Densities and isothermal compressibilities from perturbed hard-dimer-chain equation of state: application to nanofluids

4.1 Estimation of densities and isothermal compressibilities

The values of molecular parameters, ε, σ and m appearing in the PHDC EoS have to be characterized prior to use that EoS for the estimation of densities and isothermal compressibilities of nanofluids. For the case of base fluids, ε, σ and m values were fitted against the PρT data taken from the literature [54–56] and the minimization of the following objective function (OBF) based on the least-squares method:

where, n is the number of density data used in the fit procedure. Generally, from 260 literature data points [54–56] applied to 3 base fluids of interest, the minimum value of OBF was found to be 0.0070. But in case the nanoparticles of studied, the three molecular parameters (ε, σ, m) were fitted from true density data of compacted nanopowder [57, 58] over the whole temperature range of interest. Table 1 reports the required molecular parameters (ε, σ, m) as well as the molecular weights of studied systems.

First of all, the PHDC EoS was assessed for the densities of several base fluids comprising ethylene glycol (EG), water and poly ethylene glycol (PEG) as well as their mixtures as they are widely used for cooling purposes. Some binary interaction parameters (k ₁₂) have also to be introduced to the model as the mixtures of base fluids mentioned above are strongly solvating systems and therefore these binary systems are non-ideal mixtures. Table 2 also compares the AADs (in %) of the calculated densities of mixed base fluids studied from the PHDC EoS (this work), and the preceding Pak–Cho + TM EoS model of Yousefi et al. [35], both of which were compared with the experimental data [59–61]. As it is clear from Table 2, for the case of mixed base fluids, the accuracy of PHDC EOS (with AADs within 0.76%–3.04%) was comparable with Pak–Cho + TM EoS model [35] (with AADs within 0.97%–2.98%).

Despite Table 2 showing that the proposed PHDC model improved the results given by the Yousefi et al. model in general, in the case of ZnO/PEG the contrary occurs. For this case, it is somewhat attributed to the type of fit procedure used for each model. The PHDC EoS parameters for the base fluid (i.e., PEG) were fitted in the extended temperature and pressure ranging (as reported in Table 1) in contrast to the model of Yousefi et al. [35] that fitted λ-parameter of TM EoS by the help of several correlation coefficients at limited pressure (0.1 MPa) and temperature ranging in which the nanofluid densities were studied and then it obviously would lead to less AADs in the densities of PEG and ZnO/PEG nanofluid, consequently.

Negative values obtained for k ₁₂ are not against the expectations of intermolecular forces and thermodynamic models [50], “in the case of strongly solvating systems (such as EG + water studied herein), the real crossed energy term is expected to be larger than the value provided by the geometric mean rule and this is why a negative k ₁₂ is needed”. Anyhow, such binary interaction parameters sometimes do not make any sense from the molecular viewpoint as they are simply adjustable parameters.

Then, we extended the mixture version of PHDC EoS to nanofluids with the help of several combining rules as mentioned in Eqs. (1)–(6). Regarding this, the binary interaction parameter appearing in Eq. (5) was fitted against the density data at 0.1 MPa and then PHDC EoS was extended to estimate the density values in wider ranges of pressure. All the calculations on the mass density of 9 nanofluids (including TiO₂-R/EG, TiO₂-A/EG, SnO₂/EG, Co₃O₄/EG, CuO/water, ZnO/PEG, ZnO/PEG + water, ZnO/EG + water and Al₂O₃/EG + water) were carried out in 273–363 K range and pressures up to 45 MPa, then the results were compared with the experimental data [29, 54, 59, 62], [63], [64], [65]. As indicated in Table 2, for 1397 data points examined, the mean AAD of the calculated densities from the literature values was found to be 0.90%. The numerical values of k _ij parameter for each nanofluid have also been included in Table 2. The sign of that parameter indicated if the operating forces between nanoparticle surface-base fluid and base fluid-base fluid molecules were of repulsive or attractive type. As can be seen from Table 2, the type of interaction of nanoparticle surface-base fluid molecule is dominantly repulsive. But in case the base fluid-base fluid molecules, an attractive interaction is the dominant type. The uncertainty (or that of error propagation) of the calculated densities of all studied systems was of the order of ±3.09%.

Table 2 also provided a summary of the AADs obtained for the densities from the PHDC EoS (present work), and those obtained from Pak–Cho + TM EoS model [35], both of which were compared with the literature data [29, 54, 59, 62], [63], [64], [65]. The present work outperforms (with the AAD of 0.90% for 1397 data points examined) Pak–Cho + TM EoS model [35] (i.e., with AAD of 1.99% for 1397 examined data points) which used the extended Tao–Mason EoS, together with the Pak-Cho equation.

Graphically, Figure 1 shows how are the nanofluid densities coming from the PHDC EoS when compared with the experimental values. It depicts the isothermal variation of densities against the pressure for some studied nanofluids including TiO₂-A/EG (a-plot), SnO₂/EG (b-plot) and ZnO/EG + water (c-plot). The markers represent the experimental isotherms of Cabaleiro et al. [62], Mariano et al. [54] and Cabaleiro et al. [59], the solid lines are those obtained using the PHDC EoS. As it is clear from Figure 1, the linear trends associated with isotherms of Cabaleiro et al. [59, 62] and Mariano et al. [54] can be shown well by the present model with the help of crossed interaction parameters. Figure 1 also demonstrates that the density of nanofluid decreases as the temperature increases (along the isobaric curve). As can be seen from 1(a) and 1(b) plots, the calculated nanofluid densities from PHDC EoS showed larger deviations from the measurements when compared with those reported in 1(c) for the case of EG + water-based nanofluid. Then using two adjustable parameters, k ₁₂ and k _bf-np improved the accuracy of PHDC EoS for the densities of nanofluids containing mixed base fluids.

Figure 1:

The linearity of the nanofluid densities versus pressure for TiO₂-A/EG (at x _np = 0.014) at several isotherms ((a)-plot), SnO₂/EG (at x _np = 0.01045) at several isotherms ((b)-plot) and ZnO/EG + water (at x _np = 0.0183) at several isotherms ((c)-plot). (a)-plot represents the results at 283 K (◊), 313 K (∆) and 343 K (⚬). (b)-plot represents the results at 283 K (◊), 303 K (∆) and 323 K (⚬) and (c)-plot represents the results at 278 K (◊), 303 K (∆), 323 K (⚬) and 343 K (□). The markers represent the experimental isotherms of Cabaleiro et al. [62] Mariano et al. [54] and Cabaleiro et al. [59], and solid lines are coming from PHDC EoS.

The isothermal compressibility is calculated using the isothermal pressure derivative of density according to the following classical thermodynamic relation, viz.;

Then, the values of isothermal compressibility coefficient of some selected nanofluids were predicted using the PHDC EoS. We summarized our calculation results in Table 3 for several nanofluids including TiO₂-R/EG, TiO₂-A/EG, SnO₂/EG, Co₃O₄/EG, CuO/water, and ZnO/EG + water as the AAD (in %) of our estimations from those reported by Cabaleiro et al. and Mariano et al. [54, 59, 62, 63], who utilized Tammann–Tait equation to correlate the high-pressure volumetric behavior of nanofluids. We assessed the PHDC EoS by taking 1095 literature data points [54, 59, 62], [63], [64] over the pressure ranging from 0.1 to 45 MPa and temperature range from 283 to 343 K. The overall AAD for the tested data points calculated by PHDC EoS was found to be 5.74%. It should be mentioned that the uncertainty of calculated k _T values by the present work was of the order of ±8.98%. The predicted values of that property have also been tabulated as supplementary material to this article.

As can be seen from Table 3, the AAD obtained for CuO/water nanofluid is greater than 10%. Maybe it should be searched the source of that result in Table 1, where the fit parameters along with the correlated densities and isothermal compressibilities were reported for pure water as highly associating/H-bond type fluid. As Table 3 reveals, the proposed PHDC EoS worked well for the correlation of densities of pure water with the AAD of 0.6%, but in case the isothermal compressibility, the deviation of results from Tait equation were increasing which indicates the loss of accuracy of PHDC EoS for highly associating/H-bond fluids like, indeed the present PHDC EoS was capable of correlating the density and isothermal compressibility of less associating fluids (e.g., EG containing both aliphatic and associating characters) more accurately.

To see how the PHDC EoS passes through the literature data, Figure 2 has been provided. Figure 2 depicts four isothermal compressibility versus pressure plots for TiO₂-R/EG at x _np = 0.039 ((a)-plot), SnO₂/EG at x _np = 0.01472 ((b)-plot), Co₃O₄/EG at x _np = 0.0047 ((c)-plot), and ZnO/EG + water at x _np = 0.009 ((d)-plot). The solid lines are the modeling values and markers are those reported by Cabaleiro et al. and Mariano et al. [54, 59, 62, 63]. Figure 2 shows systematic predictions for isothermal compressibility over the whole pressure range. Figure 2 also demonstrates fairly linear trend for the isothermal compressibility against the pressure (as indicated in a–d plots) for some selected EG- and EG + water-based nanofluids. The trend of isothermal compressibility in terms of the pressure is well established by the present model over different nanoparticle mole fractions up to 0.040.

Figure 2:

Plots of isothermal compressibility versus pressure for some selected nanofluids including TiO₂-R/EG at x _np = 0.039 ((a)-plot), SnO₂/EG at x _np = 0.01472 ((b)-plot), Co₃O₄/EG at x _np = 0.0047 ((c)-plot) and ZnO/EG + water at x _np = 0.009 ((d)-plot). The solid lines are the modeling values and markers are isotherms of Cabaleiro et al., and Mariano et al. [54, 59, 62, 63], (a)-plot represents the results at 283 K (◊), 313 K (∆) and 343 K (⚬). (b)- and (c)-plots represent the results at 283 K (◊), 303 K (∆) and 323 K (⚬) and (d)-plot represents the results at 303 K (◊), 323 K (∆) and 343 K (⚬).

4.2 Common intersection isotherms

The present PHDC EoS has also been employed to predict some known regularities, the pressure dependence of the isotherms of isobaric expansion, α _P and isothermal compressibility coefficients, κ _T both of which were passing through a common intersection point. This intersection can be useful to evaluate the reliability of a given EoS for producing the equilibrium properties of matter [66].

The PHDC EoS under consideration has been tested for predicting those behaviors for a typical base fluid, EG, and nanofluid, SnO₂/EG at x _np = 0.021 in 0.1–350 MPa range. The results were compared with those were calculated from Tait-type equations [38]. The results for isotherms of κ _T and α _P were respectively shown by Figures 3(a) and 4(b) for EG and SnO₂/EG nanofluid at x _np = 0.021, in which the solid lines are the isotherms of our PHDC EoS and markers represent the literature isotherms [38]. As Figures 3(b) and 4(b) depict, the EoS was not able to predict the isotherms of α _P versus pressure as the deviations from the results of Tait-type equation [38] were significantly large (within 46%–68%), even though neither PHDC EoS nor Tait-type equations of literature did not predict the common intersection points in the isotherms of α _P. This fact can be regarded as the drawback of both PHDC EoS and Tait-type equation. But in case the pressure dependence of κ _T isotherms, both PHDC EoS and Tait-type equation were able to predict the common intersection points in the isotherms of κ _T , even though the trend of modeling values (solid lines) were not in accord with the literature data which reveals another drawback of the present PHDC EoS.

Figure 3:

Isotherms of κ _T ((a)-plot), and α _P ((b)-plot) versus pressure for EG. Solid lines are calculated based on the PHDC EoS and markers are literature isotherms [38].

Figure 4:

Isotherms of κ _T ((a)-plot), and α _P ((b)-plot) versus pressure for SnO₂/EG at x _np = 0.021. Solid lines are calculated based on the PHDC EoS and markers are literature isotherms [38].

4.3 Excess molar volumes

Calculating the excess properties, in particular the excess volumes, V ^E as a function of nanoparticle fractions, one can easily determine how the nanoparticle impacts the volumetric behavior of nanofluids. To do so, some typical calculations were performed on the excess volumes of three selected nanofluids and the results were shown in Figure 5. The results were obtained from the following equation:

where, x _np represents the mole fraction of nanoparticle; M _np and M ₀ are the molar masses of nanoparticle and base fluid, respectively; and ρ _nf, ρ _np, and ρ ₀ are the densities of nanofluid, nanoparticle, and base fluid, respectively. Figure 5 clearly demonstrates the sensible deviations from non-ideal liquid-solid mixture behavior for Co₃O₄/EG (5a), CuO/water (5b) and ZnO/PEG (5c). The solid lines are our modeling values and markers are the values obtained from the isotherms of Mariano et al. [63], Pastoriza-Gallego et al. [64] and Zafarani-Moattar et al. [65]. The excess volumetric trend of the non-ideal mixtures can be mentioned in terms of the contractive and expansive behavior of those mixtures, e.g., for the case of nanofluids, that non-ideal behavior is usually assessed against the nanoparticle concentration. As can be seen from Figure 5(a)–(c), both modeling values and literature values for the excess molar volume were approaching greater negative values as the nanoparticle concentration increased, under this circumstance the volumetric behavior of that system is then contractive in contrast to the expansive type, for which the excess molar volumes would approach greater positive values concerning the nanoparticle concentration loading. That contractive trend in the excess volumes is attributed to the strong attractive interactions between the base fluid molecules and nanoparticle surface. According to the results obtained from the PHDC EoS (solid lines), the addition of nanoparticles to the base fluid led to the excess molar volumes of “−0.12” cm³ mol⁻¹ at most which were somewhat greater than those obtained from the experimental density data of Mariano et al. [63], Pastoriza-Gallego et al. [64], and Zafarani-Moattar et al. [65] (with V ^E of “−0.08 cm³ mol⁻¹” at most). Then that is why the trend of excess volumes of PHDC EoS were found to be more contractive (i.e., towards greater negative values) than the literature values. The numerical values of the calculated excess volumes along with the relative deviations from the literature data were also reported in Table 4 for 3 selected nanofluids. As can be seen from Table 4, the present EoS modeling was not capable of estimating the excess volumes accurately, as the relative deviations were within −34.30%–51.31%.

Figure 5:

Excess volumes against nanoparticle mole fractions for three selected nanofluids including Co₃O₄/EG (5a), CuO/water (5b) and ZnO/PEG (5c). The solid lines are our modeling values and markers are values obtained from the isotherms of Mariano et al. [63], Pastoriza-Gallego et al. [64] and Zafarani-Moattar et al. [65]. (5a): data were taken from 0.1 MPa at 283 (□) and 45 MPa at 323 K (▲). (5b): Data were taken from 0.1 MPa at 293 (▲) and 25 MPa at 313 K (□). (5c): Data were taken from 0.1 MPa at 293 (▲) and 0.1 MPa at 308 K (□).

4.4 Artificial neural network modeling results

To be able to predict the nanofluid density over a pressure range of 0.1–45 MPa, it was decided to design a network with only one hidden layer in addition to the input and output layers. The number of input parameters has been fixed in 3, namely temperature, pressure, and nanoparticle mole fraction. At this point, a clarification of the database is necessary. In fact we preferred to use only the points not at atmospheric pressure. This choice was driven by the desire to demonstrate for the first time in the literature as the pressure is one of the essential parameters for characterizing the density of nanofluids.

The training of the network was done several times by varying the number of neurons in a range between 1 and 30 recording at each step all the variations of 3 statistical parameters: AAD%, RMSE, and R ²

At the end was chosen the best neuron in the various training that has been done.

Another particular attention that was kept was to divide the database into 3 parts. In fact, one of the biggest problems of neural networks is the over-fitting. In addition, choosing networks that are not very flexible, it is possible to have excellent results that are however very compressed on the starting data. To overcome this, the database has been broken in:

1. Training dataset: in which the network carries out its own training calculations

2. Validation set: in which the network tests the results

3. Test set: formed by a set of points that the network does not see while it performs the calculations. However, the results of the calculations were also tested on this dataset to verify not only the network’s ability to obtain results on the dataset, but also its ability to predict new density data.

The split across the three databases and the relative results are shown in Table 5.

Uncertainty can also be considered when doing network training. In this case, since the uncertainty of the data is very low, it was considered, and to reduce the impact of it during the training process, the data was normalized. This provided a more uniform distribution of the data.

As can be seen from Figure 6, increasing the number of neurons lead to the significant drop in the deviations. Given this trend we decided to choose the neuron number 17 as the minimum to which the network can reach. The minimum AAD reached by the network in the chosen neuron was 0.07 on the entire database. All other statistical parameters were given in Table 5.

Figure 6:

AAD% values for each the entire dataset, for the training, the validation and test set, as a function of the number of neurons.

The results calculated for each fluid under consideration using the present ANN were given in Table 2. As mentioned before, the results were only for fluids with the pressure greater than atmospheric pressure.

In addition, Figure 7 represents the scatterplot between the experimental and calculated densities. As can be seen, all the points are located along the first and third quadrant bisector signifying the excellent approximation of the neural network calculation.

Figure 7:

Calculated density values versus values in each dataset.

List of symbols

AAD

average absolute deviation (in %)

k _B

Boltzmann’s constant (J K⁻¹)

NP

number of data points

P

pressure (Pa)

T

absolute temperature (K)

M_W

molecular weight (g mol⁻¹)

m

segment number

R

universal gas constant (J mol⁻¹K⁻¹)

k _ij

binary interaction parameter

Greek letters

g(σ ⁺)

Pair radial distribution function at contact

η

Segment packing fraction

ρ

Segment density (mol m⁻³)

σ

Segment diameter (nm)

ε

Non-bonded interaction energy between spheres and dimmers (J)

λ

Range parameter of attractive forces

k _T

isothermal compressibility coefficient

Superscripts

HDC

Hard-dimer chain reference system

Pert.

Perturbed system

HS

Hard-sphere

HD

Hard-dimer

Calc.

Calculated

Exp.

Experimental data

Tait.

refers to values from Tait equation

Subscripts

CS

Carnahan–Starling

mix.

Mixture