Predicting the effective thermal conductivity of nanofluids for intensification of heat transfer using artificial neural network

Powder Technology 301 (2016) 288–309

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Predicting the effective thermal conductivity of nanofluids for intensification of heat transfer using artificial neural network Ali Aminian Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, PO Box 35195-63, Semnan, Iran

a r t i c l e

i n f o

Article history: Received 28 December 2015 Received in revised form 15 April 2016 Accepted 20 May 2016 Available online 25 May 2016 Keywords: Nanofluids Effective thermal conductivity Artificial neural network Nanoparticles Prediction

a b s t r a c t Nanoparticles are one of the most promising materials with significant capability to increase the heat transfer coefficient and the thermal conductivity of a base fluid, while their addition to the fluid will decrease viscosity and friction factor. Some of the nanoparticles have high thermal conductivity, which candidate them for practical application as nanoadditives in process industries. This work presents a neuromorphic model for predicting the thermal conductivity of nanofluids, which takes into consideration the effects of size, volume fraction, temperature, and thermal conductivity of nanoparticles as well as the properties of base fluids. The presented model is found to correctly predict the trends observed in experimental data for different combinations of nanoparticles-base fluids with varying concentrations. Twenty six different types of nanofluids, namely, Al2O3water/EG, CuO-water/TO/EG/MEG/paraffin, Cu-water/EG/oil, Al-water/EG/EO/TO, TiO2-water/EG, ZnO-water/ EG, SiO2-water/EG/oil, MWCNT-water/EG/oil/R113, and Ag-water are used to assess the effectiveness of the proposed neuromorphic model. The overall predicted effective thermal conductivities regarding twenty six different nanofluids are in excellent agreement with experimental data with the AAD of 3.06% and R2-value of 0.9309. © 2016 Elsevier B.V. All rights reserved.

1. Introduction By adding nanometer-sized particles into base fluids, the thermal and transport properties of the base fluids are improved. Because of the poor heat transfer and transport properties of the most conventional heat transfer fluids, nanoparticles are added for their significant effect on the thermal and transport properties of the heat transfer fluids [1–3]. The term nanofluid refers to a system composed of a base fluid in which extremely fine metallic/metal oxide particles are dispersed. Nanoparticles have outstanding properties due to their small sizes and very large specific surface areas. It is well known that the addition of conducting nanoparticles to the base fluids does enhance the heat transport due to increase in the thermal conductivity and thermal diffusivity values of the base fluids [4]. Nanofluids are under investigation to be the next generation heat transfer augmentation fluids. For example, copper nanoparticles have been added into two commercial organic heat transfer fluids concerning production of electricity by concentrated solar power as a renewable energy source [5]. The heat removal in electronic chips, laser applications, cars radiator, process heat exchangers, HVAC, and cooling fluid for mirrors are few examples where such systems

E-mail address: [email protected].

http://dx.doi.org/10.1016/j.powtec.2016.05.040 0032-5910/© 2016 Elsevier B.V. All rights reserved.

are used. In addition, the uses of nanoparticles greatly improve industrial limitations imposed by using milli/micro-meter sized particles including abrasive action, tube plugging, and high pressure drop [6]. These exclusive features of nanoparticles have been motivated many researchers to investigate the thermal/physical performance of various nanoparticles in different base fluids regarding variety of variables affecting thermal properties of the nanofluids. The thermal conductivity enhancement is one of the noteworthy effects originating from the suspension of nanoparticles into a base fluid. The thermal conductivity enhancement ratio is defined as the ratio of thermal conductivity of the nanofluid to that of the base fluid (knf/kbf). One of the most important correlations proposed for solid particles dispersed in fluids is the Maxwell's model based on the assumptions that the solid particles are spherical in shape and the thermal conductivity of nanofluid depends on the particle volume fraction and the thermal conductivities of solid particles/base fluid as follows [7]:
 knf kp þ 2kbf þ 2φ kp −kbf
 ¼ kbf kp þ 2kbf −φ kp −kbf

ð1Þ

where kp and kbf are the thermal conductivities of the solid particles and the base fluid, respectively, and φ is the solid volume fraction.

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Maxwell's model predicts that the effective thermal conductivity of suspensions containing spherical particles increases with the volume fraction of the solid particles as a result of higher surfacearea-to volume ratio solid particles have. It proposed that if nanometer-sized particles could be suspended in base fluids, the thermal and fluid flow properties of the resulting nanofluid would be improved. Since the surface-area-to-volume ratio is much more larger for nano-sized particles than that of particles with diameters in micron and millimeter, a much more dramatic improvement in the effective thermal conductivity is expected as a result of decreasing the particle size. Hamilton and Crosser (HC) modified Maxwell's model when they focused on the possible effects of increasing particle surface area by controlling particle shapes to be nonspherical [8]. However, Hamilton and Crosser analysis does not take into account the dependency of conductivity on the particle size. Hamilton and Crosser analysis predicts that the conductivity of two-component mixtures can be described by
 kp þ k0 ðn−1Þ−αðn−1Þ k0 −kp k
 ¼ k0 kp þ k0 ðn−1Þ þ α k0 −kp

ð2Þ

where k is the mixture thermal conductivity, k0 is the base fluid thermal conductivity, kp is the thermal conductivity of solid particles, α is the particle volume fraction, and n is an empirical scaling factor for particle other than spherical shapes. Several researches have been confirmed enhancement in the thermal conductivity with nanoparticles such as Al2O3, CuO, Al, Cu, and carbon nanotubes dispersed in water and EG based nanofluids [9]. The enhancement in the thermal conductivity depends on several factors including type, size and shape of the nanoparticles, volume fraction of the nanomaterials, properties of the base fluid, working temperature, Brownian effect, and so on. Detailed literature reviews of fluids containing nanoparticles over the heat transfer and flow behavior provided by Saidur et al. [10] and Murshed et al. [41]. Carbon nanotubes (CNTs) are one of the most valuable materials with high thermal conductivity (above 1800 W/m·K compared to the thermal conductivity of Ag 419 W/m·K). CNTs are one of the most suitable nanoadditives for fabricating the nanofluid with the thermal conductivities that are significantly higher than those of the parent liquids even when the CNTs concentration is negligible [12]. The researches have been shown that CNTs could have a thermal conductivity an order of magnitude higher than that of copper. This suggests that nanofluids made of CNTs could have a very high thermal conductivity. For example, at a CNT loading of 1% by volume, the thermal conductivity enhancement obtained as high as 160% for MWCNTs-synthetic poly(α-olefin) oil nanofluids [13]. However, this is not the case for the other nanofluids containing CNTs as reported by Xie et al. [14]; for MWCNTs-water, MWCNTs-ethylene glycol and MWCNTs-decane. Also, Assael et al. [15] observed only 20%–40% in the thermal conductivity enhancement for 0.6% CNTs in water by volume. An important finding of the Wen and Ding [16] shown that the effective thermal conductivity increases with increasing the concentration of CNTs; the dependence was nonlinear even at very low concentrations, which was different from the results for metal/metal oxide nanofluids. The effect of temperature on the effective thermal conductivity of CNTs found to be linear at temperatures lower than 30 °C. Also, at temperatures greater than 60–70 °C, failure of dispersant led to destabilization of CNTs nanofluids. There are some models for predicting the thermal conductivity of CNT-based nanofluids. Traditional composite models, such as Maxwell [7] and Hamilton–Crosser [8] proved to be inadequate, as they normally underpredicted the experimental data. A formula for cylindrical particles with aspect ratios greater than 100, as for CNTs,

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developed for the effective thermal conductivity by using effective medium theory [17]:
 knf 3 þ 2φ kp =kbf ¼ 3−2φ kbf

ð3Þ

where k nf, k p , k bf and φ are the thermal conductivities of the nanofluid, the particle, the base fluid and particles volume fraction, respectively. The effective thermal conductivity of CNTs-water nanofluids with 0.01 wt% of CNTs in 1 wt% of Gum Arabic stabilizer increased with increasing temperature and the effect is more pronounced after 40 °C. An enhancement of 4.0%–125.6% in the effective thermal conductivity of CNTs-water nanofluids observed with CNTs diameter of 20 nm [18]. Also, a modified model for the thermal conductivity of carbon nanotubes-nanofluids presented by Thang et al. [19]. They started from the fact that CNTs are very good thermal conductors along the tube, but good insulators laterally to the tube axis and they disperse in nanofluids in all directions randomly. Their model took into account the cylindrical shape of the CNTs and the assumption that both heat flow through base fluid and CNT is one dimensional and parallel to each other. Therefore, the modified model for the thermal conductivity of carbon nanotubes-nanofluids represented by: keff 1 kCNT εrl ≈1 þ kl 3 kl ð1−εÞrCNT

ð4Þ

where rl and r CNT are the liquid molecule and CNT radii, ε is the volume fraction of the nanoparticles and kl and k CNT are the thermal conductivity of base fluid and that of CNT, respectively. The modified model (Eq. (4)) for CNTs works reasonably for different base fluids such as water, ethylene glycol, olefin oil and refrigerant R113 [19]. Moreover, an analysis of particle size distribution employed to predict the thermal conductivity of nanofluids by considering the influence of nanoparticle clusters [20]. They defined nanoparticle aggregates with equivalent spherical diameter d more than a cut-off value dcut as clusters, while the remaining particles or aggregates with d less than dcut defined as primary particles. Finally, the volume fraction of nanoparticles in cluster spheres can be estimated from particle size distribution and the number percentage of clusters with equivalent spherical diameter. Thus, they proposed the following equation for nanofluids containing nonspherical shapes [20]:
 kp þ kpm ðξ−1Þ−φc ðξ−1Þ kpm −kp knf
 ¼ kpm kp þ kpm ðξ−1Þ þ φc kpm −kp

ð5Þ

where ξ is the shape factor given by ξ = 3/ψ with ψ denoting the sphericity of clusters. They tested their model for CuO, Ag and Fe3O4 nanoparticles for volume fractions up to 6% in ethylene glycol, hexadecane, kerosene and deionized water base fluids. Masuda and co-workers demonstrated that Al2O3 nanoparticles dispersed in water has improved the effective thermal conductivity of water [21]. Eastman et al. [22] shown that significantly higher effective thermal conductivity can be obtained for nanofluids containing smaller sized Cu nanoparticles. The Cu nanoparticles with an average diameter of less than 10 nm produced and loadings of up to approximately 0.5 vol% into ethylene glycol can be used to test the thermal behavior of resulting nanofluids. The silver-deionized water nanofluids thermal conductivity measured for volume fractions up to 0.9% and temperature between 50 °C and 90 °C [23]. They reported a minimum and maximum thermal conductivity enhancement of 27% at 0.3 vol% and 80% at 0.9 vol% at an average temperature of 70 °C for 60 nm silver nanoparticles.

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The thermophysical properties and laminar flow convective heat transfer of nanodiamond–engine oil nanofluids measured with different weight fractions ranging from 0 to 2% under constant heat flux thermal boundary condition. An increase of about 35% in thermal conductivity of the base fluid reported [24]. Turbulent convective heat transfer of grapheme-water nanofluids inside a uniformly heated circular tube examined by Akhavan-Zanjani et al. [25]. The maximum enhancement of 10.3% for the thermal conductivity obtained using graphene-water nanofluids. Said et al. [26] reported the applicability of the effective medium theory for ethylene glycol based alumina nanofluids, while considerable deviation found for water based alumina nanofluids. The effective medium theory declares that the thermal conductivity of nanofluid suspensions linearly increase in proportion to the volume concentration of spherical nanoparticles as follows:
 3 kp −kbf φ knf ¼1þ kbf kp þ 2kbf

ð6Þ

An equation for the effective thermal conductivity of oil-based nanofluids containing Cu nanoparticles with negligible Brownian motion of the Cu nanoparticles is as follows [27]:   βkp knf ¼1þ −1 φ kbf kbf

ð7Þ

The value for empirical constant β obtained between 0.003 and 0.0035 at 60 °C. The following equations obtained for the water/CuO and water/ Al2O3 nanofluids both at ambient temperature with R2-value of 95% [28]: knf ¼ 0:99 þ 1:74φ kbf

ð8Þ

knf ¼ 1 þ 1:72φ kbf

ð9Þ

The most notable drawback of the developed models for predicting the effective thermal conductivity of nanofluids is that important physical parameters such as temperature and particle size have not been considered. Although there are some models taking into account the particle size and temperature, but they developed over a relatively small particle and temperature ranges: knf ¼ 1 þ 3:761088φ þ 0:017924T−0:30734 kbf

ð10Þ

knf ¼ 1 þ 0:7644815φ þ 0:018689T−0:46215 kbf

ð11Þ

The above equations [29] developed for water/alumina nanofluids and water/copper oxide nanofluids, respectively. Another relation considered both temperature and particle size developed by Chon et al. [30] was based on the measurements of waterAl2O3 nanofluids:  0:369  0:7476 knf d kp ¼ 1 þ 64:7φ0:746 bf Pr0:9955 Re1:2321 kbf dp kbf

ð12Þ

kBrownian

2. The proposed neuromorphic model Neural networks are computational systems, either hardware or software which mimics the computational abilities of biological systems by using numbers of interconnected artificial neurons. One major advantage with the use of neural networks is that this model can simply learn the relation between input and output variables. Neural network models shown to be extremely accurate in applications such as process control, modelling, simulation and system identification [32–36]. The powerful function approximation properties of neural networks make them useful for representing nonlinear models or controllers. A neural net consists of numbers of simple processing elements called neurons in different layers. Each layer of the proposed cascade-forward neural network model except input layer, receive an input from each neuron (s) in the previous layer (s) and deliver an output to the neurons in the next layer (s) after passing its weighted sum inputs plus a bias value through an activation function according to the following equation regarding input layer and the first hidden layer: 0 1 n X
 @ wji xi þ b j A yj ¼ f h aj ¼ f h

ð13Þ

ð14Þ

j¼1

where wji is the weight between the i-th neuron and the j-th neuron, xi denotes the i-th input, bj is the bias value of the j-th node, n is the number of hidden neurons, f represents the hidden transfer function, and yj is the output of the j-th neuron. In this work, the hyperbolic tangent sigmoid activation function is used for the hidden layer, which can be written in the following form:
 ea j −ea j f h aj ¼ aj e þ ea j

ð15Þ

For the output layer, a linear transfer function is used: f o ðak Þ ¼ ak

The Reynolds number is based on the Brownian motion velocity. The contribution from Brownian motion to the effective thermal conductivity can be written as [31]: sffiffiffiffiffiffiffiffi κT ¼ 5  104 f 1 f 2 φρbf Cp bf ρdp

where f1 = f(φ), f2 = f(φ,T), Cpbf is the specific heat of the base fluid, κ is the Boltzmann constant, and dp is the average particle diameter. It may be noticed that the traditional theories like Maxwell's theory largely fails when applied to nanofluids. In fact, the traditional equations tend either to under-estimate or to over-estimate the value of keff, according to the small or large nanoparticle diameter, respectively, and the temperature of the suspension is high or low, respectively. On the other hand, although the presented models do seem to agree with certain sets of experimental data, they are most likely to be in considerable disagreement with data and correlations collected by other authors. Also, theses correlations developed for certain kind of nanofluids without considering the effects of temperature and nanoparticle size. As a whole, the cost of experimentations and the lack of experimental thermal conductivity values somewhere make valuable the use of neurocomputing. In this work, a neuromorphic model has been developed for predicting the nanofluid effective thermal conductivity, k eff , normalized by the thermal conductivity of the base fluid, k f , from a wide variety of experimental data relative to nanofluids consisting of alumina, copper oxide, titania, copper, Al, Ag, SiO 2 , ZnO and carbon nanotubes with a diameter in the range between 10 nm and 150 nm; suspended in water, Ethylene Glycol (EG), Transformer Oil (TO), Mono Ethylene Glycol (MEG), refrigerants or oils.

ð16Þ

where ak is the output from the k-th output node and fo denotes the transfer function of the output layer. The backpropagation algorithm has been utilized for learning the proposed cascade-forward network. It uses the error between the real network output and the target values

A. Aminian / Powder Technology 301 (2016) 288–309

to adjust the weight and bias of the layers. The error function can be expressed as: E¼

N 1X ðO −t Þ2 N i¼1 i i

ð17Þ

where N is the number of elements in the output vector, Oi is the i-th element of the network output and ti represents the according target value. The backpropagation algorithm minimizes the error function so that the weight and bias values can be updated using an updating rule. Consequently, a gradient of the error is considered, which is shown as below: ∇E ¼

∂E ∂wkj

ð18Þ

Now, by the steepest gradient the weight adjustment can be written as m ¼ wm wmþ1 kj þ Δwkj kj

ð19Þ

where m is the iteration number, wkj denoting the weight between the k-th neuron in the subsequent layer and neuron j in the preceding layer, Δwkj can be defined by the steepest descent procedure: Δwkj ¼ −η

∂E ∂wkj

ð20Þ

where η is the learning rate, Δw is the weight change, and E is the sum of the squares of errors. Once the neural network model is developed, trained, validated and tested, according to a different set of inputs, it can be used for prediction purposes. In this work, a cascade-forward neural network has been used as neural network model, which includes a connection from the input and every previous layer to the following layers. A cascade-forward neural network model configuration is shown in Fig. 1.

Fig. 1. Cascade-forward neural network architecture.

291

The proposed cascade-forward neural network model has been trained and validated by using 70% and 15% of nanofluids thermal conductivity datasets, respectively. The model predictions are also compared with 15% of data points not used in the model development. The selection for activation function is based on a trial and error approach [37]. The most commonly used transfer functions are logarithmic sigmoid, tan sigmoid, linear and radial basis functions, which is selected in such a way that the overall error between the network output and target ones becomes minimum. During developing the neural network model, as in Eq. (14), weighted sum of all inputs (in this case size, volume fraction, temperature, and thermal conductivities of nanoparticle and base fluid) plus the bias of neuron will become the input of activation function. Therefore, the choice of both inputs and activation function is essential regarding accurate predictions. A total of 1273 experimental data from twenty six nanofluids divided into three sets (training, validating and testing datasets) concerning different activations functions of hidden layers with their unknown neurons. Activation function, number of hidden layers and their neurons optimized when the model has minimum validation error. Then, the performance of the model evaluated for the unseen dataset during the model development. As a result, an artificial neural network model containing one hidden layer of nine neurons with tan sigmoid activation function among the wellknown activation functions selected for its lower prediction error concerning twenty six nanofluids while take into account the most influencing parameters as inputs of the model and effective thermal conductivities of nanofluids as output variable. The selection of the most influencing parameters was based on the theoretical background through different theoretical models, which are employed volume fraction, thermal conductivities of particles and base-fluids, size and temperature as well. The organizational chart representing the steps of calculating the effective thermal conductivity of nanofluids by using the neural network model is depicted in Fig. A1.

Fig. 2. The proposed cascade-forward neural network model.

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Fig. 3. Neural network training result for the effective thermal conductivity of nanofluids.

The optimal network architecture for predicting the nanofluids thermal conductivity (output of the network) as a function of the state variables (input of the network), namely, temperature, nanoparticle diameter, nanoparticle volume fraction, the thermal conductivity of both base fluid and nanoparticle is shown in Fig. 2. The training and testing results of the proposed cascade-forward network are displayed in Figs. 3 & 4, respectively. As shown, there is a good agreement between the experimental data and the tested ones. The correlation coefficient (R2-value) of the training and testing phases are 0.9303 and 0.9333, respectively, suggesting the accuracy of the proposed network. In order to have a model with high generalizeability, the model performance concerning unseen datasets during model development is essential. As can be seen from Figs. 3 & 4, the performance of the proposed model for the unused data points is as high as the data points used during the training phase. It is important to test the model for the new datasets, which are unseen to the model. Therefore, the model has been validated with 15% of data points for establishing the model and further testing by the remaining data points which are completely unseen to the proposed model. By doing this, we can be assured that the model has enough generalizeability for the new data points. It should be noted that the theoretical models such as the correlations used in Figs. 5–21 [7–9,29–31,38,56–61,66–69], could not provide an adequate prediction of the effective thermal conductivities for the wide range of operating conditions (ambient temperature, volume fraction…). However, the proposed neural network model as an adequate powerful tool can be used to predict the effective thermal conductivities of the nanofluids in the wide range of available experimental data as well as for some new data points, which are excluded and do not used in the training phase to show the predictability of the model. The optimum calculated values of the neural network parameters including weights and biases are given in Table 1.

Fig. 4. Neural network testing result for the effective thermal conductivity of nanofluids.

Fig. 5. Comparison of the empirical correlation Eq. Eq. (22) by Chon et al. [30] and neural network model with experimental data for the thermal conductivity of Al2O3-water nanofluids.

3. Results and discussions In this section, the proposed neuromorphic model is compared to those of experimental data and semi-empirical correlations to demonstrate its effectiveness for predicting the thermal conductivity of twenty six nanofluids. A total of 1273 experimental data are used for training, validating and testing the proposed neural network model. The most influencing physical properties of nanofluids including nanoparticle diameter (dp), nanoparticle volume fraction (φ), temperature (T), the thermal conductivities of base fluids (kbf) and solid particles (knf) and their ranges are shown in Table 2. Accordingly, the whole experimental data points with corresponding references are provided in Table A1. Due to significant differences in dimension of different parameters, all the data should be normalized to improve the predicting-ability of the proposed model. The accuracies of the presented and the existing theoretical models are determined by using the average absolute deviation in thermal conductivity prediction (AAD%) as follows:  

   ND  knf =kbf − knf =kbf i; exp  1 X i;pred
  100 AAD% ¼ ND i¼1 knf =kbf i; exp

ð21Þ

where (knf/kbf)i , pred and (knf/kbf)i , exp are the predicted and measured thermal conductivity ratios, respectively, and ND is the number of data points. For all the systems AAD is below 3.1% and R2-value is above 0.93 indicating very good accuracy of the proposed cascade-forward model.

Fig. 6. Effect of particle concentration on the thermal conductivity enhancement of waterCuO nanofluids [6] at different temperatures and comparison between different models.

A. Aminian / Powder Technology 301 (2016) 288–309

Fig. 7. Comparisons of measured and predicted effective thermal conductivities of Al-EO and Al-EG nanofluids [42] with particle volume fraction.

As mentioned before, the thermal conductivity of nanofluids is affected by different factors namely, nanoparticle type and size, nanoparticle volume fraction, temperature and the base fluid properties. On the other hand, there are some discrepancies between reported experimental data points, while the developed correlations for them do not take into account the whole influencing factors. In order to consider different nanoparticles, the thermal conductivity of the nanoparticle is added in the input of the neural network model. Fig. 5 compares the experimental data of Chon et al. [30] with the predicted thermal conductivity of different models for the Al2O3-water nanofluids with a nanoparticle diameter of 11–150 nm and a nanoparticle volume fraction of 1% and 4% over the temperature range of 294 K to 344 K. As shown, the comparison shows that the proposed neural network has better prediction accuracy over a wide range of temperature and particle size/volume fraction regarding the effective thermal conductivity of Al2O3-water nanofluids [30]: kp þ PrðTÞ0:9955 T1:2321 keff ¼ 1 þ const: 0:369 kbf dp kbf ðTÞ0:7476 μ 2 ðTÞ

! ð22Þ

Although Eq. (22) is written in terms of nanoparticle diameter (dp) and suspension temperature (T) for the Brownian motion effects, the slightly nonlinear function of Eq. (22) for temperature and size make its prediction deviate more than the proposed model from experimental data.

Fig. 8. Thermal conductivity ratio for ZnO-DI water [46] and TiO2-DI water nanofluids [46] with 3% and 1% volume concentration.

293

Fig. 9. Comparison between different models regarding ambient temperature CuO-water nanofluids [28] effective thermal conductivity.

The measurement of enhancement of the thermal conductivity with particle concentration at different temperatures for CuO-water system is presented in Fig. 6. For CuO-water system, the effect of temperature on the thermal conductivity enhancement is more severe as functions of particle concentration and temperature. The effect of particle concentration on the enhancement found to be less for CuO-water than that of Al2O3-water as there is a similar pattern for the effect of temperature on the enhancement. The results presented in Fig. 6 show a very good agreement of the proposed neural network model with the experimental data of Das et al. [6] and Lee et al. [38], while the HC models considerably underestimate the experimental ones. Thus, Fig. 6 confirms the necessity for a new high-precision model for the entire range of the dependent variables. Fig. 7 presents the results of the thermal conductivity predictions of ethylene glycol- and engine oil-based nanofluids with 80 nm Al nanoparticles. The comparison of the neural network model with the HC model [8] shows the superiority of the proposed model over the HC model. As can be seen in Fig. 7, the effective thermal conductivities of the Al-EG and Al-EO nanofluids increase significantly with increasing the nanoparticle volume fraction from 1% to 5%. The HC model considerably underpredicts the measured effective thermal conductivities of both Al-EG and Al-EO nanofluids. As Fig. 7 shows, the neural network model predicts more accurate the effective thermal conductivity of Al-

Fig. 10. Comparison between different models regarding ambient temperature Al2O3water nanofluids [28] effective thermal conductivity.

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Fig. 13. Comparison of experimental and model predictions for the effective thermal conductivity as a function of volume fraction for CuO-water nanofluids [44] at 299 K. Fig. 11. Predicted thermal conductivity of Cu-oil nanofluids at 333 K.

EG nanofluids than do for Al-EO nanofluids. In order to have a more accurate model it is essential to obtain unique and enough data points for various nanofluids for a wide range of operating conditions. The lack of high precise and sufficient data for some systems, experimental conditions imposed by researchers, data dividing during model development, and different nanoparticles-base fluids will affect the performance of the model [40]. Therefore, such discrepancies might be aroused from the above mentioned reasons. However, as can be seen from comparison between different theoretical or semi-empirical models with the neural network model the overall error of the neural network model for both seen and unseen data are much higher than that of alternative models over a wide range of operating conditions including temperature and size of nanoparticles. Fig. 8 shows the comparison of the measured and predicted thermal conductivity data [46,50] as a function of particle diameter for ZnO and TiO2 nanoparticles in DI water. From Fig. 8, it is apparent that the Jang's model [54] is intrinsically incapable of representing the particle size dependence of the both nanofluids particularly for small- sized nanoparticles. However, Fig. 8 reveals that the experimental thermal conductivity is not remarkably different from the predictions of the proposed neural network model. This results exhibit that the thermal conductivity of the ZnO-DI water

Fig. 12. Plot of thermal conductivity ratio at 303.15 K of 70 nm ZnO in EG [50] as a function of percentage volume fraction.

and TiO2-DI water nanofluids is strongly dependent on the size of the suspended particles and confirm the need for more advanced models such as the developed neural network model for nanofluids thermal conductivity prediction. Fig. 9 illustrates the thermal conductivity enhancement of the CuOwater nanofluids [28] versus volume concentration ranging from 0% to 14%. It is clear that the effective thermal conductivity enhances for higher particle concentration, however, existing models like Chon et al. [30], Koo and Kleinstreuer [31], HC, and Li & Peterson [29] overpredict the thermal conductivities over the whole nanoparticle concentration ranges. Similarly, Fig. 10 demonstrates the thermal conductivity enhancement for Al2O3-water nanofluids [28] with 36 nm particle diameter over the temperature range of 293 K to 313 K. As Figs. 9 & 10 illustrates, the proposed neural network model matches experimental data best, although other models have poorer results when temperature changes. This can be concluded that temperature dependency of the aforementioned models is poor, leading to inaccurate results for different temperatures. Temperature affects the movement of nanoparticles by Brownian motion as well as the interactive forces between them that leads to inaccurate prediction of the effective thermal conductivities by using the empirical correlations. Also for Cu-Oil nanofluids at 333 K [27], as shown in Fig. 11, the developed model by Jang and Choi [54] contains some inaccuracy for higher solid particle concentrations while the Brownian motion of the Cu nanoparticles ignored. The Jang and Choi model fail at higher

Fig. 14. Comparison of experimental and model predictions for the effective thermal conductivity as a function of volume fraction for CuO-water nanofluids [44] at 299 K.

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Fig. 15. Thermal conductivity enhancement for 12 nm SiO2 nanoparticles in water at 333 and 353 K [43].

temperatures because of the particles aggregation due to temperature rising, which lead to higher effective thermal conductivity predictions by this model. The interactions complexity between nanoparticles aggregate and particle concentration/temperature made the introduction of Kapitza resistance parameter β for better predictions [54]. Additional mechanisms such as the near-field radiation are needed to describe the temperature effect on the effective thermal conductivity of nanofluids at higher temperatures. From Fig. 11, the neural network predicted data agrees well with the experimental values over the particle concentration of 0% to 2.3%. Experimental thermal conductivity values of ZnO-EG nanofluids [50] at 303 K are compared with the Maxwell [7], HC [8], Jeffrey [55], Bruggeman [56], Turian [57] and the neural network predictions, as shown in Fig. 12. The Bruggeman model calculates the thermal conductivity of nanoparticle aggregates as follows: 2 " #0:5 3 2 kp kp kp ka 14 5 ¼ ð3φin −1Þ þ ð3ð1−φin Þ−1Þ þ ð3φin −1Þ þ ð3ð1−φin Þ−1Þ þ 8 kbf 4 kbf kbf kbf

ð23Þ where ka and φin are the thermal conductivity and the volume fraction

Fig. 16. Comparison between experimental thermal conductivity values and models result regarding TiO2-water nanofluids [49] at different temperatures and concentrations for 72, 73 and 76 nm particles.

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Fig. 17. The comparison between Eq. (4) and the neural network model with the experimental data of Chen et al. [63], Hwang et al. [64] and Jiang et al. [65] in the case of dispersing MWCNTs in distilled water, EG and refrigerant R113, respectively.

of aggregates, respectively. The Jeffrey model is the modified version of the HC model by considering the pair interactions of randomly dispersed spheres: knf ¼ 1 þ 3βφ þ k0

! 3β3 9β3 α þ 2 3β4 þ þ … φ2 3β þ þ 4 16 2α þ 3 64 2

ð24Þ

k

p where β ¼ α−1 αþ2 and α ¼ k0 is the thermal conductivity ratio. Also, Turian

et al. [57] presented a number of models using thermal conductivity data on dispersions of silica, alumina, coal, and other particles. As shown, the Maxwell, Jeffrey and Bruggeman models underpredict the experimental enhancement of ZnO nanofluids while the Turian and HC models overpredict the experimental data when the sphericity taken as 0.55 for the HC model. On the other hand, the neural network model represents a reasonable result compared to estimations provided by the theoretical models. Figs. 13 and 14 compares the values of the effective thermal conductivity determined experimentally to those predicted by the Bruggeman [56], Maxwell [7], HC [8] and the neural network model. The comparison for 25 nm CuO in water [44] can be seen in Fig. 13, whereas that for the same size of CuO in MEG [44] illustrated in Fig. 14. Figs. 13 and 14 obviously indicates that the Maxwell and HC models underpredict the experimentally determined effective thermal conductivity values while the Bruggeman model had better prediction than did

Fig. 18. The comparison between Eq. (4) and the neural network model with the experimental data of Choi et al. [13] and Jiang et al. [65] in the case of dispersing MWCNTs in olefin oil and refrigerant R113, respectively.

296

A. Aminian / Powder Technology 301 (2016) 288–309

Fig. 19. Comparison of experimental thermal conductivity ratio [23] with the existing models at 323 K.

Maxwell and HC models, because the former considers the interactions between randomly distributed particles. The Maxwell and HC models do not take into account the effects of nanoparticle-fluid interface, the size of the nanoparticles and the temperature effect. On the other hand, from the comparison of the proposed neural network model results and those predicted by the theoretical models, it is obvious that the neural network model outperforms the other models both in accuracy and generality. This is probably because the theoretical models do not regard the particle size, the Brownian motion, the nano layering and the effect of nanoparticles clustering, which are important to nanoparticles in nanofluids. In addition, the comparison between Figs. 9 & 13 shows some discrepancies regarding the neural network model predictions for CuO-water nanofluids with 25 nm and 29 nm particles. One reason for this deviation might be due to the randomness of data dividing during the model development. Accordingly, for better comparison in the wide ranges of volume fraction the whole datasets of each nanofluid system at constant temperature is figured out. Also, as can be seen from those figures, smaller particles of Cu nanoparticles have much higher effective thermal conductivities than those of bigger Cu nanoparticles of 29 nm. The difference in nanoparticle size can greatly examine the prediction accuracy of every model as reported by several references [42–46]. However, the neural network model shows better performance compared to that of other alternatives, as shown in Figs. 9 & 13.

Fig. 21. Comparison of models for thermal conductivity with experimental data [39] for CuO-water, 18 nm, T = 300 K.

Fig. 15 shows the evolution of the thermal conductivity enhancement of SiO2-water nanofluids [43] at 333 and 353 K. The available models namely, Fricke [58], Nan [59], Krischer [62], Bruggeman [56], Leong [61] and the proposed neural network model are used to predict the evolution of the thermal conductivity enhancement with nanoparticle content. It can be seen that for 12 nm silica nanofluids at 333 K, the deviation of the theoretical models from the experimental ones are somehow obvious. However, the error increases as solid content increases for Fricke, Bruggeman and Leong models. This is probably because of the change in the arrangement of particles inside the base fluid due to the higher number of interactions between them as a result of increasing the number of particles in the base fluid. The Leong et al. [61], Fricke [58], Nan [59] and Krischer [62] models for the thermal conductivity of nanofluids have the following forms: Leong,
knf ¼



 

  kp −klr φklr 2γ 31 −γ 3 þ 1 þ kp þ 2klr γ31 φγ3 klr −kbf þ kbf


 γ 31 kp þ 2klr − kp −klr φ γ 31 þ γ 3 −1 ð25Þ

where k lr , h and σ are the thermal conductivity of the interfacial layer, the interfacial layer thickness and a parameter which characterizes the diffuseness of interfacial boundary, respectively [62]. Krischer,  knf ¼

  −1 1− f 1−φ φ þf þ ð1−φÞkbf þ φkp kbf kp

ð26Þ

where f is the fraction of parallel resistances in a rectangular array of elements, with f = 0 and 1 equivalent to series and parallel particles, respectively. Fricke,  1 5kbf þ kp
knf ¼ kbf þ φ kp −kbf 3 kbf þ kp

ð27Þ

Nan, knf ¼ kbf

Fig. 20. Comparison of experimental thermal conductivity ratio [23] with the neural network and Godson models as a function of temperature.


 3 þ φ kp =kbf 3−2φ

ð28Þ

Also, the prediction of the thermal conductivity enhancement of TiO2-water nanofluids [49] as a function of temperature and nanoparticle concentration is depicted in Fig. 16. As shown in Fig. 16, the HC model underpredicts the thermal conductivity enhancement with

A. Aminian / Powder Technology 301 (2016) 288–309

297

Table 1 The optimum calculated values of the neural network model parameters. w2ji, j = 1, …, 9; i = 1, …, 5 0.037617 2.0613 0.40292 1.5929 0.49669 2.0354 −1.626 −1.6004 −0.56312

b2j , j = 1, …, 9 0.61086 1.1336 0.041619 −3.0137 −1.7591 −1.7301 −0.81263 2.9246 0.41962

−1.7683 −1.1986 −1.7127 −3.0494 −2.4788 0.48535 0.38843 2.877 1.7707

4.1434 −1.1758 3.8698 0.10781 −0.64387 −1.1412 −1.0145 −1.1444 −3.3543

2.7105 1.0017 3.4574 −5.3952 1.3917 −1.1846 −0.02592 4.962 −4.0994

7.3126 −1.5433 7.0332 −1.2898 0.19826 −0.65052 1.4011 0.10069 −6.6533

w3ji, j = 1; i = 1, …, 5 −0.17634 w3ji,

0.28871

0.54104

0.014639

0.43402

j = 1; i = 1, …, 9

−2.0892

0.25853

4.1898

−2.8935

0.43037

0.18952

1.2275

−2.7248

2.6137

b3j , j = 1 −0.27824

respect to experimental data at temperatures ranging between 342 K and 352 K, while its predictions do not differ much over the temperature range of 323 K to 352 K indicating the temperature independency of the HC model for predicting the effective thermal conductivities of TiO2-water nanofluids with 72, 73 and 76 nm particle diameters [49]. The deviations between the neural network predictions and experimental values over the particle concentration range of 0.24% to 11.22% and at temperatures ranging between 323 K and 352 K is about 2% representing the accuracy of the proposed model for the TiO2-water nanofluids.

The comparison between Eq. (4) and the neural network model with the experimental data of Chen et al. [63], Hwang et al. [64] and Jiang et al. [65] in the case of dispersing MWCNTs in distilled water, EG and refrigerant R113 is illustrated in Fig. 17. In this figure, the average diameters of these CNTs are about 15, 20 and 80 nm, respectively. The accuracy of the theoretical models for predicting the thermal conductivity of CNTs-based nanofluids is depend on the shape that they considered in their models. As shown in Fig. 17, Eq. (4) underpredicts the experimental values for CNTs-water and CNTs-R113, while overpredicts the experimental ones for CNTs-EG

Table 2 The ranges of experimental conditions for different nanofluid systems studied in this work. Systems Al-Water Al-TO Al-EG Al-EO Cu-Water Cu-EG Cu-Oil Al2O3-Water Al2O3-EG CuO-TO CuO-Water CuO-EG CuO-EMG CuO-MEG CuO-Paraffin Al2O3-DI Water TiO2-EG TiO2-DI Water TiO2-Water ZnO-EG ZnO-DI Water SiO2-Oil SiO2-Water SiO2-EG MWCNT-Water MWCNT-EG MWCNT-Oil MWCNT-R113 Ag-Water

dp (nm)

T (K)

Φ (%)

No. of data points

Ref.

25, 80

293–333

0.02–5

65

[18,17,27,39,42]

80 80 20 11, 13, 20, 25, 30, 33, 36, 38.4, 45, 47, 100, 150 11, 28, 35, 38, 45, 80, 150 31 18, 23.6, 25, 28.5, 29, 33 12, 23.6, 29, 35 25 30, 40 30, 40 11, 47, 80, 150 10, 15, 34, 40, 70 10, 34, 70 18, 35, 40, 52, 72, 73, 76 30, 60, 70 10, 30, 60 15 10, 12 12 9.2, 15 20 25 15, 80 60, 63

293–323 293–323 303–483 273–353

0.1–3 0.1–3 0.01–2.2 0.015–9

75

[27,40,41]

293–324

0.01–14

294–344 298 298 283–352 283–343

1–4 1–5 1–3 0.005–11.22 1–6.2

455 69 16 191 17 6 10 10 28 17 9 54 309

296–380 298–353 298 293–313

1.2–7 0.015–5 0.05–0.4 0.005–0.8 0.25–1 0.25–1 0.195–1 0.3–1.2

25 24 4 17 4 5 10 135

323–363

[28,6,38–42,35] [43,37,17] [36] [33,34,28,47,44,39,6] [39] [38,22] [48] [48] [30,41] [42,46] [46] [44,49,39] [46,50] [51] [43,39] [39] [19,52,63] [19,64] [13,19,65] [19,65] [23,53]

298

A. Aminian / Powder Technology 301 (2016) 288–309

Brownian motion decreases with an increase in the bulk viscosity of nanofluids due to higher particles concentration. Therefore, Godson's model at high concentrations and at high temperatures has lower accuracy when compared with the proposed neural network model. Fig. 21 compares the different models for predicting the thermal conductivity ratio versus volume fraction for CuO-water nanofluids [39] at 300 K. The diameter and the thermal conductivity of the particles are 18 nm and 78 W/m·K, respectively. As can be seen from Fig. 21, five models namely the proposed neural network, Chen [68], Zhu [69], semi-BM [39] and Maxwell [7] models are compared against volume fraction at 300 K. Chen et al. [68] applied the Maxwell's model with some modifications in term of the aggregation mechanism as follows: Fig. 22. Comparison between the proposed neural network model predictions and experimental thermal conductivity ratios.

nanofluids. Also, Fig. 18 depicts the comparison between our model and Eq. (4) with the experimental data of Choi et al. [13] and Jiang et al. [65] in the case of dispersing MWCNTs in olefin oil and refrigerant R113, respectively. Fig. 18 illustrates that our model has correctly predicted the trends observed in the experimental data of Choi et al. [13] and Jiang et al. [65] for MWCNTs-olefin oil and MWCNTs-R113. The AAD of the proposed model for predicting the effective thermal conductivities of the CNTs-based nanofluids is obtained 1.89%. As shown in Figs. 17 and 18, calculated results of Eq. (4) are somehow worsen as particles concentration increases and the temperature effects on dispersing CNTs does not considered in their model. The comparison between various models regarding the thermal conductivity enhancement of 60 nm Ag nanoparticles in water [23] as a function of particle volume concentration and temperature is shown in Figs. 19 and 20, respectively. The measured thermal conductivity enhancement of nanofluids is much higher than the predicted value of the theoretical models namely, HC [8], Timofeeva et al. [66], and Wasp [67]. Timofeeva et al. [66] implemented the effective medium theory to calculate the thermal conductivity of nanofluids as follows: knf ¼ 1 þ 3φ kbf

ð29Þ

Also, Wasp [67] used the following equation for calculating the thermal conductivity of nanofluids:
 kp þ 2kbf ‐2φ kbf ‐kp knf
 ¼ kbf kp þ 2kbf þ φ kbf ‐kp

ð30Þ

Since the aforementioned models are far deviating from the experimental values over the particles concentration and temperature ranges, the results of the proposed neural network model are also compared with Godson et al. [23] model for the thermal conductivity of Ag-water nanofluids with the coefficients regressed against temperature, as shown in Fig. 20. The Godson's model is as follows [23]: knf ¼ 0:9692φ þ 0:9508 kw

ð31Þ

As shown in Fig. 20, as particles concentration increases the deviation between Godson's model and experimental ones become more apparent, because at higher temperatures the effectiveness of the

keff ka þ 2kbf −2∅a ðkbf −ka Þ ¼ kbf ka þ 2kbf þ ∅a ðkbf −ka Þ

ð32Þ

where ka and ϕa are the thermal conductivity of aggregates and effective volume fraction of aggregates, respectively. Zhu et al. [69] suggested a model for calculating the effective thermal conductivity of nanofluids as follows: keff ¼ kbf

      Cp;nf a ρnf b M c Cp ρ Mnf

ð33Þ

where ρnf, Cp , nf and Mnf are the density, specific heat and molecular weight of a nanofluid, as given by the following equations: ρnf ¼ ð1−φÞρ f þ φρp


ρCp

 nf

¼ ð1−φÞρ f Cp; f þ φρp Cp

Mnf ¼ ð1−φÞM f þ φMp

ð34Þ ð35Þ ð36Þ

Prasher et al. [70] modified Maxwell's model by taking into account the Brownian motion of the nanoparticles: keff ¼ kbf

 1þ

Re: Pr 4

 ð1 þ 2αÞ þ 2φð1−αÞ ð1 þ 2αÞ−φð1−αÞ

ð37Þ

where Re is Reynolds number, Pr is Prandtl number, and α is equal to 2Rbkf/dp in which Rb is the interfacial resistance and dp is the nanoparticle diameter. The semi-empirical Brownian model (semi-BM) can be obtained by modifying Eq. (37) by a Nu number correlation, which can be written as: ð1 þ 2αÞ þ 2φð1−αÞ keff

¼ 1 þ ARem : Pr0:333 φ kbf ð1 þ 2αÞ−φð1−αÞ

ð38Þ

where A is 40000 for both water and EG base fluids and m can be chosen as 2.5 for water and 1.6 for EG. From Fig. 21, both Chen with r-ratio equal to 3 and Zhu models overpredict the experimental findings over the studied volume fraction. The semi-BM and Maxwell models had better results compared to those of Chen and Zhu models for CuO-water nanofluids. As shown in Fig. 21, the neural network model yields the best prediction at low and high volume fractions, whereas the aforementioned models were all poor as result of particles aggregation at high concentrations. The proposed neural network model for finding the thermal conductivity ratios agrees well with the experimental values from the present study. The proposed neuromorphic model predictions are compared with all experimental values via a parity plot in Fig. 22. The coefficient of determination (R 2 -value) of about 0.9309 with the overall AAD of about 3.06% indicates the very good accuracy of the proposed model for the prediction of the thermal conductivity

A. Aminian / Powder Technology 301 (2016) 288–309

ratio of nanofluids regarding heat transfer intensification of process fluids in different industries. 4. Conclusions Accurate prediction of the thermal conductivity of nanofluids is essential for controlling the size and cost associated with manufacturing and operation of new thermal systems. Therefore the effective thermal conductivity of Al 2 O 3, CuO, TiO2 , Cu, Al, Ag, SiO2 , ZnO and MWCNTs with a diameter in the range between 10 nm and 150 nm, suspended in water, Ethylene Glycol, Transformer Oil, Mono Ethylene Glycol, refrigerants and oils are predicted using a cascade-forward neural network model, and the effect of different parameters such as nanoparticles volume fraction, size and type of the particles, temperature, and type of the base fluid on the thermal conductivity ratios are investigated and compared to other theoretical models. The neural network model is trained and validated using the Levenberg-Marquardt backpropagation algorithm and the tan-sigmoid activation function is applied to calculate the output values of the neurons of the hidden layer. There are good agreements between the predicted and measured experimental values regarding the training, validation and testing phases. The proposed model is very accurate and reliable for predicting the thermal conductivity ratios over wide ranges of dependent variables. The presented model outperforms the other existing correlations both in accuracy and generalizeability. The correlation of determination (R 2-value) of 0.9309 and AAD of 3.06% proofing the advantages of the developed neuromorphic model for accurate prediction of the effective thermal conductivity of nanofluids.

ρ κ ξ σ ϕa Δw ∇E

299

density Boltzmann constant shape factor diffuseness of interfacial boundary effective volume fraction of aggregates weight change gradient of the error

Subscript

bf CNT exp f h nf o p pred

base fluid carbon nanotube experimental value fluid hidden layer nanofluid output layer particle predicted value

Appendix

Nomenclature output from the k-th output node ak output from the j-th hidden node aj AAD% percentage average absolute deviation defined by Eq. (21) bias value of the j-th neuron bj specific heat of the base fluid Cpbf average particle diameter dp E error function f transfer function h interfacial layer thickness k index of k-th neuron in the output layer thermal conductivity of primary particles kpm thermal conductivity of the interfacial layer klr thermal conductivity of aggregates ka m iteration number molecular weight of nanofluid Mnf n hidden nodes number N number of elements in the input-output vectors overall network output for the i-th element Oi Pr Prandtl number liquid molecule radii r1 correlation coefficient R2 Re Reynolds number T temperature (C) target values of the i-th element ti weight value between the i-th neuron and the j-th neuron wji i-th input to the network xi net output from j-th neuron yj Greeks

φ, ε & α φc η

nanoparticle volume fraction volume fraction of clusters learning rate

Fig. 23. Organizational chart representing the steps of calculating the effective thermal conductivity.

300

A. Aminian / Powder Technology 301 (2016) 288–309

Table A1 The whole datasets used for training, validation and testing. d (nm)

T (K)

kp

kbf

80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80

323.15 323.15 323.15 323.15 323.15 313.15 313.15 313.15 313.15 313.15 303.15 303.15 303.15 303.15 303.15 293.15 293.15 293.15 293.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 323.15 323.15 323.15 323.15 313.15 313.15 313.15 313.15 313.15 303.15 303.15 303.15 303.15 303.15 293.15 293.15 293.15 293.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15

204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383

0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 3.00 2.00 1.00 0.50 0.11 3.00 2.00 1.00 0.50 0.11 3.00 2.01 1.00 0.50 0.11 3.00 2.00 1.00 0.50 0.11 3.00 3.00 3.00 3.00 2.00 2.00 2.00 2.00 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.10 0.10 0.10 0.10 3.00 2.00 1.00 0.50 0.10 3.00 2.00 1.00 0.50 0.10 3.00 2.00 1.00 0.50 0.10 3.00 2.00 1.00 0.50 0.10 3.00 3.00 3.00 3.00 2.00 2.00 2.00 2.00 1.00 1.00 1.00 1.00 0.50 0.50

kratio

Ref.

1.280 1.239 1.185 1.120 1.039 1.269 1.215 1.160 1.090 1.031 1.212 1.188 1.110 1.101 1.028 1.179 1.165 1.116 1.050 1.025 1.292 1.260 1.240 1.187 1.236 1.215 1.201 1.170 1.230 1.176 1.156 1.135 1.141 1.117 1.097 1.090 1.093 1.093 1.092 1.094 1.359 1.300 1.230 1.151 1.080 1.329 1.320 1.214 1.130 1.057 1.291 1.235 1.181 1.071 1.059 1.261 1.180 1.152 1.061 1.044 1.331 1.281 1.251 1.211 1.268 1.241 1.231 1.140 1.191 1.182 1.157 1.113 1.131 1.106

[40]

d (nm) 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 36 36 36 36 36 36 36 36 36

T (K) 306.56 306.43 303.90 303.18 302.98 302.71 302.71 302.40 302.19 302.02 301.96 301.75 301.55 301.27 301.03 300.69 298.58 296.53 295.30 295.06 311.99 311.61 310.52 310.59 308.51 307.86 306.77 305.61 305.16 304.52 304.17 303.90 303.90 298.37 298.37 297.93 297.62 297.28 296.94 296.94 296.32 296.09 296.02 295.78 295.23 295.16 312.78 312.06 311.21 310.08 309.36 308.95 307.25 306.63 305.95 305.54 304.99 299.12 298.88 298.13 298.51 298.13 297.62 296.77 296.56 312.48 312.00 310.57 309.44 308.42 308.25 307.87 307.77 307.33

kp

kbf 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 9.30 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00

kratio 1.147 1.129 1.145 1.144 1.132 1.145 1.131 1.129 1.147 1.129 1.113 1.134 1.135 1.115 1.118 1.113 1.116 1.111 1.111 1.097 1.190 1.208 1.194 1.179 1.194 1.161 1.145 1.144 1.111 1.094 1.095 1.094 1.115 1.077 1.061 1.076 1.061 1.027 1.027 1.044 1.045 1.045 1.029 1.029 1.050 1.016 1.127 1.129 1.115 1.100 1.097 1.095 1.113 1.113 1.129 1.079 1.127 1.047 1.013 1.011 1.047 1.044 1.047 1.063 1.044 1.242 1.240 1.239 1.192 1.177 1.195 1.210 1.174 1.160

Ref.

A. Aminian / Powder Technology 301 (2016) 288–309

301

Table A1 (continued) d (nm)

T (K)

kp

kbf

80 80 80 80 80 80 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 10 44 150 11

303.15 293.15 323.15 313.15 303.15 293.15 323.15 323.15 323.15 323.15 313.15 313.15 313.15 313.15 303.15 303.15 303.15 303.15 293.15 293.15 293.15 293.15 323.15 323.15 323.15 323.15 313.15 313.15 313.15 313.15 303.15 303.15 303.15 303.15 293.15 293.15 293.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 313.15 303.15 293.15 323.15 323.15 323.15 323.15

383 383 383 383 383 383 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.62 0.62 0.62 0.62

Vol. 0.50 0.50 0.10 0.10 0.10 0.10 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 2.00 1.00 0.50 3.00 3.00 3.00 3.00 2.00 2.00 2.00 2.00 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 3.00 3.00 3.00 3.00 2.00 2.00 2.00 2.00 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 3.00 3.00 3.00 1.00

kratio 1.096 1.070 1.055 1.051 1.038 1.036 1.105 1.070 1.041 1.042 1.095 1.065 1.034 1.030 1.090 1.045 1.040 1.024 1.079 1.049 1.024 1.015 1.260 1.175 1.150 1.094 1.194 1.159 1.130 1.078 1.175 1.136 1.113 1.069 1.124 1.109 1.070 1.048 1.318 1.240 1.210 1.159 1.250 1.189 1.173 1.134 1.158 1.140 1.125 1.116 1.140 1.097 1.082 1.071 1.112 1.104 1.095 1.095 1.079 1.070 1.079 1.054 1.055 1.047 1.037 1.025 1.043 1.033 1.024 1.013 1.181 1.140 1.105 1.130

Ref.

d (nm) 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

T (K) 306.82 307.09 306.48 306.51 306.10 305.72 305.69 305.35 304.70 303.99 303.51 303.51 303.17 302.89 302.62 302.14 301.87 301.60 301.05 300.64 300.71 300.06 299.52 299.31 298.56 298.49 295.32 294.26 293.89 312.38 311.97 311.70 311.39 311.22 310.98 310.71 310.33 309.75 307.84 307.40 305.39 304.87 303.51 297.85 296.21 295.53 294.98 303.15 303.15 303.15 303.15 303.15 333.15 333.15 333.15 333.15 333.15 363.15 363.15 363.15 363.15 363.15 393.15 393.15 393.15 393.15 393.15 423.15 423.15 423.15 423.15 423.15 453.15 453.15

kp 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11

Vol. 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 2.22 1.00 0.52 0.10 0.01 2.22 1.00 0.52 0.10 0.01 2.22 1.00 0.52 0.10 0.01 2.22 1.00 0.52 0.10 0.01 2.22 1.00 0.52 0.10 0.01 2.22 1.00

kratio 1.158 1.177 1.160 1.174 1.174 1.177 1.129 1.147 1.147 1.148 1.147 1.161 1.161 1.160 1.144 1.145 1.142 1.145 1.113 1.113 1.129 1.129 1.131 1.116 1.116 1.132 1.115 1.116 1.113 1.194 1.213 1.166 1.166 1.147 1.181 1.177 1.210 1.195 1.132 1.113 1.095 1.126 1.097 1.061 1.048 1.047 1.048 1.329 1.241 1.172 1.135 1.130 1.306 1.204 1.144 1.113 1.103 1.251 1.158 1.110 1.079 1.077 1.190 1.107 1.069 1.043 1.041 1.072 1.050 1.019 1.002 0.995 0.985 0.972

Ref.

[27]

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A. Aminian / Powder Technology 301 (2016) 288–309

Table A1 (continued) d (nm)

T (K)

45 150 10 44 150 11 45 150 11 45 150 11 45 150 11 45 150 11 45 150 47 47 47 47 47 47 11 11 11 11 11 11 47 47 47 47 47 47 150 150 150 150 150 150 38 38 38 38 24 24 24 24 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38

323.15 323.15 293.15 293.15 293.15 293.15 293.15 293.15 323.15 323.15 323.15 293.15 293.15 293.15 323.15 323.15 323.15 293.15 293.15 293.15 344.00 334.11 324.03 314.14 304.24 294.16 344.00 334.11 324.03 314.14 304.24 294.16 344.00 334.11 324.03 314.14 304.24 294.16 344.00 334.11 324.03 314.14 304.24 294.16 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 324.15 324.15 324.15 324.15 309.15 309.15 309.15 309.15 294.15 294.15 294.15 294.15 324.19 319.30 314.42 309.26 304.37 298.94 294.33 324.19 319.30 314.42 309.26

kp 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 1.00 1.00 3.00 3.00 3.00 1.00 1.00 1.00 3.00 3.00 3.00 3.00 3.00 3.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00 4.00 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00

kratio 1.090 1.040 1.110 1.079 1.080 1.069 1.050 1.024 1.319 1.170 1.109 1.160 1.090 1.093 1.158 1.100 1.053 1.114 1.059 1.024 1.290 1.239 1.193 1.163 1.116 1.081 1.151 1.159 1.127 1.118 1.100 1.089 1.106 1.095 1.068 1.053 1.043 1.027 1.081 1.047 1.047 1.024 1.015 1.011 1.097 1.067 1.048 1.020 1.143 1.122 1.097 1.063 1.242 1.171 1.144 1.107 1.160 1.118 1.101 1.068 1.093 1.067 1.050 1.019 1.243 1.220 1.194 1.158 1.126 1.113 1.094 1.106 1.095 1.086 1.066

Ref.

[30]

[6]

d (nm)

T (K)

20 20 20 20 20 20 20 20 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 25 25 25 25 25 25

453.15 453.15 453.15 483.15 483.15 483.15 483.15 483.15 296.15 296.15 296.15 296.15 380.29 379.08 377.67 376.05 373.43 370.61 367.59 363.35 358.92 353.07 347.02 340.97 335.13 329.68 323.84 317.79 311.54 305.69 301.25 299.84 299.24 343.34 323.27 303.19 283.12 343.34 323.10 303.02 283.12 343.34 323.18 303.28 283.29 343.17 323.18 303.19 283.12 343.51 323.18 303.19 283.29 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 273.15 296.15 296.15 296.15 296.15 296.15 296.15

kp 383 383 383 383 383 383 383 383 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78

kbf 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 0.52 0.10 0.01 2.22 1.00 0.52 0.10 0.01 7.00 5.00 3.60 1.20 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 6.20 6.20 6.20 6.20 4.70 4.70 4.70 4.70 3.40 3.40 3.40 3.40 2.10 2.10 2.10 2.10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.60 0.60 0.60 0.60 0.45 0.45 0.45 0.45 0.15 0.15 0.15 0.15 7.50 5.01 4.01 3.01 2.00 1.01

kratio 0.966 0.959 0.954 0.921 0.911 0.903 0.897 0.894 1.185 1.126 1.082 1.016 1.146 1.145 1.147 1.137 1.138 1.138 1.140 1.129 1.130 1.129 1.129 1.122 1.121 1.122 1.122 1.121 1.122 1.121 1.121 1.120 1.120 1.492 1.384 1.298 1.233 1.346 1.258 1.202 1.159 1.248 1.175 1.140 1.101 1.165 1.100 1.074 1.047 1.093 1.044 1.022 1.002 1.271 1.174 1.096 1.016 1.183 1.122 1.065 0.996 1.162 1.101 1.044 0.987 1.118 1.081 1.036 0.980 1.320 1.177 1.151 1.126 1.108 1.057

Ref.

[51]

[50]

[45]

[47]

A. Aminian / Powder Technology 301 (2016) 288–309

303

Table A1 (continued) d (nm)

T (K)

kp

kbf

38 38 38 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 38 38 38 38 24 24 24 24 15 15 15 15 15 40 40 40 80 80 80 80 80 80 80 80 80 80 80 80 80 28 28 28 28 28 38 38 38 38 38 38 38 38 38 10 10

304.37 298.94 294.33 324.19 319.30 314.42 309.26 304.37 298.94 294.33 324.19 319.30 314.42 309.26 304.37 298.94 294.33 324.15 324.15 324.15 324.15 309.15 309.15 309.15 309.15 294.15 294.15 294.15 294.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15

27 27 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 27 27 27 27 78 78 78 78 8 8 8 8 8 8 8 8 204 204 204 204 204 204 204 204 204 204 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 29 29

0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.15 0.15 0.15 0.15 0.15 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61

Vol. 1.00 1.00 1.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 1.00 5.00 4.00 3.00 2.00 1.00 4.00 3.00 2.00 5.00 4.00 3.00 2.00 1.00 5.00 4.00 3.00 2.00 1.00 4.00 3.00 1.50 5.00 4.00 3.00 2.00 1.00 3.00 2.00 1.00 3.00 2.00 1.50 0.80 0.50 0.30 1.00 2.00

kratio 1.052 1.044 1.021 1.358 1.295 1.252 1.255 1.230 1.190 1.141 1.290 1.248 1.231 1.214 1.155 1.136 1.062 1.352 1.345 1.306 1.290 1.253 1.244 1.241 1.213 1.139 1.126 1.102 1.063 1.097 1.067 1.048 1.020 1.143 1.122 1.097 1.063 1.176 1.130 1.106 1.066 1.040 1.136 1.118 1.083 1.446 1.359 1.288 1.192 1.109 1.308 1.259 1.207 1.138 1.081 1.169 1.145 1.069 1.175 1.148 1.118 1.091 1.050 1.107 1.070 1.029 1.080 1.053 1.038 1.019 1.014 1.010 1.049 1.096

Ref.

[42]

[46]

d (nm)

T (K)

kp

kbf

25 25 25 25 25 25 12 12 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 13 13 13 13 13 13 13 13 13 13 11 11 11 11 11 11 11 11 11 11 13 13 13 33 33 33 33 40 40 40 150 150 150 150 150 47 47 47 47 47 11 11 11 11 11 9 9 9

296.15 296.15 296.15 296.15 296.15 296.15 313.15 313.15 313.15 333.15 333.15 333.15 333.15 353.15 353.15 353.15 313.15 313.15 313.15 333.15 333.15 333.15 333.15 353.15 353.15 353.15 313.15 313.15 313.15 333.15 333.15 333.15 333.15 353.15 353.15 353.15 313.15 313.15 313.15 333.15 333.15 333.15 333.15 353.15 353.15 353.15 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 330.23 320.17 310.10 300.03 290.11 330.23 320.17 309.95 300.03 289.96 330.23 320.02 309.95 300.18 290.11 313.15 313.15 313.15

78 78 78 78 78 78 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 8 8 8 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 1800 1800 1800

0.27 0.27 0.27 0.27 0.27 0.27 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 7.50 5.02 4.01 2.99 2.00 1.01 5.01 1.01 0.50 5.02 1.00 0.50 0.02 5.01 0.99 0.50 5.02 1.02 0.50 5.02 1.01 0.50 0.02 5.02 1.00 0.51 5.01 1.01 0.51 5.00 1.01 0.51 0.02 5.01 1.00 0.51 5.02 1.02 0.52 5.01 1.02 0.52 0.02 5.01 1.02 0.51 6.00 5.58 5.15 4.02 3.02 2.11 1.10 3.22 2.22 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.28 0.11 0.06

kratio 1.210 1.167 1.159 1.121 1.074 1.039 1.014 1.001 0.985 1.070 1.037 0.972 0.999 1.087 1.002 1.016 1.001 0.966 0.995 0.997 0.921 1.001 1.001 1.081 0.947 1.005 1.054 1.019 1.006 1.032 1.037 0.961 1.001 1.122 0.938 0.949 1.047 1.018 1.016 1.100 0.981 0.988 1.001 1.185 1.096 1.148 1.240 1.189 1.109 1.138 1.108 1.093 1.063 1.092 1.078 1.059 1.076 1.056 1.018 0.986 0.949 1.115 1.075 1.039 1.007 0.973 1.200 1.149 1.120 1.064 1.011 1.390 1.282 1.217

Ref.

[43]

[44]

[52]

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304

A. Aminian / Powder Technology 301 (2016) 288–309

Table A1 (continued) d (nm)

T (K)

kp

kbf

10 30 30 30 60 60 60 30 30 30 60 60 60 10 10 10 34 34 34 70 70 70 10 10 10 34 34 34 70 70 70 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 38 38 38 38 38 38 38 47 47 47 47 47 24 24 24 24 24 24 24

298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 333.84 323.79 313.68 303.59 294.56 333.84 323.79 313.68 303.59 294.56 333.84 323.79 313.68 303.59 294.56 333.84 323.79 313.68 303.59 294.56 333.84 323.79 313.68 303.59 294.56 325.79 320.63 315.49 310.18 304.95 299.93 294.48 333.84 323.79 313.68 303.59 294.56 298.15 298.15 298.15 298.15 298.15 298.15 298.15

29 29 29 29 29 29 29 29 29 29 29 29 29 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 27 27 27 27 27 27 27 27 27 27 204 204 204 204 204 204 204 204 204 204 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78 78

0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.25 0.25 0.25 0.25 0.25 0.25 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.61 0.61 0.61 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25

Vol. 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 1.00 2.00 3.00 0.50 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.00 3.00 3.00 3.00 3.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.50 2.50 2.00 1.69 1.00 4.00 3.00

kratio 1.142 1.033 1.071 1.115 1.018 1.049 1.073 1.060 1.127 1.210 1.032 1.071 1.107 1.033 1.077 1.114 1.028 1.063 1.087 1.020 1.043 1.064 0.266 0.278 0.292 0.262 0.273 0.284 0.259 0.266 0.272 1.370 1.345 1.292 1.246 1.221 1.208 1.167 1.134 1.108 1.094 1.123 1.111 1.102 1.082 1.061 1.091 1.075 1.062 1.046 1.037 1.122 1.099 1.085 1.063 1.032 1.107 1.097 1.088 1.068 1.054 1.046 1.024 1.140 1.111 1.082 1.058 1.028 1.122 1.093 1.070 1.057 1.039 1.213 1.142

Ref.

[38]

d (nm) 9 9 9 9 9 9 9 9 9 76 76 76 76 76 76 76 72 72 72 72 72 72 72 73 73 73 73 73 73 73 73 73 73 73 73 73 73 2 2 2 15 15 15 15 15 20 20 20 20 25 25 25 25 25 15 15 15 15 15 80 80 80 80 80 35 35 35 35 35 35 35 35 35 25 25

T (K)

kp

kbf

Vol.

kratio

313.15 303.15 303.15 303.15 303.15 293.15 293.15 293.15 293.15 293.70 303.70 313.50 323.20 332.50 342.50 352.40 294.20 303.60 313.50 323.30 330.40 341.80 352.40 294.80 303.70 313.40 323.00 333.00 341.20 352.30 294.20 303.60 313.70 323.20 332.70 341.90 352.80 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 327.99 317.84

1800 1800 1800 1800 1800 1800 1800 1800 1800 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 78 78 78 78 27 27 27 27 27 27 27

0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.62 0.62

0.01 0.28 0.11 0.05 0.01 0.28 0.11 0.05 0.00 0.24 0.24 0.24 0.24 0.24 0.24 0.24 2.54 2.54 2.54 2.54 2.54 2.54 2.54 5.54 5.54 5.54 5.54 5.54 5.54 5.54 11.22 11.22 11.22 11.22 11.22 11.22 11.22 0.38 0.18 0.09 1.00 0.80 0.60 0.50 0.30 1.00 0.75 0.50 0.25 1.00 0.80 0.60 0.40 0.20 1.00 0.80 0.60 0.40 0.20 1.00 0.79 0.60 0.40 0.20 3.99 2.98 2.01 1.00 5.00 4.01 2.99 1.99 1.00 5.00 5.00

1.144 1.316 1.221 1.180 1.111 1.156 1.074 1.065 1.047 1.012 1.039 1.023 1.071 1.086 1.103 1.177 1.031 1.034 1.044 1.078 1.095 1.169 1.247 1.083 1.099 1.135 1.127 1.154 1.200 1.296 1.213 1.226 1.234 1.247 1.312 1.331 1.332 1.402 1.250 1.149 1.121 1.090 1.074 1.055 1.028 1.291 1.155 1.085 1.054 2.575 2.102 1.617 1.452 1.202 2.030 1.909 1.700 1.578 1.452 1.413 1.370 1.257 1.135 1.061 1.226 1.142 1.090 1.050 1.181 1.136 1.112 1.061 1.030 1.345 1.300

Ref.

[49]

[19]

[22]

[45]

A. Aminian / Powder Technology 301 (2016) 288–309

305

Table A1 (continued) d (nm)

T (K)

24 24 29 29 29 29 29 29 29 29 29 29 29 29 29 13 13 13 20 20 20 20 33 33 33 33 20 20 20 20 38 38 38 38 38 38 38 38 38 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29

298.15 298.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 296.15 311.30 311.35 310.99 310.08 309.47 307.80 307.64 305.61 305.21 304.85 303.79 302.22 301.20 299.57 299.37

kp 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78

kbf

Vol.

kratio

0.25 0.25 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

2.00 1.00 14.03 12.83 12.25 9.03 8.00 7.10 6.10 5.10 4.00 3.08 2.05 1.18 0.00 4.40 2.72 1.29 5.51 4.29 3.01 0.99 4.95 4.06 1.92 0.93 14.58 5.97 4.09 2.74 4.22 2.84 1.98 1.22 4.55 4.16 2.94 1.98 0.99 18.18 16.99 16.20 15.24 13.92 13.00 12.04 10.06 8.05 7.09 5.05 3.17 1.98 1.03 9.97 8.39 6.83 4.82 3.43 1.55 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61

1.091 1.047 1.163 1.168 1.129 1.042 1.024 1.048 1.034 1.008 1.000 0.994 0.968 0.952 0.937 2.094 1.934 1.756 1.863 1.858 1.815 1.692 1.890 1.819 1.706 1.661 1.935 1.803 1.768 1.726 1.774 1.734 1.697 1.656 1.760 1.755 1.715 1.684 1.635 2.113 2.100 2.097 2.082 2.042 2.027 2.010 1.861 1.850 1.808 1.744 1.674 1.719 1.698 2.124 2.035 1.945 1.845 1.773 1.684 1.148 1.129 1.129 1.116 1.161 1.129 1.111 1.145 1.113 1.098 1.098 1.081 1.095 1.097 1.097

Ref.

[28]

d (nm) 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 30 30 30 30 30 40 40 40 40 40 30 30 30 30 30 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 35 35

T (K) 308.06 297.91 288.01 322.85 317.96 312.95 307.94 302.93 297.06 292.90 328.11 318.09 308.06 298.04 287.89 333.12 323.10 312.83 302.80 293.02 328.11 317.96 307.94 297.91 287.77 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 362.98 357.79 353.06 347.96 343.05 337.76 333.03 327.93 322.92 362.98 357.98 352.88 347.87 342.86 337.95 332.94 327.84 322.92 363.08 357.98 352.97 347.87 343.05 337.85 332.94 327.84 323.02 300.00 300.00

kp 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 8 8

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 5.00 5.00 5.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 3.00 3.00 3.00 3.00 3.00 4.00 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00 1.00 0.05 0.04 0.03 0.02 0.01 0.05 0.04 0.03 0.02 0.01 0.05 0.04 0.03 0.02 0.01 0.05 0.04 0.03 0.02 0.01 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.05 0.02

kratio 1.238 1.202 1.157 1.253 1.235 1.204 1.163 1.133 1.122 1.093 1.244 1.192 1.156 1.130 1.063 1.242 1.192 1.161 1.111 1.090 1.161 1.102 1.073 1.062 1.017 1.478 1.422 1.320 1.232 1.088 1.398 1.331 1.215 1.122 1.056 1.517 1.448 1.359 1.237 1.100 1.485 1.389 1.299 1.139 1.060 2.300 2.173 2.054 1.923 1.800 1.685 1.554 1.427 1.300 1.854 1.765 1.685 1.600 1.523 1.438 1.362 1.277 1.196 1.435 1.400 1.354 1.315 1.265 1.227 1.181 1.142 1.100 1.230 1.181

Ref.

[48]

[23]

[39]

(continued on next page)

306

A. Aminian / Powder Technology 301 (2016) 288–309

Table A1 (continued) d (nm)

T (K)

29 29 29 29 29 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 38 38 38 38 38 13 13 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36

297.80 296.94 296.63 294.86 294.25 320.99 320.64 314.47 311.78 311.68 311.12 310.51 309.76 309.40 300.59 300.29 299.43 299.28 300.54 299.93 299.22 299.78 299.32 295.89 295.84 306.81 323.07 313.20 303.08 297.71 293.19 319.64 304.51 313.05 310.71 309.43 308.19 307.21 306.74 305.72 305.08 304.66 304.31 303.89 303.50 302.99 300.09 298.09 296.00 296.13 295.66 295.11 308.61 306.78 304.56 302.17 308.01 306.39 304.30 302.00 312.04 311.63 310.54 310.67 308.59 307.84 306.75 305.66 305.22 304.57 304.23 303.92 303.99 298.39 298.22

kp 78 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 0.61 0.61 0.61 0.61 0.61 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 2.00 2.00 2.00 2.00 4.00 4.00 4.00 4.00 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10

kratio 1.079 1.047 1.047 1.045 1.061 1.913 1.861 1.934 1.768 1.794 1.821 1.802 1.798 1.756 1.684 1.679 1.685 1.711 1.661 1.663 1.660 1.640 1.640 1.697 1.671 1.665 1.974 1.866 1.813 1.790 1.755 2.113 2.123 1.974 1.923 1.897 1.897 1.897 1.915 1.895 1.895 1.895 1.900 1.895 1.895 1.898 1.840 1.752 1.719 1.810 1.723 1.760 2.319 2.253 2.187 2.187 2.319 2.279 2.242 2.158 1.190 1.208 1.192 1.176 1.192 1.158 1.144 1.145 1.113 1.100 1.097 1.098 1.111 1.079 1.063

Ref.

d (nm) 35 35 35 30 30 30 30 30 25 25 25 25 25 18 18 18 18 30 30 30 30 30 18 18 18 18 18 25 25 25 25 20 20 20 12 12 12 29 29 29 29 29 40 40 40 18 18 18 18 18 35 35 35 35 35 18 18 18 18 18 12 12 12 12 12 12 12 12 52 52 52 52 29 29 29

T (K) 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 283.00 283.00 283.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 298.00 323.06 313.06 302.98 292.98 323.90 319.02 313.98

kp 8 8 8 27 27 27 27 27 204 204 204 204 204 78 78 78 78 27 27 27 27 27 78 78 78 78 78 27 27 27 27 27 27 27 78 78 78 78 78 78 78 78 8 8 8 8 8 8 8 8 8 8 8 8 8 78 78 78 78 78 1 1 1 1 1 1 1 1 8 8 8 8 78 78 78

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.25 0.25 0.25 0.25 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 0.02 0.01 0.00 0.05 0.02 0.02 0.01 0.00 0.03 0.03 0.02 0.01 0.01 0.04 0.03 0.02 0.01 0.05 0.03 0.02 0.01 0.00 0.03 0.02 0.02 0.01 0.01 0.04 0.03 0.02 0.01 0.04 0.03 0.01 0.02 0.00 0.00 0.05 0.04 0.03 0.02 0.01 0.03 0.01 0.01 0.05 0.02 0.02 0.01 0.00 0.05 0.02 0.02 0.01 0.00 0.04 0.03 0.02 0.01 0.01 0.36 0.18 0.09 0.05 0.36 0.18 0.09 0.05 2.00 2.00 2.00 2.00 4.00 4.00 4.00

kratio 1.161 1.090 1.050 1.221 1.170 1.120 1.070 1.030 1.120 1.100 1.070 1.060 1.040 1.100 1.084 1.080 1.050 1.218 1.168 1.119 1.070 1.030 1.120 1.100 1.071 1.062 1.041 1.088 1.069 1.050 1.029 1.100 1.070 1.040 1.050 1.020 1.011 1.229 1.189 1.139 1.109 1.041 1.060 1.030 1.011 1.269 1.248 1.229 1.210 1.129 1.231 1.180 1.162 1.090 1.060 1.121 1.101 1.070 1.060 1.042 1.120 1.110 1.079 1.050 1.110 1.090 1.060 1.040 1.230 1.200 1.171 1.121 1.362 1.289 1.259

Ref.

A. Aminian / Powder Technology 301 (2016) 288–309

307

Table A1 (continued) d (nm)

T (K)

36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36

297.95 297.71 297.37 296.93 296.93 296.45 296.14 296.01 295.29 295.29 312.25 311.32 311.05 310.57 310.06 309.48 308.32 308.15 307.84 307.64 307.40 307.09 306.55 306.07 303.54 303.75 303.34 300.34 300.23 299.07 299.96 299.52 299.21 299.14 298.67 297.47 295.63 295.39 293.55 312.44 312.00 310.47 309.51 308.42 308.22 307.94 307.81 307.19 307.02 306.78 306.48 306.48 306.10 305.76 305.28 304.67 304.02 303.54 303.58 303.34 302.89 302.52 302.25 301.94 301.53 301.02 300.85 300.74 300.13 299.45 298.46 298.63 295.83 295.25

kp 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 3.10 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00

kratio 1.081 1.063 1.029 1.029 1.047 1.047 1.047 1.029 1.047 1.015 1.144 1.179 1.163 1.210 1.174 1.145 1.205 1.144 1.129 1.129 1.176 1.160 1.113 1.131 1.098 1.129 1.129 1.100 1.129 1.115 1.081 1.061 1.063 1.079 1.047 1.081 1.063 1.061 1.063 1.244 1.242 1.242 1.194 1.176 1.194 1.206 1.176 1.177 1.177 1.161 1.160 1.181 1.177 1.177 1.144 1.144 1.142 1.145 1.161 1.160 1.158 1.147 1.147 1.147 1.144 1.115 1.115 1.131 1.132 1.129 1.129 1.115 1.116 1.113

Ref.

d (nm) 29 29 29 29 39 39 39 39 39 39 39 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63

T (K) 308.71 303.98 298.94 293.82 323.67 318.86 313.90 309.02 304.13 299.02 293.90 362.88 358.08 352.87 347.86 342.95 337.94 332.93 327.82 322.92 362.88 357.97 352.97 347.86 342.85 338.04 332.93 327.82 322.92 362.98 358.08 352.97 347.96 342.95 338.04 332.93 327.92 322.92 362.88 358.08 352.97 347.96 342.95 338.04 332.93 327.92 322.92 362.88 358.08 353.07 347.96 343.05 338.04 332.93 327.92 322.92 362.88 357.97 352.77 347.96 342.95 337.94 332.93 327.82 322.92 323.00 323.00 323.00 323.00 323.00 323.00 328.00 328.00 328.00

kp 78 78 78 78 27 27 27 27 27 27 27 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 4.00 4.00 4.00 4.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 1.20 0.90 0.79 0.60 0.40 0.29 1.20 0.90 0.80

kratio 1.250 1.229 1.191 1.140 1.109 1.100 1.090 1.068 1.050 1.040 1.020 2.865 2.692 2.514 2.329 2.163 1.991 1.812 1.634 1.455 2.298 2.169 2.052 1.929 1.794 1.677 1.560 1.431 1.302 2.102 1.991 1.886 1.788 1.677 1.578 1.462 1.351 1.246 1.849 1.782 1.683 1.609 1.523 1.443 1.363 1.289 1.197 1.597 1.535 1.468 1.418 1.363 1.302 1.246 1.191 1.135 1.437 1.400 1.357 1.302 1.271 1.228 1.191 1.135 1.092 1.453 1.302 1.248 1.187 1.127 1.097 1.634 1.423 1.363

Ref.

[53]

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308

A. Aminian / Powder Technology 301 (2016) 288–309

Table A1 (continued) d (nm)

T (K)

36 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29

294.09 313.08 310.73 309.36 308.30 307.28 306.77 305.81 305.20 304.38 303.97 303.49 303.12 300.15 298.03 296.02 295.61 295.13 296.15 311.31 311.24 310.97 309.98 309.33 307.76 307.66 305.54 305.20 304.82 303.73 302.16 301.21 299.53 299.29 297.76 296.87 296.60 294.75 294.17 313.08 312.71 311.85 310.11 309.16 308.47 307.11

kp 27 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 9.00 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 3.30 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 9.30 9.30 9.30 9.30 9.30 9.30 9.30

kratio

Ref.

1.115 1.147 1.115 1.098 1.098 1.098 1.113 1.098 1.097 1.100 1.098 1.098 1.098 1.063 1.015 0.998 0.997 1.019 1.047 1.150 1.132 1.131 1.116 1.163 1.131 1.116 1.145 1.115 1.098 1.100 1.081 1.098 1.097 1.098 1.081 1.048 1.048 1.048 1.065 1.195 1.194 1.179 1.163 1.177 1.147 1.163

References [1] H. Loulijat, H. Zerradi, A. Dezairi, S. Ouaskit, S. Mizani, F. Rhayt, Effect of Morse potential as model of solid-solid inter-atomic interaction on the thermal conductivity of nanofluids, Adv. Powder Technol. 26 (2015) 180–187. [2] H. Loulijat, H. Zerradi, S. Mizani, E.M. Achhal, A. Dezairi, S. Ouaskit, The behavior of the thermal conductivity near the melting temperature of copper nanoparticle, J. Mol. Liq. 211 (2015) 695–704. [3] H. Zerradi, S. Mizani, H. Loulijat, A. Dezairi, S. Ouaskit, Population balance equation model to predict the effects of aggregation kinetics on the thermal conductivity of nanofluids, J. Mol. Liq. 218 (2016) 373–383. [4] J.H. Lee, K.S. Hwang, S.P. Jang, B.H. Lee, J.H. Kim, S.U.S. Choi, C.J. Choi, Effective viscosities and thermal conductivities of aqueous nanofluids containing low volume concentrations of Al 2 O 3 nanoparticles, Int. J. Heat Mass Transf. 51 (2008) 2651–2656. [5] D. Singh, E.V. Timofeeva, M.R. Moravek, S. Cingarapu, W. Yu, T. Fischer, S. Mathur, Use of metallic nanoparticles to improve the thermophysical properties of organic heat transfer fluids used in concentrated solar power, Sol. Energy 105 (2014) 468–478. [6] S.K. Das, N. Putra, P. Thiesen, W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, J. Heat Transf. 125 (4) (2003) 567–574. [7] J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press, Oxford, UK, 1873. [8] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two-component systems, Ind. Eng. Chem. Fundam. 1 (1962) 187–191. [9] L.S. Sundar, K.V. Sharma, Heat transfer enhancements of low volume concentration Al2O3 nanofluid and with longitudinal strip inserts in a circular tube, Int. J. Heat Mass Transf. 53 (2010) 4280–4286.

d (nm) 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63

T (K) 328.00 328.00 328.00 333.00 333.00 333.00 333.00 333.00 333.00 338.00 338.00 338.00 338.00 338.00 338.00 343.00 343.00 343.00 343.00 343.00 343.00 348.00 348.00 348.00 348.00 348.00 348.00 353.00 353.00 353.00 353.00 353.00 353.00 358.00 358.00 358.00 358.00 358.00 358.00 363.00 363.00 363.00 363.00 363.00 363.00

kp 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420

kbf 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

Vol. 0.60 0.40 0.30 1.20 0.90 0.80 0.60 0.40 0.30 1.20 0.90 0.79 0.60 0.40 0.29 1.20 0.90 0.80 0.60 0.40 0.30 1.20 0.90 0.80 0.60 0.40 0.30 1.20 0.90 0.80 0.60 0.40 0.29 1.20 0.89 0.80 0.60 0.39 0.30 1.20 0.90 0.80 0.60 0.39 0.29

kratio

Ref.

1.272 1.181 1.145 1.804 1.544 1.459 1.356 1.236 1.169 1.991 1.683 1.580 1.447 1.302 1.230 2.178 1.804 1.677 1.514 1.356 1.266 2.341 1.931 1.798 1.610 1.411 1.308 2.517 2.039 1.882 1.683 1.465 1.350 2.698 2.184 1.985 1.761 1.532 1.411 2.891 2.299 2.118 1.840 1.586 1.441

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