New version of Tammann-Tait equation: Application to nanofluids

Journal of Molecular Liquids 220 (2016) 404–408

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New version of Tammann-Tait equation: Application to nanofluids S.M. Hosseini ⁎, M.M. Alavianmehr Department of Chemistry, Shiraz University of Technology, Shiraz 71555-313, Iran

a r t i c l e

i n f o

Article history: Received 20 February 2016 Received in revised form 13 April 2016 Accepted 20 April 2016 Available online xxxx Keywords: New Tammann-Tait Nanofluids Density

a b s t r a c t It has been one hundred years, since Tammann-Tait equation was widely employed to correlate high-pressure density behaviour of liquids against the temperature. Nevertheless, in the case of suspensions containing dispersed nanoparticles, called nanofluids, the applicability of above-mentioned equation is limited to the fitting of numerous coefficients at any nanoparticle concentration of interest. This paper aims to overcome this problem through developing a Tammann-Tait like equation, in which the number of adjustable coefficients is reduced by introducing a nanoparticle concentration-dependent function to the original scheme. High pressure densities of some ethylene glycol, water, and ethylene glycol + water-based nanofluids were predicted using new version of Tammann-Tait equation over the temperature range within 273–363 K and pressures up to 45 MPa. From 1215 data points examined, the AARD of the correlated and predicted densities from the experimental values was found to be 0.061%. The thermodynamic coefficients of studied nanofluids were also estimated by the proposed equation. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The suspension of nanoscale particles in a fluid is a colloidal system usually referred to as nanofluid. Some nanofluids are suspensions of metallic nanoparticles in conventional base fluids (denoted as nanoparticle/base fluid) which possess better thermal performance compared to single-phase fluids (e.g., water, ethylene glycol, and propylene glycol) [1]. In nanofluids, the particle diameters generally lie within 100 nm. They became important significantly in the last decade because of their enhanced heat transfer capabilities. Concerning their most relevant thermophysical properties of nanofluids, although their viscosities and thermal conductivities are crucial for use in the heat and mass transferring phenomena [2–4], other thermophysical properties such as density could become as critical as thermal conductivity in engineering systems requiring fluid flow [5–8]. In order to evaluate the fluid dynamic and heat transfer performances of nanofluids in thermal cycles, the density must be known accurately to compute matter and energy balances [9]. Since the available experimental nanofluid densities are not much, there is a need for accurate correlations to supplement the available experimental data at various nanoparticle concentrations. Engineers often demand generalized simple models without sophisticated and long-time computations. Concerning nanofluids, such models are still scarce for the density correlation. So far, the densities of nanofluids have been correlated based on the various approaches. The most widely used one is the Tammann-Tait equation [10,11] and ⁎ Corresponding author. E-mail address: [email protected] (S.M. Hosseini).

http://dx.doi.org/10.1016/j.molliq.2016.04.088 0167-7322/© 2016 Elsevier B.V. All rights reserved.

those were adopted for nanometer sized particles by Sharma-Kumar [12], Cheremisinoff [13] and Pak-Cho [14]. Sharma and Kumar developed an EOS for nano-materials as follows [12]:

 V V 2 þ a2 1− P ¼ a1 1− V0 V0

ð1Þ

where, P is the pressure, V/V0 is the relative change in volumes and subscript 0 refers to the initial condition. a1 and a2 are the size-dependent parameters, which have been proposed to be determined from the definition of bulk modulus and its first order pressure derivative. Pak and Cho [14] presented the general equation for representing the density of nanofluids as follows: ρnf ¼ ρnp φ þ ρbf ð1−φÞ

ð2Þ

where, ρnf is the density of the nanofluid, ρnp is the nanoparticle density, φ is the nanoparticle volume fraction, and ρbf is the base fluid density. Vajjha et al [15] employed Pak-Cho equation to calculate the nanofluid densities. However, Eq. (2) needs an accurate knowledge of dry powder density, ρnp a parameter that is usually known with poor accuracy and this issue limits the applicability of the Pak-Cho equation. The artificial neural network method has also been employed to correlate the density data of nanofluids. For instance, Karimi and Yousefi [16] predicted the density of four nanofluids in the temperature range of 273–323 K and the nanoparticle volume fraction up to 10% with average absolute deviation (AAD) equal to 0.13%.

S.M. Hosseini, M.M. Alavianmehr / Journal of Molecular Liquids 220 (2016) 404–408

In this paper, we aim to present a new equation based on TammannTait scheme to correlate (at 0.1 MPa) and predict high-pressure densities of ethylene glycol-, water-, and ethylene glycol + water-based nanofluids. Although the original Tammann-Tait equation leads to the quite accurate results, the weak point of this equation is that there are numerous coefficients in it that need to be adjusted to each nanoparticle concentration. On the other hand, Tammann-Tait equation is correlative for nanofluid densities rather than predictive. The novelty of the present work is the development of new predictive equation for nanofluid densities and derived properties by introducing a nanoparticle concentrationdependent function to the original Tammann-Tait equation. That nanoparticle concentration-dependent function is determined by correlating the nanofluid densities at 0.1 MPa and employed to predict highpressure densities. Another significance of the proposed equation is that it has two generalized adjusting temperature-dependent parameters which cause the high-pressure densities of a given base fluid and its corresponding nanofluid be predicted accurately. Our correlation and prediction results are also compared with those obtained from the original Tammann-Tait equation. 2. Original version of Tammann-Tait equation The “Tammann-Tait equation,” [10,17] which is now widely used to fit liquid density data over wide pressure ranges, is a modification of the Tait equation [11]: ρðT; P Þ ¼

ρ0 ðT Þ ¼

ρ0 ðT; P 0 Þ ð1−C Lnð1 þ ðP=BðT ÞÞÞÞ

X

Ai T i

ð3Þ

ð4Þ

i

BðT Þ ¼

X

405

3. Results and discussion 3.1. New version of Tammann-Tait equation The new version of Tammann-Tait equation is proposed as follows:     ρ0 ðT; P 0 ÞF xnp
 ρ T; P; xnp ¼
BðT Þ þ P 1−C Ln BðT Þ þ 0:1

ð7Þ

where, F(xnp) is the nanoparticle mole fraction-dependent function which is expressed as:   F xnp ¼ 1 þ λxnp

ð8Þ

where, λ is a specific constant that needs to be fixed for each nanoparticle concentration suspended in the base fluid. It should be mentioned that the adjustable parameters of the original version of the TammannTait cannot be directly applied to the nanofluids. We propose a linear relationship for ρ0(Τ) and B(T) in terms of temperature: ρ0 ðT Þ ¼ A0 þ A1 T

ð9Þ

BðT Þ ¼ B0 þ B1 T

ð10Þ

The fitting parameters of the new Tammann-Tait equation, ρ0(T), B(T) and C are determined by minimizing the following objective function and least-squares method: AARD ¼ min

n jρðP; T ÞExp: −ρðP; T ÞCorr: j 1X i i n i¼1 ρðP; T ÞExp:

ð11Þ

i

Bi T i

ð5Þ

i

Where, T is the absolute temperature and P is the pressure. The subscript 0 refers to the low-pressure, usually 0.1 MPa or saturation pressure, B(T) and C are parameters derived from the fit. Here, ρ0(T) represents the atmospheric density of liquid being a function of temperature. B(T) is usually a generalized polynomial function of the temperature. C is a constant that needs to be adjusted for any liquid type. Recently, another expression of Tammann-Tait has also been employed by numerous researchers, in which the second part of the denominator is slightly different from the original one (Eq. (3)) although fit parameters are similar to Eqs. (4) and (5) [10], viz: ρ0 ðT; P 0 Þ
 ρðT; P Þ ¼
BðT Þ þ P 1−C Ln BðT Þ þ 0:1

ð6Þ

Nevertheless, in the case of suspensions containing dispersed nanoparticles, the applicability of above equations is limited to the fitting of numerous coefficients at any nanoparticle concentration. For instance, assume that we want to fit density results for each nanofluid concentration and for the base fluid. We need to adjust three parameters, ρ0(T), B(T) and C for the base fluid as well as nanofluid containing various nanoparticle concentrations. This issue needs long-time fitting for the determination of Tammann-Tait correlation and subsequently derivation of thermodynamic coefficients become very lengthy. In the PρTxnp version of Tammann-Tait equation, not only the fast evaluation of high-pressure densities and derived properties is feasible for nanofluids but also the number of adjustable coefficients is reduced significantly due to introducing a nanoparticle concentrationdependent function to Eq. (6).

where n represents the number of density data points and AARD is the average absolute relative deviation for a given base fluid. The remaining parameter (λ) is related to effect of nanoparticle mole fraction on the base fluid densities as demonstrated by Eq. (7). The specific constant, λ is adjusted by minimizing the following objective function (OF) and the least-squares method:  Exp:  Corr: 1 n jρ P 0 ; xnp i −ρ P 0 ; xnp i j OF ¼ min ∑  Exp: n i¼1 ρ P 0 ; xnp

ð12Þ

i

Where n is the number of correlated isothermal density data at 0.1 MPa for a given nanofluid. The fitting parameters of new Tammann-Tait equation (Eq. (7)) and the AARD (in %) of correlated (at 0.1 MPa) and predicted (at elevated pressures) density of studied nanofluids together with fitting standard errors are reported in Table 1. The numerical values of constant λ for each nanofluid are also included in Table 1. Generally, the correlated and predicted results for 1251 experimental data points [18–23] led to AARD equal to 0.061% over the temperature range 273–363 K and pressures up to 45 MPa. It should be mentioned that, the uncertainties of our correlation results were of the order of ± 0.60% which is almost equal to those obtained from Eq. (6) (i.e., ±0.64%) [18]. To show graphically, how the new version of Tammann-Tait equation passes through the experimental points, two plots indicating high-pressure density behaviour of ZnO/EG + water nanofluid at several isotherms have been shown in Fig. 1. Further, the correlated densities using the original Tammann-Tait equation [10,17] are also depicted in Fig. 1. The markers are the experimental densities [18], the dashed lines are correlated data from Tammann-Tait equation (Eq. (6)) [18] and solid lines are those correlated (at 0.1 MPa) and predicted (at elevated pressures) from the proposed equation (Eq. (7)). As it is obvious from Fig. 1, the accuracy of new version of Tammann-Tait equation is

406

S.M. Hosseini, M.M. Alavianmehr / Journal of Molecular Liquids 220 (2016) 404–408

Table 1 AARD (in %)a of the correlated (at 0.1 MPa) and predicted high-pressure densities of studied nanofluids together with fitting coefficients to be used in Eqs. (7)–(10). The values of standard errors of fit parameters are also included. NPb

Nanofluid EG-based nanofluids Δxnp:0.0041–0.048 TiO2-A/EG TiO2-R/EG SnO2/EG Co3O4/EG

30 30 270 270

EG + water-based nanofluid Δxnp:0.009–0.0378 ZnO/EG + water 165 Water-based nanofluid Δxnp: 0.002–0.012 CuO/water 450 Overall 1215 a

ΔT/K

ΔP/MPa

λ

C·102

A0 g·cm−3

A1·103 g·cm−3·K−1

SE·102

283–343 283–343 283–323 283–323

0.1–45 0.1–45 0.1–45 0.1–45

0.870 0.951 2.010 0.835

6.108 6.108 6.108 6.108

1.3188 1.3188 1.3188 1.3188

−0.7019 −0.7019 −0.7019 −0.7019

0.017 0.017 0.017 0.017

278–363

0.1–45

2.599

7.110

1.2619

−0.65047

0.062

283–323

0.1–45

3.825

3.79

1.084

2.94

0.073

B1 MPa·K−1

SE·102

AARDa

298.98 298.98 298.98 298.98

−0.4875 −0.4875 −0.4875 −0.4875

0.0461 0.0461 0.0461 0.0461

0.035 0.048 0.020 0.022

313.93

−0.44126

0.082

0.150

0.0559

0.079 0.061

B0 MPa

3064.9

−19.96

NP

. AARD ¼ 100=NP∑ jρCalc: −ρExp: j=ρExp: i i i i¼1

b

NP represents the number of data points examined.

as almost accurate as its original version, which has been previously employed by Cabaleiro et al. [18] in literature. Generally, the density of nanofluid decreases with increasing temperature (along the isobaric

curve), which agrees well with the behaviour of the ordinary fluids. Also, densities of nanofluids increase with nanoparticle concentration. 3.2. Thermodynamic coefficients The isothermal compressibility (κT) and thermal expansion coefficient (αP) are respectively related to the fluctuations in volume/density versus pressures and temperature. Having an PρT correlation like Tammann-Tait, one can easily estimate the isothermal compressibility and thermal expansion coefficients, both of which closely related to the isobaric temperature derivative as well as the isothermal density derivative of PρT equation of interest. The relevant expressions to the isothermal compressibility (κT) and thermal expansion coefficient (αP) derived by new Tammann-Tait equation can be read as: 1 κT ¼ ρ

0 1  

  ∂ρ 1 B ρ0 ðT; P 0 ÞF xnp C C ¼ @h A  i2 P þ BðT Þ ∂P T ρ BðT ÞþP 1−CLn BðT Þþ0:1


 −1 ∂ρ ρ ∂T p     1 0 A1 F xnp ρ ðT; P 0 ÞF xnp

 þ h 0   i 2 C B BðT Þ þ P ÞþP C 1−CLn BBðTðTÞþ0:1 1B C B 1−CLn BðT Þ þ 0:1 ¼− B C ! C ρB @ 1=ðBðT Þ þ P Þ−ðBðT Þ þ P Þ=ðBðT Þ þ 0:1Þ2 A CB1 ðBðT Þ þ P ÞðBðT Þ þ 0:1Þ

ð13Þ

αp ¼

Fig. 1. High-pressure density behaviour of ZnO/EG + water nanofluid at xnp = 0.0183 (aplot) and xnp = 0.0378 (b-plot), both were shown at several isotherms; a-plot at 278 K (◊), 303 K (Δ), 323 K (□), 343 K (○), and 363 K (▲). b-plot at 278 K (◊), 303 K (▲), 323 K (□), 343 K (Δ), and 363 K (●). The markers represent the experimental data [18], the dashed lines are the correlated results from Tammann-Tait equation (Eq. (6)) [18] and sold lines are those correlated (at 0.1 MPa) and predicted (at elevated pressures) from the proposed equation (Eq. (7)).

ð14Þ

The isothermal compressibility (κT) and thermal expansion coefficient (αP) of studied nanofluids were calculated using new TammannTait equation. Typically, Fig. 2 shows isothermal and isobaric behaviour of thermodynamic coefficients of ZnO/EG + water nanofluid at xnp = 0.0378 derived by the new Tammann-Tait equation, a-plot shows the trend of αP against the temperature at several isobars and b-plot depicts the variation of κT versus pressure at several temperatures. As it is clear from Fig. 2, the isobaric behaviour of αP is almost similar to those reported for ordinary liquids, for which αP increases as the temperature increases. The isothermal behaviour of κT has also been investigated and seemed to be in agreement with ordinary liquids, for which κT decreases as the pressure increases. In other words, no irregular trend is observed for κT values using new Tammann-Tait equation. This observation is contrary to the results previously reported by Cabaleiro et al. [18], for example, for xnp = 0.0378, the estimated κT values (from Eq. (6)) increases as the pressure increases. We predicted thermodynamic coefficients of nanofluids at various temperatures and pressures. The whole

S.M. Hosseini, M.M. Alavianmehr / Journal of Molecular Liquids 220 (2016) 404–408

407

Fig. 2. Isothermal and isobaric behaviour of thermodynamic coefficients of ZnO/ EG + water nanofluid at xnp = 0.0378 derived by the new Tammann-Tait equation, aplot shows the trend of αP against the temperature at several isobars and b-plot depicts the variation of κT versus pressure at several isotherms.

predicted data for thermodynamic coefficients of interest were tabulated as supplementary material to this article. It should be added that, from 1215 data points examined, the AARD of estimated isothermal compressibilities and thermal expansion coefficients using proposed equation (Eq. (7)) from those obtained by the use of Tammann-Tait equation [18–23] were found to be 6.77% and 12.42%, respectively. To assess further the degree of reliability of proposed equation for the estimation of derived properties, some typical comparisons were made between the predicted values from Tammann-Tait (Eq. (6)) and those obtained from the proposed equation in this work. The results were also compared with those obtained from the experimental data, for which their values were determined from the slope of experimental densities [18,21] versus temperature and pressure. The comparison results were shown by the Fig. 3 for some studied nanofluids. The markers represent experimental thermodynamic coefficients, dashed lines are prediction results of Tammann-Tait equation and solid lines are those obtained from the proposed equation in this work (Eq. (7)). Fig. 3 reveals that a little difference between the slopes of experimental density data versus pressure/temperature and those obtained from both Tammann-Tait and new equation result in huge differences in the estimated derivative properties. 4. Conclusion This work showed that using Tammann-Tait scheme and a nanoparticle concentration-dependent function one can develop a new equation for correlating (at 0.1 MPa) and predicting (high-pressure) nanofluid densities. In the present work, a mole fraction-dependent version of the Tammann-Tait equation was proposed for nanofluids. In this equation the adjustable parameters were generalized for a given base fluid by introducing a nanoparticle mole fraction-dependent function. Also,

Fig. 3. Predicted κT and αP values for some studied nanofluids using new version of Tammann-Tait (solid lines) compared with those obtained from original Tammann-Tait (dashed lines), both were compared with the experimental values (markers) [18,21]. aplot depicts the predicted κT values for ZnO/EG + water at xnp = 0.0090. b-plot depicts the predicted αP values for ZnO/EG + water at xnp = 0.0090. c-plot depicts the predicted κT values for Co3O4/EG at xnp = 0.033.

the number of adjustable coefficients was reduced to almost one fifth. It should be mentioned that the accuracies of the correlated (at 0.1 MPa) and predicted (at elevated pressures) densities was quite good, although the proposed scheme showed some deviations from Tammann-Tait correlation when predicted the derivative properties. Nomenclature and units P T ρ x φ

pressure/MPa absolute temperature/K density/kg·m−3 mole fraction volume fraction

408

κT αP λ C B(T) ρ0(T) B0 B1 A0 A1

S.M. Hosseini, M.M. Alavianmehr / Journal of Molecular Liquids 220 (2016) 404–408

isothermal compressibility/MPa−1 thermal expansion coefficient/K−1 adjustable constant used in Eq. (8) adjustable constant used in Eqs. (3), (6) and (7) temperature-dependent parameter used in Eqs. (3), (6) and (7)/MPa temperature-dependent density at 0.1 MPa used in Eqs. (3), (6) and (7)/kg·m−3 adjustable coefficient used in Eq. (10)/MPa adjustable coefficient used in Eq. (10)/MPa·K−1 adjustable coefficient used in Eq. (9)/g·cm−3 adjustable coefficient used in Eq. (9)/g·cm−3·K−1

Superscripts Exp Corr Calc

experimental value correlated value calculated value

Subscript np 0

nanoparticle reference state or at 0.1 MPa

Acknowledgment The research committee of Shiraz University of Technology is acknowledged. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.molliq.2016.04.088. References [1] S.U. Choi, Z.G. Zhang, P. Keblinski, Nanofluids, Encyclopedia of Nanoscience and Nanotechnology, American Scientific Publishers, 2004 757–773. [2] S.K. Das, S.U. Choi, H.E. Patel, Heat transfer in nanofluids—a review, Heat Transfer Eng. 27 (2006) 3–19. [3] E.E. Michaelides, Transport properties of nanofluids. A critical review, J. Non-Equilib. Thermodyn. 38 (2013) 1–79.

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