Figure captions:
Fig.S1 S1a: nanofluid image for Fe3O4 in PEG at x1=0.3995×10-2. S1b: nanofluid image for Fe3O4 nanoparticles coated with oleic acid - PEG with PEG: oleic acid ratio of 1:1at different volume fractions of Fe3O4.
Fig. S2S2a: Shear stress versus shear rate for PEG-oleic acid solution with PEG mole fraction of 0.44446. S2b: Shear viscosity versus shear rate for PEG-oleic acid solution with PEG mole fraction of 0.44446.
Fig. S3 S3a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=0.301%, different magnetic field and T=298.15 K. S3b: Shear viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=0.301%, different magnetic field and T=298.15 K.
Fig. S4 S4a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=1.502%, different magnetic field and T=298.15 K. S4b: Shear viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=1.502%, different magnetic field and T=298.15 K.
Fig. S5 S5a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=4.440%, different magnetic field and T=298.15 K. S5b: Shear viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=4.440%, different magnetic field and T=298.15 K.
Fig. S6 S6a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=310.15 K. S6b: Shear viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=310.15 K.
Fig. S7 S7a: Shear stress versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=323.15 K. S7b: Shear viscosity versus shear rate for nanofluid of Fe3O4 nanoparticles coated with oleic acid dispersed in PEG (PEG: oleic acid ratio of 1:1) at φ1=6.483%, different magnetic field and T=323.15 K.
Fig. S8Experimental and calculated isentropic compressibility, κs, plotted against mole fraction of Fe3O4 nanoparticle, x1, or PEG, x2, for nanofluids of Fe3O4 – PEG and Fe3O4 coated with oleic acid– PEG (PEG: oleic acid ratio of 1:1) and PEG-oleic acid solution at different temperatures.
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Fig. S1
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Fig. S2
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Fig. S3
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Fig. S4
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Fig. S5
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Fig. S6
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Fig. S7
Fig. S8
Eyring-NRTL-Carreau-Yasuda model used in this work is as follow
(S1a)
, . (S1b)
In above relations V1and V2 are the molar volume of Fe3O4nanoparticles and PEG, respectively; η1and η2 are the viscosity of Fe3O4nanoparticles and PEG, respectively; is high shear rate viscosity,φI is the volume fraction of component I, equal to in which xI is the mole fraction of component I. T is temperature and R is the universal constant of gases. a12, b12, a21, b21, a, n and λ are empirical parameters of Eyring-NRTL-Carreau-Yasuda model. Subscripts 1 and 2 stand for Fe3O4nanoparticles and PEG molecules, respectively. α is the non randomness factor which was set to 0.2 in this work. Viscosity of Fe3O4nanoparticles is treated as an adjustable parameter and set to 1000 for obtaining good result.
Eyring-mNRF-Carreau-Yasuda model is as following equation
(S2a)
,(S2b)
, ,(S2c)
where approximates the ratio of the molar volume of the PEG and Fe3O4nanoparticles; and ; ,, , a, n and λ are the parameters of Eyring-mNRF-Carreau-Yasuda model. Z is the nonrandom factor which was set to 8 in this work.
The Krieger-Dougherty-Carreau-Yasuda model is as follow
(S3)
where is the maximum particle volume fraction and is the intrinsic viscosity. These quantities along with a, n and λ were set as empirical parameters in this work.
Table S1
Parameters of Eyring-NRTL-Carreau-Yasuda, Eyring-mNRF-Carreau-Yasuda and Krieger-Dougherty-Carreau-Yasuda models along with absolute average relative deviation, AARD,a and standard deviation, σ, b obtained from fitting the viscosity values of Fe3O4-PEG nanofluids
Eyring-NRTL-Carreau-Yasuda modela12 / b12 / a21 / b21 / λ / a / n / 100.AARD
(σ (Pa.s))
0.06305 / 0.3766 / 0.06305 / 0.6404 / 0.5569 / 0.1193 / -0.7828 / 9.32
(0.012)
Eyring-mNRF-Carreau-Yasuda model
/ / / / λ / a / n / 100.AARD
(σ (Pa.s))
26.54 / 0.2555 / 6.284 / 0.3691 / 1.505 / 0.1467 / -0.964 / 9.64
(0.013)
Krieger-Dougherty- Carreau-Yasuda model
/ / λ / a / n / 100.AARD
(σ (Pa.s))
0.03733 / 1.945 / 880.8 / -0.808 / 8229 / 3.89
(0.009)
a in which N is the total number of data points and Y is the number of parameters. b.
1