32
D. P. Barai et al.
model along with the Bruggeman model (1935a, b) as given in Eq. (28) for Al 2 O 3
nanofluids prepared in water, but both the models could not predict the experimental
data.
1 − φ =
k p − k eff
k p − k bf
k p
k eff
1/3
(28)
So, they presented a new model, by performing a regression analysis, having
a correlation coefficient of 0.9975 which is given in Eq. (29) relating the thermal
conductivity of the Al 2 O 3 /water nanofluids (λ) with temperature (T) and volume
fraction (φ).
λ = 176.69 + 588.41φ − 13.64T − 86.31φ
2
+ 0.36T
2
+ 1.07T φ
+ 11.06φ
3
− 0.003T
3
+ 0.18T
2
φ − 1.01T φ
2
(29)
Again, as found out by Shen et al. (2012) for ZnO nanofluids prepared using insulated oil as a basefluid, the Maxwell model underpredicts the electrical conductivity
of the nanofluid. They concluded that the electrical conductivity of the nanofluid
depend on two additional factors along with Maxwell electrical conductivity (λ M ).
Those two factors are the electrical conductivity due to electrophoresis (λ E ) and
due to the Brownian motion (λ B ). The equation thus derived by them to predict
the electrical conductivity of the ZnO-insulated oil nanofluid is as given in Eq. (30)
where λ bf is the electrical conductivity of the basefluid, φ is the volume fraction
of the nanoparticles in the nanofluid, ε r is the relatively dielectric constant of the
nanofluid, ε 0 is the dielectric constant of the vacuum, U 0 is the zeta potential of the
nanoparticles relative to the basefluid, r is the radius of the spherical nanoparticle, R
is the thermodynamic constant, t is the temperature, L is the Avagadro’s constant, λ
is the viscosity index of the fluid, T 0 is the temperature of the nanofluid at which the
viscosity is measured, ρ is the nanofluid density and ν is the nanofluid kinematic viscosity. This equation is valid for particles with higher electrical conductivity than the
basefluid as the first term, that is the term for Maxwell conductivity, is approximated
for such solid–liquid systems.
λ = λ M + λ B + λ E = λ b f (1 + 3φ)
+
3φε r ε 0 U 0
r 3/2
RT
L
ε
e λ(T −T 0 )
3Πρν(1 + 25φ + 625φ 2 )
1/2
+
2φε 2
r ε 2
0 U 2
0
ρν(1 + 25φ + 625φ 2 r 2 )
e
λ(T −T 0 )
(30)
Similar study was done by Dong et al. (2013) for transformer oil-based aluminium
nitride (AlN) nanofluids in which they reported that the experimental electrical conductivity is in good agreement with the one predicted using the model given in
Eq. (31), which is similar to the previous one by Shen et al. (2012) ignoring the
effect of Brownian motion.
D. P. Barai et al.
model along with the Bruggeman model (1935a, b) as given in Eq. (28) for Al 2 O 3
nanofluids prepared in water, but both the models could not predict the experimental
data.
1 − φ =
k p − k eff
k p − k bf
k p
k eff
1/3
(28)
So, they presented a new model, by performing a regression analysis, having
a correlation coefficient of 0.9975 which is given in Eq. (29) relating the thermal
conductivity of the Al 2 O 3 /water nanofluids (λ) with temperature (T) and volume
fraction (φ).
λ = 176.69 + 588.41φ − 13.64T − 86.31φ
2
+ 0.36T
2
+ 1.07T φ
+ 11.06φ
3
− 0.003T
3
+ 0.18T
2
φ − 1.01T φ
2
(29)
Again, as found out by Shen et al. (2012) for ZnO nanofluids prepared using insulated oil as a basefluid, the Maxwell model underpredicts the electrical conductivity
of the nanofluid. They concluded that the electrical conductivity of the nanofluid
depend on two additional factors along with Maxwell electrical conductivity (λ M ).
Those two factors are the electrical conductivity due to electrophoresis (λ E ) and
due to the Brownian motion (λ B ). The equation thus derived by them to predict
the electrical conductivity of the ZnO-insulated oil nanofluid is as given in Eq. (30)
where λ bf is the electrical conductivity of the basefluid, φ is the volume fraction
of the nanoparticles in the nanofluid, ε r is the relatively dielectric constant of the
nanofluid, ε 0 is the dielectric constant of the vacuum, U 0 is the zeta potential of the
nanoparticles relative to the basefluid, r is the radius of the spherical nanoparticle, R
is the thermodynamic constant, t is the temperature, L is the Avagadro’s constant, λ
is the viscosity index of the fluid, T 0 is the temperature of the nanofluid at which the
viscosity is measured, ρ is the nanofluid density and ν is the nanofluid kinematic viscosity. This equation is valid for particles with higher electrical conductivity than the
basefluid as the first term, that is the term for Maxwell conductivity, is approximated
for such solid–liquid systems.
λ = λ M + λ B + λ E = λ b f (1 + 3φ)
+
3φε r ε 0 U 0
r 3/2
RT
L
ε
e λ(T −T 0 )
3Πρν(1 + 25φ + 625φ 2 )
1/2
+
2φε 2
r ε 2
0 U 2
0
ρν(1 + 25φ + 625φ 2 r 2 )
e
λ(T −T 0 )
(30)
Similar study was done by Dong et al. (2013) for transformer oil-based aluminium
nitride (AlN) nanofluids in which they reported that the experimental electrical conductivity is in good agreement with the one predicted using the model given in
Eq. (31), which is similar to the previous one by Shen et al. (2012) ignoring the
effect of Brownian motion.