is validated by a large number of experimental data from literatures. 2. Model formulation In this section, a quadratic distribution of thermal conductivity in nanolayer is proposed. Based on the quadratic distribution, the equivalent thermal conductivity of the nanolayer can be calculated. Then, a corrected thermal conductivity model of nanofluid is developed based on Yu-Choi model. 2.1. Thermal conductivity of nanolayer with quadratic distribution In the nanofluid, the liquid molecules around a solid nanoparticle surface are attracted by the nanoparticle, thus there is a nanolayer formed by the ordered liquid molecules. The sketch of

Fig. 1. Sketch of the single spherical nanoparticle with an interfacial nanolayer.

nanoparticle with an interfacial nanolayer is shown in Fig. 1. The radius of the nanoparticle is rp and the nanolayer thickness is t. A new nonlinear thermal conductivity profile in nanolayer (kl(r)) is proposed which is expressed as Eq. (7). In this nanolayer, it assumes that the kl(r) varies from kp at the nanoparticle surface to kbf at outer nanolayer surface, and the slope of kl(r) should be contiguous at the outer nanolayer surface. The boundary conditions are determined by Eqs. (8) and (9). The thermal conductivity distribution of the nanoparticle with a nanolayer is shown in Fig. 2.

kl ðr Þ ¼ ar 2 þ br þ c

ð7Þ


 kl rp ¼ kp

ð8Þ



 dkl rp þ t ¼0 kl rp þ t ¼ kbf ; dr

ð9Þ

From Eqs. (7)–(9), the kl(r) can be expressed as Eq. (10).


      kp  kbf
kp  kbf 2 r þ 2 kl ðr Þ ¼ r p þ t r þ kp 2 2 t t


kp  kbf 2 þ r p þ 2rp t t2

ð10Þ

The heat flow across the nanolayer (Q) is assumed as a constant, and the steady-state heat transfer across the nanolayer in spherical coordinates can be calculated using the Fourier’s law of heat conduction which is shown as Eq. (11). The integration of Eq. (11) from the nanoparticle surface to the outer nanolayer surface is expressed as Eq. (12).

Q ¼ kl ðrÞAr Z

r p þt

Q rp

dT dr

ð11Þ

dr ¼ 4pr 2 kl ðr Þ

Z

T2

dT ¼

Q 

kl

T1

Z

r p þt

rp

dr 4pr 2

ð12Þ

where Ar is the surface area of the nanolayer at the coordinate (r), which is evaluated as Ar = 4pr2. T1 and T2 represent the nanolayer temperature at the nanoparticle surface and the outer nanolayer, 

respectively. kl is the equivalent thermal conductivity of the nano

layer. Based on Eqs. (10) and (12), kl is expressed as Eq. (13). 

kl ¼



t rp ðr p þt Þ

2 r þt b 2ac M  bln prp þ r rtcþt  pb ffiffiffiffiffiffiffiffiffiffiffi ðN  P Þ 2 Þ pð p 4acb2

a

where :     2 aðr p þt Þ þbðr p þt Þþc bþ2aðrp þt Þ bþ2arp p p ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ; N ¼ arctan ; P ¼ arctan M ¼ ln ar 2 þbr þc 2 2 p

k k a ¼ p t2 bf

p

; b ¼ 2

4acb

ðkp kbf Þ
t2



r p þ t ; c ¼ kp þ

ðkp kbf Þ t2



r 2p þ 2rp t

4acb

ð13Þ

Fig. 2. The distribution of thermal conductivity in the nanolayer.

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Fig. 3. Comparison of various models and experimental data with volume fraction of nanoparticles for H2O/EG based nanofluid.

Fig. 4. Comparison of the predicted thermal conductivities using various models and experimental data with volume fraction of nanoparticle for nanofluid based on diathermic oil.

2.2. Corrected thermal conductivity model of nanofluid In this study, to simplify the mathematical model, the assumptions are presented as follows. (1) Each equivalent nanoparticle formed by the nanolayer and nanoparticle is the same in the nanofluid. (2) The volume fraction of nanoparticles is so low that there is no overlap of the equivalent nanoparticles. The steps to get the corrected thermal conductivity model of nanofluid are described as follows. (1) To calculate the equivalent thermal conductivity of nano

layer (kl ) with the quadratic distribution by Eq. (13). (2) To calculate the thermal conductivity of the equivalent nanoparticle (kpe) by Eq. (14). Instead of treating the nanolayer thermal conductivity (kl) to be same with nanoparticle

kpe

   

3    2 1  kkpl þ 1 þ rtp 1 þ 2 kkpl    kl ¼   

3  kl k 1 þ 2 kpl  1  kp þ 1 þ rtp

ð14Þ

(3) To calculate the thermal conductivity of nanofluid (knf) based on Yu-Choi model by Eq. (15). In this equation, the kpe employs the value calculated by step 2.

3
 kpe þ 2kbf þ 2 kpe  kbf 1 þ rtp / knf ¼

3 kbf
 kpe þ 2kbf  kpe  kbf 1 þ rtp /

ð15Þ

3. Model validation



thermal conductivity (kp) in Yu-Choi model, kl employs the value calculated by step 1 in Eq. (14).

The corrected thermal conductivity model of nanofluid proposed in this study is validated by comparing the knf calculated

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Fig. 5. Effect of nanolayer thickness on the predicted thermal conductivity using present model [40].

Fig. 6. Effect of volume fraction of particle on t0 /rp with different nanofluid temperature (a) (b), type of nanoparticle (c), and size of nanoparticle (d).

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M.-J. Li et al. / International Journal of Heat and Mass Transfer 138 (2019) 1002–1013 Table 1 The fitting parameters for Section 4.1.1. Variable

a

c1

T = 30 °C T = 50 °C T = 90 °C T = 120 °C T = 150 °C

9.0e-5 9.0e-5 9.0e-5 9.0e-5 9.0e-5

0.436 0.342 0.330 0.292 0.222

Al2O3 CuO ZnO

6.5e-5 2.6e-4 3.2e-4

0.108 0.289 0.272

Variable

a

c1

Fig. 6(a)

T = 20 °C T = 30 °C T = 40 °C T = 50 °C

2.6e-4 1.5e-3 1.5e-3 1.5e-3

0.023 0.026 0.190 0.382

Fig. 6(b)

Fig. 6(c)

rp = 12.5 nm rp = 25 nm rp = 50 nm

3.3e-3 2.2e-4 6.6e-5

0.493 0.393 0.327

Fig. 6(d)

Fig. 7. Effect of particle size on t0 /rp with different volume fraction of nanoparticle.

Table 2 The fitting parameters for Section 4.1.2. Variable

m

n

/ = 0.005 / = 0.010 / = 0.015 / = 0.020

1.43 1.06 1.30 1.12

0.366 0.296 0.354 0.311

Fig. 7(a)

Variable

m

n

T = 30 °C T = 40 °C T = 50 °C T = 60 °C

3.76 4.10 3.32 2.85

0.126 0.141 0.097 0.066

Fig. 7(b)

by this model with the experiments and the predicted values by several widely-used models [35–37]. The experimental results include the knf of water-based, ethylene glycol (EG)-based, and diathermic oil (DO)-based nanofluid. In Yu-Choi model, Jiang model, and Xie model, the nanolayer thickness (t) is assumed as 2 nm.

cles (/). However, it can be observed that the experimental data has good agreement with the predicted results calculated by the other four models, which consider the effect of nanolayer and regard the nanolayer thickness as 2 nm.

3.1. Comparisons with experimental data for H2O/EG-based nanofluid

In this section, it is worth noting that the t in present model is also regard as 2 nm. The predicted results and experimental data for DO-based nanofluid with CuO nanoparticles (rp = 15 nm, T = 40 °C) and ZnO nanoparticles (rp = 30 nm, T = 60 °C) with an increasing / are shown in Fig. 4(a) and (b), respectively [37]. It is seen that the experimental data is higher than all the results predicted by the models in the DO-based nanofluid. Many experimental and theoretical studies had shown that the nanolayer thickness of H2O-based/EG-based nanofluid was around 1–4 nm [33,38,39]. So when the Present model, Jiang model, YuChoi model, and Xie model employ 2 nm as the nanolayer thickness in H2O-based and EG-based nanofluid, the results predicted

Fig. 3 shows the experimental data and predicted results calculated by the present corrected model and other widely-used models [35,36]. In this section, the t in present corrected model is also regard as 2 nm to compare the accuracy of those models fairly. Fig. 3(a) and (b) show the predicted results and experimental data for H2O-based nanofluid with CuO nanoparticles (rp = 3 nm) and EG-based nanofluid with Cu nanoparticles (rp = 3 nm) at room temperature, respectively. It is seen that the results predicted by the Maxwell model and Bruggeman model are much lower than the experimental data with increasing volume fraction of nanoparti-

3.2. Comparisons with experimental data for DO-based nanofluid

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Fig. 8. Effect of nanofluid temperature on t0 /rp with different volume fraction of particle.

Table 3 The fitting parameters for Section 4.1.3. Variable

b

c2

Variable

b

c2

/ = 0.001 / = 0.005 / = 0.010 / = 0.020

-1.13e-3 -1.67e-3 -2.13e-3 -2.47e-3

0.498 0.464 0.570 0.564

Fig. 8(a)

/ = 0.007 / = 0.010 / = 0.015 / = 0.020

-2.80e-3 -6.67e-4 -6.67e-4 -1.00e-3

0.278 0.259 0.265 0.263

Fig. 8(b)

/ = 0.002 / = 0.004 / = 0.006 / = 0.008 / = 0.010

3.27e-2 2.30e-2 1.64e-2 1.77e-2 1.48e-2

-0.427 -0.342 -0.166 -0.323 -0.268

Fig. 8(c)

/ = 0.002 / = 0.004 / = 0.006 / = 0.008

2.03e-2 9.13e-3 1.43e-2 1.18e-2

-0.007 0.319 0.237 0.232

Fig. 8(d)

by the four models have good agreement with experimental data as shown in Section 3.1. However, it is seen from Fig. 4 that the poor agreement between predicted results and experimental data indicates that it is improper to treat nanolayer thickness as 2 nm in the DO-based nanofluid.

3.3. Effect of nanolayer thickness on the corrected model In this section, to analyze the effect of different nanolayer thickness (t) on the present model for DO-based nanofluid, the results predicted by the corrected model with t are compared, which are

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Fig. 9. Schematic flowchart for modeling the novel semi-empirical thermal conductivity model of nanofluid.

Table 4 The conditions needed and results calculated during the process of predicting knf with different / [37]. Nanofluid

Conditions

Variable

knf/kbf (experiment)

t0 /rp

Empirical equation of t0

ZnO/DO

rp = 30 nm, T = 40 °C kp = 29 W/(mK) kbf = 0.115 W/(mK)

/ = 0.001 / = 0.010 / = 0.020

1.0111 1.0640 1.1290

0.6000 0.3133 0.3067

t0 rp

¼

3:1310 /

- 4

þ 0:287

CuO/DO

rp = 15 nm, T = 50 °C kp = 20 W/(mK) kbf = 0.117 W/(mK)

/ = 0.001 / = 0.010 / = 0.020

1.0111 1.0640 1.1290

0.6200 0.2333 0.2333

t0 rp

¼

4:1710 /

- 4

þ 0:202

shown in Fig. 5. Firstly, it can be seen from Fig. 5 that knf/kbf rises with the increasing t. It is probably because the nanolayer could reduce the contact thermal resistance between the solid nanoparticle and the base fluid. Thus, the contact thermal resistance decreases and the thermal conductivity of nanofluid increases with the improvement of t. Secondly, it can be observed from Fig. 5(a) that when the DO-based nanofluid temperature (T) is 60 °C, the predicted results with t = 4.5 nm have good agreement with experimental data [40]. When T rises to 120 °C, it is seen in Fig. 5(b) that t should be 3.5 nm to get the good predicted results. It indicates that the nanolayer thickness varies with nanofluid temperature.

From above results, it indicates that the nanolayer thickness is the key point for the thermal conductivity model of nanofluid to predict the results accurately. So it is necessary to study the effects on nanolayer thickness. In the next section, the effects of volume fraction, radius of nanoparticle, and nanofluid temperature on nanolayer thickness are analyzed. 4. Analysis and discussions In this section, based on experimental data of the nanofluid thermal conductivity (knf) in literatures, the nanolayer thickness

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can be calculated using the corrected model proposed in Section 2. It is worth noting that the calculated nanolayer thickness (t0 ) is not actual, but a semi-analytical value based on the corrected model with the quadratic profile of the nanolayer thermal conductivity. Using this method, the effects of the radius (rp), the volume fraction of nanoparticles (/), and the temperature (T) on t0 in the DO-based nanofluid are analyzed, and the empirical correlation of the nanolayer thickness is proposed. Based on the empirical correlation and the corrected model, a novel semi-empirical model is developed for predicting the thermal conductivity of DO-based nanofluid. Finally, the novel semi-empirical model is validated by a large number of experimental data from literatures. 4.1. Effects on nanolayer thickness 4.1.1. Effect of volume fraction of nanoparticle Based on the experimental data of nanofluid thermal conductivity (knf) in literatures [37,40,41], the semi-analytical nanolayer thickness (t0 ) is calculated using the corrected model. Fig. 6 shows the variations on t0 /rp with an increasing / for different nanofluid temperature (Fig. 6(a) and (b)), nanoparticle type (Fig. 6(c)), and nanoparticle radius (Fig. 6(d)). The t0 /rp calculated based on experimental results are pointed in the figures. It is seen that t0 /rp decreases sharply at first and then there is almost no variation in t0 /rp when / is larger than 0.01. Based on this phenomenon, a reciprocal equation is proposed as Eq. (16) to express the effect of / on nanolayer thickness.

t0 a ¼ þ c1 / rp

ð16Þ

where a and c1 are the fitting parameters of Eq. (16) which are determined by the experimental data. The values of a and c1 for all the cases are shown in Table 1, and the empirical equation profiles for each case are lined in Fig. 6. The profiles have good agreement with the experimental data points.

4.1.2. Effect of nanoparticle radius Based on the experimental data of knf in literatures [37,42], t0 is calculated using the corrected model. Fig. 7 shows the variations on t0 /rp with an increasing nanoparticle radius (rp) for different / (Fig. 7(a)) and T (Fig. 7(b)). The t0 /rp calculated based on experimental results are pointed in the figures. It is seen that t0 /rp reduces exponentially with rp. Based on the exponential decay relationship, an equation is proposed as Eq. (17) to reveal the effect of rp on t0 .



n t0 ¼ m  r p  109 rp

ð17Þ

where m and n are the fitting parameters of Eq. (17) which are determined by the experimental data. The values of m and n for the cases are shown in Table 2, and the empirical equation profiles for each case are lined in Fig. 7, and it demonstrates that the exponential profiles have good agreement with the experimental data points. 4.1.3. Effect of nanofluid temperature Based on the experimental data of knf in literatures [40,41,43], t0 is calculated using the corrected model. Fig. 8 shows the variations on t0 /rp with an increasing nanofluid temperature (T). The t0 /rp based on the experimental results are pointed in the figures. It is seen in Fig. 8(a) and (b) that t0 /rp decreases with the improvement of T in the Cu/DO and CuO/DO nanofluids. However, in Fig. 8(c) and (d), t0 /rp increases with the improvement of T in the TiO2/DO and SiC/DO nanofluids. It can be observed that almost all the t0 /rp varies linearly with T. Based on this linear relationship, an equation is proposed as Eq. (18) to express the effect of T on t0 .

t0 ¼ bT þ c2 rp

ð18Þ

where b and c2 are the fitting parameters of Eq. (18) which are determined by the experimental data. The values of k and c2 for all the cases are shown in Table 3, and the empirical equation pro-

Fig. 10. Comparison of the predicted knf with / using various models and experimental data [37].

Table 5 The conditions needed and results calculated during the process of predicting knf with different rp [37,43]. Nanofluid

Conditions

Variable

knf/kbf (experiment)

t0 /rp

Empirical equation of t0

TiO2/DO

/ = 0.002, kp = 8.4 W/(mK) T = 20 °C, kbf = 0.055 W/(mK)

rp = 15 nm rp = 60 nm

1.3824 1.2000

3.00 2.30

t0 rp



0:192 ¼ 5:04  r p  109

Cu/DO

/ = 0.015, kp = 390 W/(mK) T = 30 °C, kbf = 0.117 W/(mK)

rp = 12.5 nm rp = 50 nm

1.1685 1.1085

0.536 0.334

t0 rp



0:341 ¼ 1:26  r p  109

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files for each case are lined in Fig. 8. The linear profiles have good agreement with the experimental data points. From above analysis, it is found that the semi-analytical nanolayer thickness (t0 ) varies with volume fraction of particle (/), nanoparticle radius (rp), and nanofluid temperature (T). There are a reciprocal relationship between / and t0 /rp, an exponential decay relationship between rp and t0 /rp, a linear relationship between T and t0 /rp, respectively.

4.2. The semi-empirical model for predicting the thermal conductivity considering the variation of nanolayer thickness In this section, based on the empirical correlation summarized in Section 4.1 and the corrected model proposed in Section 2, a novel semi-empirical model is developed for predicting the thermal conductivity of DO-based nanofluid including the effects of / , rp, and T. Then, the thermal conductivities of some DO-based nanofluids are predicted using this semi-empirical model, and the predicted results are compared with the experimental data in literatures to validate the accuracy of the model.

(1) First decided which purpose of the model, predicting knf with /, rp or T? (2) Measure few knf of nanofluid with / (rp or T) by experiment. (3) Calculate t0 using the corrected model based on these experimental data, and get the fitting parameters based on the empirical equation of nanolayer thickness (Eqs. (16)–(18)). (4) Employ the empirical equation instead of a constant value as the nanolayer thickness in the corrected model (Eqs. (13)– (15)), then get the semi-empirical thermal conductivity model for DO-based nanofluid. (5) Gain the thermal conductivity of nanofluid with / (rp or T) by this semi-empirical thermal conductivity model. 4.2.2. Predicting the thermal conductivity of DO-based nanofluid by the semi-empirical model In this section, the thermal conductivity of DO-based nanofluid with varying / (rp or T) is predicted by using the semi-empirical thermal conductivity model proposed before. Then, the predicted results are compared with the experimental data from literatures to validate the accuracy of the model. (1) Predicting knf with varying /

4.2.1. The semi-empirical thermal conductivity model Based on the above study, a semi-empirical thermal conductivity model for DO-based nanofluid is developed. The processes of developing this semi-empirical model to predict the thermal conductivity of DO-based nanofluids are as follows. The detailed steps for modeling are shown in Fig. 9.

In this part, the thermal conductivity of ZnO/DO and CuO/DO nanofluid with varying / are predicted, respectively. Then, the predicted results are compared with the experimental data. First, three DO-based nanofluids with different / are selected and their thermal conductivities are obtained from the experimental data in literature. And the nanolayer thickness (t0 ) is calculated

Fig. 11. Comparison of the predicted knf with rp using various models and experimental data.

Table 6 The conditions needed and results calculated during the process of predicting knf with different T [40,43]. Nanofluid

Conditions

Variable

knf/kbf (experiment)

t0 /rp

Empirical equation of t0

Cu/DO

/ = 0.01, rp = 10 nm, kp = 401 W/(mK) kbf = 1.299e4T + 0.1289 W/(mK)

T = 30 °C T = 150 °C

1.1022 1.0590

0.5000 0.2500

t0 rp

¼ - 2:08  10

SiC/DO

/ = 0.04, rp = 15 nm, kp = 220 W/(mK) kbf = 1.0e4T + 0.131 W/(mK)

T = 20 °C T = 50 °C

1.0395 1.0648

0.5000 0.7733

t0 rp

¼ 9:11  10

-3

-3

T þ 0:563

T þ 0:318

1012

M.-J. Li et al. / International Journal of Heat and Mass Transfer 138 (2019) 1002–1013

Fig. 12. Comparison of the predicted knf with T using various models and experimental data.

by the corrected model based on these experimental data. Then, according to the calculated t0 and the reciprocal relationship between / and t0 /rp, the empirical equations of t0 for the nanofluids are obtained. These conditions needed for the modeling and results calculated during the modeling process are shown in Table 4. Finally, the empirical equation of t0 is employed in the corrected model of Eqs. (13)–(15) to get the semi-empirical thermal conductivity model for these nanofluid. The thermal conductivity of ZnO/DO and CuO/DO nanofluid with varying / are predicted by the semi-empirical model. Fig. 10(a) and (b) show the predicted results and experimental data of ZnO/DO and CuO/DO nanofluids, respectively. In Fig. 10, the black hollow points are the experimental data used for developing the semi-empirical model, and the red solid points are the experimental data to validate the accuracy of the model. It is seen that the predicted results calculated by the semi-empirical model have good agreements with the experiment, and the present model is obviously more precise than the other models for predicting the thermal conductivity of DO-based nanofluid with varying /.

test conditions and results calculated during the modeling process are shown in Table 6. The thermal conductivity of Cu/DO and SiC/DO nanofluid with different T are predicted by the semiempirical model, which are shown in Fig. 12(a) and (b), respectively. It is seen in Fig. 12 that the predicted results calculated by the semi-empirical model have good agreements with the experiment. It is also observed that the predicted results by using other models are independent of T, and the models can not reveal the effect of nanofluid temperature on knf. So it is obvious that the semi-empirical model is proposed in this study is superior to the other models for predicting knf of DO-based nanofluid with different T. From previous analysis, it is seen that by using few experimental data, the semi-empirical model can reveal the effect of volume fraction of nanoparticle, nanoparticle radius and nanofluid temperature on thermal conductivity of DO-based nanofluid, and greatly improves the accuracy of thermal conductivity prediction. Therefore, the semi-empirical model proposed in this study is superior to the other models for predicting thermal conductivity of DO-based nanofluid.

(2) Predicting knf with varying rp In this part, the thermal conductivity of TiO2/DO and Cu/DO nanofluid with varying rp are predicted, respectively. And the predicted results are compared with the experimental data. The method used in this part is the same as the way for predicting knf with varying /. The needed test conditions and results calculated during the modeling process are shown in Table 5. The thermal conductivity of TiO2/DO and Cu/DO nanofluids with different rp are predicted by the semi-empirical model, which are shown in Fig. 11(a) and (b), respectively. It is seen in Fig. 11 that the predicted results calculated by the semi-empirical model have good agreements with the experiment, but the predicted results from other models are much smaller than the experimental data. It is obvious that the present model is more precise than other models for predicting the thermal conductivity of DO-based nanofluid with varying rp. (3) Predicting knf with varying T In this part, the thermal conductivity of Cu/DO and SiC/DO nanofluid with varying T are predicted, respectively. The predicted results are compared with the experimental data. The procedure used in this part is the same as the way in last part. These needed

5. Conclusions In this work, a novel semi-empirical model for predicting the thermal conductivity of Diathermic Oil (DO)-based nanofluid is developed. The main works are as follows. (1) A corrected thermal conductivity model of nanofluid (knf) is developed based on Yu-Choi model. In this corrected model, the thermal conductivity distribution of nanolayer is treated as a quadratic form instead of a constant value along with the distance from nanoparticle. (2) When the corrected model adopts a constant value (2 nm) as the nanolayer thickness (t), it gives good predictions for knf of H2O-based and EG-based nanofluids, but very poor predictions for knf of DO-based nanofluid. After analyzing the effect of t on the corrected model, it shows that knf calculated by this model is significantly influenced by t. (3) Utilizing the corrected model and the experimental data in literature, the semi-analytical nanolayer thicknesses (t0 ) for types of DO-based nanofluid are calculated, it is found that t0 of DO-based nanofluid is not a constant value but varies with volume fraction of particle (/), nanoparticle radius

M.-J. Li et al. / International Journal of Heat and Mass Transfer 138 (2019) 1002–1013

(rp), and nanofluid temperature (T). And the results show that there are a reciprocal relationship between / and t0 /rp, an exponential decay relationship between rp and t0 /rp, and a linear relationship between T and t0 /rp, respectively. (4) By adopting the empirical equation of the nanolayer thickness in the corrected model, a novel semi-empirical thermal conductivity model is developed for predicting thermal conductivity of DO-based nanofluid. This semi-empirical thermal model is validated by a large number of experimental data from literatures.

[16]

[17] [18] [19] [20] [21]

In conclusion, the semi-empirical model can be used for predicting the thermal conductivity of DO-based nanofluid for solar thermal applications based on limited experimental data.

[22] [23]

Conflict of interest [24]

The authors declared that there is no conflict of interest. [25]

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