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Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder
Advanced Powder Technology xxx (2016) xxx–xxx
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Original Research Paper
Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder A. Malvandi a,⇑, Amirmahdi Ghasemi b, D.D. Ganji c, I. Pop d,⇑ a
Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA c Mechanical Engineering Department, Babol University of Technology, Babol, Iran d Department of Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 22 June 2016 Accepted 24 June 2016 Available online xxxx Keywords: Film condensation Nanofluids Nanoparticle migration Thermophoresis Brownian motion
a b s t r a c t The change in concentration and direction of nanoparticle migration can control the thermophysical properties of nanofluids. This dynamic is useful since it is able to improve the cooling performance by tuning the flow and heat transfer rate. In the current study, a theoretical investigation on the impact of nanoparticle migration on heat transfer enhancement of nanofluids condensate film over a vertical cylinder has been conducted. The Brownian motion and thermophoretic diffusivity have been considered by using the modified Buongiorno model which can take into account the effect of nanoparticle slip velocity. The results have been obtained for different parameters, including the Brownian motion to thermophoretic diffusivities NBT, the saturation nanoparticle volume fraction /sat, and the normal temperature difference c = ðT sat T w Þ=T w . It is shown that nanoparticle migration has significant impact on the flow and thermal fields and considerably affects the heat transfer rate. Furthermore, heat transfer enhancement in film condensation is strongly depended on the thermophysical properties of nanoparticles such that alumina-water nanofluid exhibits higher cooling performance than titania-water. Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction The condensation process is the change of the physical state of matter from gas to liquid phase. It usually releases a significant amount of heat due to large internal energy differences between the liquid and vapor states. Thus, it is ideal for several heat exchange equipments, especially in heat transfer enhancement for high-power cooling, food processing, biotechnology, and miniature electronic devices like microelectromechanical systems (MEMS) [1,2]. In addition, thermal performance of solar systems will be enhanced with the inclusion of nanoparticles to the working fluid. Consequently, it can be stated that nanofluids are able to participate in considerable reduction of carbon emissions. Generally, the heat will transfer inside the condensate film merely by conduction depending on the thickness of the film and the condensation rate. Either the thickness of the condensate film or the condensation rate of vapor depends on thermophysical properties which can be tuned by nanoparticle inclusion. Inclusion of
nanoparticle improves the heat exchangers performance since it improves the thermal conductivity of regular cooling fluids such as water, oil, ethylene-glycol, and power-law fluid [3]. Notably, nanoparticles have higher thermal conductivity relative to the working fluids and due to their similar size to the molecules of the base fluid, they would not induce any significant problems (abrasion, clogging, fouling and additional pressure loss in heat exchangers) relative to larger particles. However, the stability of nanofluids is an extreme challenge in front of scientists and researchers. The stability of nanofluids is commonly depended to the method of nanoparticle preparation. Nevertheless, inclusion of surfactants is a useful technique to avoid aggregation of nanoparticles and to improve the stability. Their impact on the environment is another concern of using nanofluids that we have to face with. Public concerns about their safety forced the engineers to be prudent to produce green nanofluids by biodegradable and nontoxic nanoparticles. 1.1. Film condensation
⇑ Corresponding authors. E-mail addresses: [email protected] (A. Malvandi), [email protected], [email protected] (I. Pop).
Nusselt [1] originally presents a theoretical model for the film condensation of pure vapors over tubes and plates. Then, several
http://dx.doi.org/10.1016/j.apt.2016.06.025 0921-8831/Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Nomenclature cp dp CB CT DB DT g hp HTE Jp k kB NBT q q00w R T u x; r
specific heat (m2/s2 K) nanoparticle diameter (m) reduced Brownian diffusion coefficient (1/K), C B ¼ DB =T reduced thermophoretic diffusion coefficient, C T ¼ DT =/ Brownian diffusion coefficient thermophoretic diffusion coefficient gravity (m/s2) specific enthalpy of nanoparticles (J/Kg) heat transfer enhancement nanoparticle mass flux (kg/m2 s) thermal conductivity (W/m K) Boltzmann constant (=1:3806488 1023 m2 kg=s2 K) ratio of the Brownian to thermophoretic diffusivities energy flux relative to the nanofluid (W) surface heat flux (W/m2) cylinder radius (m) temperature (K) axial velocity (m/s) coordinate system
Greek symbols b proportionality factor d condensate film thickness e ratio of condensate film thickness to cylinder radius e = d/R
publications are devoted to develop our understanding on film condensation which are fully reviewed and presented in literature such as [4–9]. Furthermore, Shang [10] conducted a comprehensive study of theory of film condensation and introduced novel similarity solutions for a wide range of basic problems. Nevertheless, few studies have been conducted for the condensation of nanofluid vapor. Recently, Avramenko et al. [11] introduced a model for the heat transfer of nanofluids condensate film near a vertical plate. Their model developed the classical model of Nusselt by adding an equation for the nanoparticle concentration and a dependence of the nanofluid density on the nanoparticle concentration. In another study [12], they studied the heat transfer rate at condensate film of moving vapor with nanoparticles over a flat surface. They concluded that increasing the nanoparticle volume fraction enhances the processes of momentum and heat transfer. They also employed their model to study the stable film boiling of nanofluids over vertical surfaces [13]. Malvandi et al. [14] considered the laminar film condensation of nanofluids over a vertical flat plate considering nanoparticles migration. Turkyilmazoglu [15] investigated the effects of considering the slip velocity of nanoparticles for the condensate film of nanofluids and indicated that slip mechanisms can be responsible for additional heat transfer enhancement. It should be noted that study of the film condensation of nanofluids is important due to the wide range of applications in nanoparticle synthesis, thermosyphons cooling, and heat transfer in nano and micro heat exchangers [16]. 1.2. Migration of nanoparticles There are several papers in literature deal with the steady or unsteady motion of particles in regular fluids [17,18]. However, for nanofluids, nanoparticles migrate because of nano-scale slip mechanisms that intensify the thermal conductivity and heat transfer rate of nanofluids. According to Buongiorno [19], Brownian
/
c g l q s
nanoparticle volume fraction normal temperature difference, c = ðT sat T w Þ=T w transverse direction dynamic viscosity (kg/m s) density (kg/m3) shear stress (Pa)
Subscripts B Brownian index BT ratio of Brownian to thermophoretic index bf base fluid k thermal conductivity index p nanoparticle sat saturation condition t time index T thermophoresis index w condition at the wall l dynamic viscosity index q density index Superscripts t transpose of a matrix ⁄ dimensionless variable
diffusion and thermophoresis are merely two important slip mechanisms in nanofluids. The effects of Brownian motion and thermophoresis on the nanoparticle migration of nanofluids have been investigated by several researchers. For example, Yang et al. [20,21] consider the effects of nanoparticle migration on forced convective heat transfer of alumina/water and titania/water nanofluids in circular, parallel plate, and tube-in-tube channels. Malvandi et al. [22] conducted a numerical analysis on mixed convection of nanofluids in a vertical concentric annulus. In another study, Malvandi and Ganji [23] investigated the effects of the nanoparticle migration on alumina/water nanofluids in a parallelplate channel. They demonstrated that nanoparticles move from the adiabatic wall (nanoparticles depletion) toward the cold wall (nanoparticles accumulation) and construct a non-uniform nanoparticle distribution. In addition, the anomalous heat transfer rate occurs when the Brownian motion takes control of the nanoparticle migration. Hedayati and Domairry [24,25] studied the effects of nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. They indicated that nanoparticles migration has significant effects on heat transfer characteristics of nanofluids. More details can be found in different scientific researches, for example [26–33]. 1.3. Motivation Recently, the thermophysical behaviors of film condensation of ferrofluids (magnetic nanofluids) as a class of metallic nanofluids in the presence of variable-directional magnetic field have been considered theoretically [34,35]. Their results indicated that anisotropic behaviors of thermophysical properties of ferrofluids in the presence of magnetic field are able to control the heat transfer behavior. However, film condensation of non-magnetic (regular) nanofluids (simply nanofluids) over a vertical cylinder has not been a subject of a study yet. Consequently, there is a need to consider the different behavior of heat transfer rate at film condensation
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
of isotropic nanofluids. Also, the effects of nanoparticle migration on heat transfer enhancement at film condensation of nanofluids are completely a novel subject. Thus, in this paper, the falling film condensation of nanofluids over a vertical cylinder has been investigated theoretically considering the effects of nanoparticles migration. Modified Buongiorno’s model is employed for the nanofluids taking into account the dependency of Brownian motion and thermophoresis to the temperature and the nanoparticle concentration, respectively. The main aim of the present study is to develop a formula for the nanoparticle migration inside the film condensation over a vertical cylinder and consider how the nanoparticle migration affects the heat transfer rate. The results of this study can be applied to enhance the thermal performance on various fields such as air cooled power station condensers [36], biotechnology [37], food processing [38] and industrial machines [39]. 2. Model formulation Fig. 1 depicts the physical geometry of a continuous condensate film of nanofluids falls along the outer surface of a vertical cylinder in contact with a vapor, when the surface temperature (Tw) is cooled below the local vapor saturation temperature (Tsat). A two-dimensional coordinate system is considered where the xaxis is aligned vertically and the r-axis is normal to the walls. Because of the gravity, the formed condensate film at the top of the cylinder wall flows downward. Flow is assumed to be laminar and the density of vapor is negligible. The vapor temperature is constant in a way that the condensation is the merely heat transfer mechanism between the liquid and vapor interface. Further, the shear stress at the liquid-vapor interface is assumed to be negligible. The film thickness d is very small relative to the cylinder length (boundary layer assumptions) and heat and mass transfer across the condensate film is much larger than those in the streamwise direction.
the base fluid, as Buongiorno [19] asserted. Thermophoresis moves the nanoparticles from warmer to colder region, inducing a nonuniform nanoparticle volume fraction distribution. Brownian diffusion is directly proportional to gradient of the nanoparticle volume fraction and originate owing to random drifting of nanoparticle within the suspension. From [19], nanoparticle distribution (/) can be written as
@ t ð/Þ þ r ðu/Þ ¼
Because of the thermophoresis and Brownian diffusion in nanoscale flows, nanoparticles will have a slip velocity relative to
rðJp Þ
ð1Þ
where Jp ¼ qp ðDB r/ þ DT rTT Þ and the Brownian diffusion coefficient, DB, and thermophoresis diffusion coefficient, DT, can be written as
DB ¼
kB T ; 3plbf dp
DT ¼ b
lbf / qbf
ð2Þ
where kB is the Boltzmann’s constant, lbf is the dynamic viscosity of the base fluid, T is the local temperature, dp is the nanoparticle diameter, and b ¼ 0:26 2k
kbf
bf þkp
.
2.2. Transport equations The governing equations are the conservation equations of the mass, momentum, and thermal energy in general coordinate system which can be written as:
@ t ðqÞ þ r ðquÞ ¼ 0
ð3Þ
@ t ðquÞ þ r ðquuÞ ¼ rp þ r s
ð4Þ
@ t ðqcp TÞ þ r ðqcp uTÞ ¼ r q þ hp r Jp
ð5Þ
where
hp
is
the
s ¼ lðru þ ðruÞt Þ q¼
rfflT} k |fflfflffl{zfflffl
conduction heat flux
2.1. Modeling the nanoparticle migration
1
qp
specific is
þ
the hp Jp |{z}
enthalpy
of
nanoparticles,
shear stress, and is the energy flux relative
nanoparticle diffusion heat flux
to the nanofluid velocity. Further, q; l; k; c are the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluids respectively which can be written as:
q ¼ F q ð/Þqbf ; qbf ¼ 998:2 kg=m3 l ¼ F l ð/Þlbf ; lbf ¼ 9:93 104 kg=ðm sÞ k ¼ F k ð/Þkbf ;
kbf ¼ 0:597 W=ðm KÞ ( qp ¼ 3880 kg=m3 ; alumina qp /; F q ð/Þ ¼ 1 / þ qbf qp ¼ 4175 kg=m3 ; titania al ¼ 39:11; bl ¼ 533:9; alumina ; F l ð/Þ ¼ ð1 þ al / þ bl /2 Þ al ¼ 5:45; bl ¼ 108:2; titania ak ¼ 7:47; bk ¼ 0; alumina F k ð/Þ ¼ ð1 þ ak / þ bk /2 Þ ak ¼ 1:292; bk ¼ 11:99; titania ð6Þ For the problem illustrated in Fig. 1, using the cylindrical coordinate systems, and equating rhp ¼ cp rT, one may simply obtain governing equations for steady, laminar, and incompressible flow as follows:
Fig. 1. The geometry of physical model and coordinate system.
1 d du r lð/Þ þ qð/Þg ¼ 0 r dr dr
ð7Þ
1 d dT rkð/Þ ¼0 r dr dr
ð8Þ
@ @/ DT @T ¼0 DB þ @r @r T @r
ð9Þ
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Fig. 2. Contour lines of nanoparticle volume fraction for different values of NBT and T⁄.
It should be stated that, as Buongiorno [19] stated, the heat transfer associated with the nanoparticle diffusion can be neglected with respect to the other terms and the governing equations. In addition, the boundary conditions can be written as:
(
r¼R:
u ¼ 0; T ¼ T w ; DB @/ þ DTT @r
r ¼Rþd:
du dr
@T @r
¼ 0:
ð10Þ
¼ 0; T ¼ T sat ; / ¼ /sat :
where R is the cylinder radius. The governing equations of Eqs. (7)– (9) can reduced further with introducing the following nondimensional parameters:
rR ; g¼ d ¼
d e¼ ; R
T sat T w ; Tw
u ¼ qbf
NBT ¼
u
lbf
gd2
;
T Tw T ¼ ; T sat T w
c
CBT w CT c
ð11Þ
2
d u þ dg2
Fk
d T þ dg2 2
eF l dF l d/ du þ Fq ¼ 0 þ eg þ 1 d/ dg dg
@/ / @T þ ¼0 @ g NBT ð1 þ cT Þ2 @ g
ð13Þ
/ðgÞ ¼ /sat e
T ¼ 1;
ð14Þ
/ ¼ /sat
ð15Þ
The heat transfer rate can be formulated as
kbf ðT sat T w Þ dT dT q00w ¼ kð/Þ ¼ F k ð0Þ dr y¼R dg g¼0 d
ð16Þ
Thus, heat transfer enhancement (HTE) can be given by
HTE ¼
q00w d dT ¼ F k ð0Þ kbf ðT sat T w Þ dg g¼0
N
T ðgÞ1 BT ð1þcÞð1þcT ðgÞÞ
ð18Þ
⁄
g ¼ 0 : u ¼ 0; T ¼ 0 du ¼ 0; dg
In this section, the governing equations are solved and the effects of pertinent parameters are discussed. It should be stated that due to non-linearity of governing equations, they cannot be solved analytically and they should be solved with semianalytical methods [40–42] or numerical ones as presented here.
Eq. (14) with boundary conditions of Eq. (15) can be solved analytically. With a simple integration from Eq. (14), the nanoparticle volume fraction distribution can be written as
Substituting Eq. (11) into Eq. (10), the boundary conditions can be expressed as
g¼1:
3. Results and discussion
ð12Þ
eF k dF k d/ dT ¼0 þ eg þ 1 d/ dg dg
HTE is very useful to describe the physics of the problem since it is constant and for regular fluids becomes HTE = 1. As a result, they show the enhancement in heat transfer rate.
3.1. Distribution of nanoparticles
where C B ¼ DTB , C T ¼ D/T . Thus, Eqs. (7)–(9) can be reduced as
Fl
Fig. 3. Algorithm of the numerical method.
ð17Þ
In Eq. (18), T would change from 0 at the wall to 1 at the border of condensate film. In addition, with the nanoparticle diameter around and less than 100 nm, the ratio of Brownian motion to thermophoretic force N BT can be changed over a wide range of 0.2–10. Fig. 2 demonstrates the contour of / via T⁄ and NBT. It can be observed that for NBT > 1, nanoparticle distribution is uniform (/ /sat) and there is no significant migration among nanparticles. We call this the region of uniform nanoparticle distribution. In this region, nanoparticles are homogeneously distributed in the film which diminishes the nanoparticle concentration gradient (Brownian diffusion is dominant). For NBT < 1, however, nanoparticles migrate from the wamer region (T⁄ = 1) toward the colder region (T⁄ = 0) and form a non-uniform nanoparticle distribution (thermophoresis is dominant). The migration of nanoparticles toward the cold wall increases the nanoparticles concentration there which, in turn, intensifies the viscosity and thermal conductivity of film. It should be stated that NBT is inversely proportional to nanoparticle diameter; thus, larger nanoparticles (lower NBT) boosts the nanoparticle concentration inside the film.
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Fig. 4. Comparison of the obtained results with the reported ones of Nusselt [1] for film condensation over a vertical plate for (a) velocity and (b) temperature profiles when e = 106, /sat = 0.
5
Fig. 5. The effects of e on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when NBT = 0.5, c = 0.2, and /sat = 0.06.
3.2. Accuracy of the results
In this section, the results for the flow and the thermal fields for justified values of parameters are obtained and discussed. All w results have been carried out for 0:1 < c ffi T satTT < 0:3 and to prow
The effect of e on the nanoparticle volume fraction (/), velocity (u⁄) and temperature (T⁄) is shown in Fig. 5. It can be seen that the effect of e on the profiles is insignificant, particularly for the temperature and the nanoparticle volume fraction profiles. But, it can be observed that increasing e slightly increases the velocity of the film. From Fig. 6, it can be realized that as NBT decreases, nanoparticles significantly migrate from the warmer region (vapor-liquid interface) toward the colder one (cold cylinder wall). Thus, the nanoparticle volume fraction through the condensate film increases which pushes the nanoparticles toward at the cold wall. This dynamic leads to an increase in the viscosity of the nanofluids, thereby reducing the velocities of the film and the temperature gradient near the cold wall. Fig. 7 illustrates that increasing /sat enhances the nanoparticle volume fraction inside the condensate film. Therefore, it reduces the velocity inside the condensate film as well as the wall temperature gradient. Reducing the velocity gradient is a useful attribute since it decreases the friction at the surface. However, reducing the temperature gradient at the wall has a negative effect on the heat transfer rate. Fig. 8 shows w that rising c = T satTT reduces the migration of nanoparticles toward w
vide sufficient information, three different values of 0.2, 0.5, and 1 for NBT have been considered.
the condensate film. As a result, the viscosity and thermal conductivity decrease with increasing c and enrich the momentum
Eq. (18) can be coupled with a system of Eqs. (12) and (13) to solve the flow and thermal fields inside the film condensation. Runge-Kutta-Fehlberg method as an efficient algorithm is utilized to solve the governing equations, whose accuracy is shown elsewhere [22]. Runge-Kutta-Fehlberg is an adaptive method with the order O(h4) with an error estimator of order O(h5). It can adapts the number and the position of the grid points during each iteration to improve the accuracy. To clarify, the numerical algorithm is shown in Fig. 3. The numerical results for velocity and temperature profiles for regular fluids (/sat = 0) and e = 106 is compared with the analytical results of Nusselt [1] for film condensation of regular fluid over a vertical plate in Fig. 4. As is obvious from Fig. 4, the results are in the best agreement.
3.3. Flow and thermal fields
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Fig. 6. The effects of NBT on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, c = 0.2, and /sat = 0.06.
Fig. 7. The effects of /sat on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, c = 0.2, and NBT = 0.5.
through the film. It should be mentioned that with a constant Tsat, increasing c originates from the reduction of wall temperature Tw. Thus, as the wall temperature Tw becomes smaller, migration of nanoparticles as well as the concentration of nanoparticle volume fraction through the condensate film reduce.
negative effects on the heat transfer rate at greater nanoparticle volume fractions. Moreover, despite the titania-water nanofluid, increasing c reduces the heat transfer rate for alumina-water nanofluid. This comes from the fact that increasing c decreases the nanoparticle migration inside a condensate film and increases the temperature gradient at the wall. However, thermal conductivity of nanofluids significantly reduces at the wall. Therefore, mutual effects of thermal conductivity reduction and the wall temperature gradient increment (see Eq. (17)) determines the net enhancement/deterioration of the heat transfer rate, as c increases. Fig. 9 illustrates the different impacts of titania and alumina nanoparticles on heat transfer enhancement. The figure is divided to two sections. For NBT < 1, the heat transfer enhancement is significantly changed while for NBT > 1, the heat transfer enhancements for both titania-water and alumina water nanofluids reach a plateau. As it is obvious, the alumina and titania nanoparticles have different trends via NBT. Alumina-water nanofluid expresses a higher enhancement in the heat transfer rate relative to titaniawater nanofluid, especially for the lower values of NBT. In other words, with respect to the alumina-water nanofluid, the titaniawater nanofluid does not represent a significant enhancement in heat transfer rate. In addition, the most enhancement of the heat
3.4. Heat transfer enhancement The heat transfer enhancement (HTE) for different values of e, /sat, and c for alumina-water and titania-water nanofluids has been shown in Table 1. Clearly, increasing e improves the heat transferate for both alumina and titania nanoparticles. Physically, increasing e corresponds to reducing the cylinder radius. Thus, smaller cylinder radius results in a greater heat transfer enhancement. In addition, it can be observed that heat transfer enhancement at titania-water nanofluids is much lower than such enhancement for alumina-water nanofluid. Therefore, aluminawater nanofluid exhibits a better performance. Notably, increasing /sat intensifies the heat transfer rate for the alumina-water nanofluid while has a reverse effect for the titania-water nanofluid. Thus, it can be deduced that adding titania nanoparticles has
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Fig. 9. The effects of NBT on heat transfer enhancement (HTE) for alumina/water and titania/water nanofluids when e = 0.01, c = 0.2, and /sat = 0.06.
4. Conclusions
Fig. 8. The effects of c on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, NBT = 0.5, and /sat = 0.06.
Table 1 Heat transfer enhancement (HTE) for different values of c, /sat, e, and different nanoparticles type when NBT = 1. /sat
c
Alumina-water nanofluid
Titania-water nanofluid
e = 0.1
e = 0.01
e = 0.001
e = 0.1
e = 0.01
e = 0.001
0.02
0.1 0.2 0.3
1.301 1.288 1.277
1.246 1.234 1.224
1.241 1.228 1.218
1.079 1.078 1.077
1.033 1.033 1.032
1.029 1.028 1.027
0.06
0.1 0.2 0.3
1.805 1.765 1.734
1.729 1.691 1.661
1.721 1.683 1.653
1.055 1.062 1.067
1.010 1.017 1.022
1.006 1.013 1.017
0.1
0.1 0.2 0.3
2.309 2.242 2.190
2.212 2.148 2.098
2.202 2.138 2.089
0.920 0.947 0.966
0.881 0.907 0.926
0.877 0.903 0.922
transfer rate for alumina-water nanofluid is corresponded to larger nanoparticles (lower NBT), whereas smaller nanoparticles leads to a better enhancement for titania-water nanofluid. Thus, it can be concluded that the heat transfer behavior of the nanofluids is extremely depended to thermophysical properties of nanofluids.
Condensate falling film of nanofluids over a vertical cylinder is theoretically investigated considering the effects of nanoparticles migration. The Brownian motion and thermophoretic diffusivities have been considered via the modified Buongiorno model which takes into account the effects of nanoparticles slip velocity relative to the base fluid. Different nanoparticle types namely titania and alumina nanoparticles are considered and their thermophysical behavior on the enhancement of the heat transfer rate is studied. The results have been obtained for different values of parameters including Brownian motion to thermophoretic diffusivities NBT, saturation nanoparticle volume fraction /sat, and normal temperaw . The major findings of this paper can be ture difference c ¼ T satTT w expressed as: The migration of nanoparticles toward the cold wall increases the nanoparticle concentration there which intensify the local viscosity and thermal conductivity of the film. Migration of nanoparticles is significantly enhanced as NBT decreases. Thus, larger nanoparticles (lower NBT) boosts the nanoparticle concentration inside the film. For NBT < 1, the region of significant nanoparticle migration, heat transfer rate is significantly changes while for NBT > 1 nanoparticle distribution becomes uniform and heat transfer rate does not change considerably. Decreasing the wall temperature Tw, reduces the migration of nanoparticles as well as the concentration of nanoparticle volume fraction through the condensate film. Temperature gradient at the wall is reduced as the migration of nanoparticles increases which has a negative effect on the heat transfer rate. However, increment of thermal conductivity at the cold wall because of nanoparticles accumulation plays a positive role in the heat transfer enhancement. Thus, there is a tradeoff between thermal conductivity enhancement and a reduction in temperature gradient at the walls which determines the net enhancement/deterioration of heat transfer rate. Reducing the cylinder radius improves the heat transfer rate for both alumina and titania nanoparticles
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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For the lower values of NBT, the heat transfer enhancement of titania-water nanofluids is completely insignificant relative to such enhancement for alumina-water nanofluid. Therefore, alumina-water nanofluid exhibits a better performance for larger nanoparticles. The results of this study can be applied to a wide range of engineering applications which deal with condensation process to increase the thermal performance of the system and to reduce the maximum temperature at the surface. References [1] W. Nusselt, Die Oberflächenkondensation des Wasserdampfes, Z. Ver. Dtsch. Ing. 60 (1916) 541–569. [2] M. Javadi, A. Golshani, A.M. Ghasemi, M. Anbarsooz, M. Moghiman, Improving Power Plant Efficiency using Water Droplet Injection in Air Condensers, World Acad. Sci., Eng. Technol. 62 (2010) 619–623. [3] G.H.R. Kefayati, Heat transfer and entropy generation of natural convection on non-Newtonian nanofluids in a porous cavity, Powder Technol. 299 (2016) 127–149. [4] J.W. Rose, Dropwise condensation theory and experiment: a review, Proc. Inst. Mech. Eng. Part A: J. Power Energy 216 (2002) 115–128. [5] K. Yoshioka, T. Okuda, H. Fujii, Y. Kamimoto, H. Kamiya, Effect of TiO2 nanoparticles aggregation in silicate thin coating films on photocatalytic behavior for antifouling materials, Adv. Powder Technol. 24 (2013) 886–890. [6] Z. Abbas, T. Hayat, M. Sajid, S. Asghar, Unsteady flow of a second grade fluid film over an unsteady stretching sheet, Math. Comput. Model. 48 (2008) 518– 526. [7] M. Sajid, M. Awais, S. Nadeem, T. Hayat, The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method, Comput. Math. Appl. 56 (2008) 2019–2026. [8] M.L. Eggersdorfer, S.E. Pratsinis, Agglomerates and aggregates of nanoparticles made in the gas phase, Adv. Powder Technol. 25 (2014) 71–90. [9] T. Hayat, R. Ellahi, F.M. Mahomed, Exact solution of a thin film flow of an Oldroyd 6- constant fluid over a moving belt, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 133–139. [10] D.Y. Shang, Theory of Heat Transfer with Forced Convection Film Flows, Springer Berlin Heidelberg, 2010. [11] A.A. Avramenko, I.V. Shevchuk, A.I. Tyrinov, D.G. Blinov, Heat transfer at film condensation of stationary vapor with nanoparticles near a vertical plate, Appl. Therm. Eng. 73 (2014) 391–398. [12] A.A. Avramenko, I.V. Shevchuk, A.I. Tyrinov, D.G. Blinov, Heat transfer at film condensation of moving vapor with nanoparticles over a flat surface, Int. J. Heat Mass Transf. 82 (2015) 316–324. [13] A.A. Avramenko, I.V. Shevchuk, A.I. Tyrinov, D.G. Blinov, Heat transfer in stable film boiling of a nanofluid over a vertical surface, Int. J. Therm. Sci. 92 (2015) 106–118. [14] A. Malvandi, D.D. Ganji, I. Pop, Laminar filmwise condensation of nanofluids over a vertical plate considering nanoparticles migration, Appl. Therm. Eng. 100 (2016) 979–986. [15] M. Turkyilmazoglu, Analytical solutions of single and multi-phase models for the condensation of nanofluid film flow and heat transfer, Eur. J. Mech. B. Fluids 53 (2015) 272–277. [16] H. El Mghari, H. Louahlia-Gualous, E. Lepinasse, Numerical study of nanofluid condensation heat transfer in a square microchannel, Numer. Heat Transfer, Part A: Appl. 68 (2015) 1242–1265. [17] M. Rahimi-Gorji, O. Pourmehran, M. Gorji-Bandpy, D.D. Ganji, An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by collocation method, Ain Shams Eng. J. 6 (2015) 531– 540. [18] A. Malvandi, S.A. Moshizi, D.D. Ganji, An analytical study on unsteady motion of vertically falling spherical particles in quiescent power-law shear-thinning fluids, J. Mol. Liq. 193 (2014) 166–173. [19] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer 128 (2006) 240–250. [20] C. Yang, W. Li, A. Nakayama, Convective heat transfer of nanofluids in a concentric annulus, Int. J. Therm. Sci. 71 (2013) 249–257.
[21] C. Yang, W. Li, Y. Sano, M. Mochizuki, A. Nakayama, On the anomalous convective heat transfer enhancement in nanofluids: a theoretical answer to the nanofluids controversy, J. Heat Transfer 135 (2013) 054504. [22] A. Malvandi, S.A. Moshizi, E.G. Soltani, D.D. Ganji, Modified Buongiorno’s model for fully developed mixed convection flow of nanofluids in a vertical annular pipe, Comput. Fluids 89 (2014) 124–132. [23] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel, Adv. Powder Technol. 25 (2014) 1369–1375. [24] F. Hedayati, G. Domairry, Nanoparticle migration effects on fully developed forced convection of TiO2–water nanofluid in a parallel plate microchannel, Particuology 24 (2016) 96–107. [25] F. Hedayati, G. Domairry, Effects of nanoparticle migration and asymmetric heating on mixed convection of TiO2–H2O nanofluid inside a vertical microchannel, Powder Technol. 272 (2015) 250–259. [26] M. Bahiraei, Studying nanoparticle distribution in nanofluids considering the effective factors on particle migration and determination of phenomenological constants by Eulerian-Lagrangian simulation, Adv. Powder Technol. 26 (2015) 802–810. [27] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration on hydromagnetic mixed convection of alumina/water nanofluid in vertical channels with asymmetric heating, Physica E 66 (2015) 181–196. [28] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu–water nanofluid using CVFEM, Adv. Powder Technol. 24 (2013) 980–991. [29] A. Malvandi, S.A. Moshizi, D.D. Ganji, Effect of magnetic fields on heat convection inside a concentric annulus filled with Al2O3–water nanofluid, Adv. Powder Technol. 25 (2014) 1817–1824. [30] S. Dinarvand, R. Hosseini, M. Abulhasansari, I. Pop, Buongiorno’s model for double-diffusive mixed convective stagnation-point flow of a nanofluid considering diffusiophoresis effect of binary base fluid, Adv. Powder Technol. 26 (2015) 1423–1434. [31] A. Malvandi, D.D. Ganji, Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, Int. J. Therm. Sci. 84 (2014) 196–206. [32] M. Ghalambaz, A. Behseresht, J. Behseresht, A. Chamkha, Effects of nanoparticles diameter and concentration on natural convection of the Al2O3–water nanofluids considering variable thermal conductivity around a vertical cone in porous media, Adv. Powder Technol. 26 (2015) 224–235. [33] A. Malvandi, A. Ghasemi, D.D. Ganji, Thermal performance analysis of hydromagnetic Al2O3-water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating, Int. J. Therm. Sci. 109 (2016) 10–22. [34] S. Heysiattalab, A. Malvandi, D.D. Ganji, Anisotropic behavior of magnetic nanofluids (MNFs) at filmwise condensation over a vertical plate in presence of a uniform variable-directional magnetic field, J. Mol. Liq. 219 (2016) 875– 882. [35] A. Malvandi, S. Heysiattalab, D.D. Ganji, Effects of magnetic field strength and direction on anisotropic thermal conductivity of ferrofluids (magnetic nanofluids) at filmwise condensation over a vertical cylinder, Adv. Powder Technol. (2016), http://dx.doi.org/10.1016/j.apt.2016.05.015. in press. [36] G.P. Fieg, W. Roetzel, Calculation of laminar film condensation in/on inclined elliptical tubes, Int. J. Heat Mass Transf. 37 (1994) 619–624. [37] M. Capellas, G. Caminal, G. Gonzalez, J. López-Santín, P. Clapés, Enzymatic condensation of cholecystokinin CCK-8 (4–6) and CCK-8 (7–8) peptide fragments in organic media, Biotechnol. Bioeng. 56 (1997) 456–463. [38] D.-W. Sun, L. Zheng, Vacuum cooling technology for the agri-food industry: Past, present and future, J. Food Eng. 77 (2006) 203–214. [39] A. Dutta, S.K. Som, P.K. Das, Film condensation of saturated vapor over horizontal noncircular tubes with progressively increasing radius of curvature drawn in the direction of gravity, J. Heat Transfer 126 (2005) 906–914. [40] S. Mosayebidorcheh, Taylor series solution of the electrohydrodynamic flow equation, J. Mech. Eng. Technol. 1 (2) (2013) 9–13. [41] M. Rahimi-Gorji, O. Pourmehran, M. Hatami, D. Ganji, Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis, Eur. Phys. J. Plus 130 (2015) 1–21. [42] S. Mosayebidorcheh, Analytical investigation of the micropolar flow through a porous channel with changing walls, J. Mol. Liq. 196 (2014) 113–119.
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
Contents lists available at ScienceDirect
Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Research Paper
Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder A. Malvandi a,⇑, Amirmahdi Ghasemi b, D.D. Ganji c, I. Pop d,⇑ a
Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA c Mechanical Engineering Department, Babol University of Technology, Babol, Iran d Department of Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 22 June 2016 Accepted 24 June 2016 Available online xxxx Keywords: Film condensation Nanofluids Nanoparticle migration Thermophoresis Brownian motion
a b s t r a c t The change in concentration and direction of nanoparticle migration can control the thermophysical properties of nanofluids. This dynamic is useful since it is able to improve the cooling performance by tuning the flow and heat transfer rate. In the current study, a theoretical investigation on the impact of nanoparticle migration on heat transfer enhancement of nanofluids condensate film over a vertical cylinder has been conducted. The Brownian motion and thermophoretic diffusivity have been considered by using the modified Buongiorno model which can take into account the effect of nanoparticle slip velocity. The results have been obtained for different parameters, including the Brownian motion to thermophoretic diffusivities NBT, the saturation nanoparticle volume fraction /sat, and the normal temperature difference c = ðT sat T w Þ=T w . It is shown that nanoparticle migration has significant impact on the flow and thermal fields and considerably affects the heat transfer rate. Furthermore, heat transfer enhancement in film condensation is strongly depended on the thermophysical properties of nanoparticles such that alumina-water nanofluid exhibits higher cooling performance than titania-water. Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction The condensation process is the change of the physical state of matter from gas to liquid phase. It usually releases a significant amount of heat due to large internal energy differences between the liquid and vapor states. Thus, it is ideal for several heat exchange equipments, especially in heat transfer enhancement for high-power cooling, food processing, biotechnology, and miniature electronic devices like microelectromechanical systems (MEMS) [1,2]. In addition, thermal performance of solar systems will be enhanced with the inclusion of nanoparticles to the working fluid. Consequently, it can be stated that nanofluids are able to participate in considerable reduction of carbon emissions. Generally, the heat will transfer inside the condensate film merely by conduction depending on the thickness of the film and the condensation rate. Either the thickness of the condensate film or the condensation rate of vapor depends on thermophysical properties which can be tuned by nanoparticle inclusion. Inclusion of
nanoparticle improves the heat exchangers performance since it improves the thermal conductivity of regular cooling fluids such as water, oil, ethylene-glycol, and power-law fluid [3]. Notably, nanoparticles have higher thermal conductivity relative to the working fluids and due to their similar size to the molecules of the base fluid, they would not induce any significant problems (abrasion, clogging, fouling and additional pressure loss in heat exchangers) relative to larger particles. However, the stability of nanofluids is an extreme challenge in front of scientists and researchers. The stability of nanofluids is commonly depended to the method of nanoparticle preparation. Nevertheless, inclusion of surfactants is a useful technique to avoid aggregation of nanoparticles and to improve the stability. Their impact on the environment is another concern of using nanofluids that we have to face with. Public concerns about their safety forced the engineers to be prudent to produce green nanofluids by biodegradable and nontoxic nanoparticles. 1.1. Film condensation
⇑ Corresponding authors. E-mail addresses: [email protected] (A. Malvandi), [email protected], [email protected] (I. Pop).
Nusselt [1] originally presents a theoretical model for the film condensation of pure vapors over tubes and plates. Then, several
http://dx.doi.org/10.1016/j.apt.2016.06.025 0921-8831/Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Nomenclature cp dp CB CT DB DT g hp HTE Jp k kB NBT q q00w R T u x; r
specific heat (m2/s2 K) nanoparticle diameter (m) reduced Brownian diffusion coefficient (1/K), C B ¼ DB =T reduced thermophoretic diffusion coefficient, C T ¼ DT =/ Brownian diffusion coefficient thermophoretic diffusion coefficient gravity (m/s2) specific enthalpy of nanoparticles (J/Kg) heat transfer enhancement nanoparticle mass flux (kg/m2 s) thermal conductivity (W/m K) Boltzmann constant (=1:3806488 1023 m2 kg=s2 K) ratio of the Brownian to thermophoretic diffusivities energy flux relative to the nanofluid (W) surface heat flux (W/m2) cylinder radius (m) temperature (K) axial velocity (m/s) coordinate system
Greek symbols b proportionality factor d condensate film thickness e ratio of condensate film thickness to cylinder radius e = d/R
publications are devoted to develop our understanding on film condensation which are fully reviewed and presented in literature such as [4–9]. Furthermore, Shang [10] conducted a comprehensive study of theory of film condensation and introduced novel similarity solutions for a wide range of basic problems. Nevertheless, few studies have been conducted for the condensation of nanofluid vapor. Recently, Avramenko et al. [11] introduced a model for the heat transfer of nanofluids condensate film near a vertical plate. Their model developed the classical model of Nusselt by adding an equation for the nanoparticle concentration and a dependence of the nanofluid density on the nanoparticle concentration. In another study [12], they studied the heat transfer rate at condensate film of moving vapor with nanoparticles over a flat surface. They concluded that increasing the nanoparticle volume fraction enhances the processes of momentum and heat transfer. They also employed their model to study the stable film boiling of nanofluids over vertical surfaces [13]. Malvandi et al. [14] considered the laminar film condensation of nanofluids over a vertical flat plate considering nanoparticles migration. Turkyilmazoglu [15] investigated the effects of considering the slip velocity of nanoparticles for the condensate film of nanofluids and indicated that slip mechanisms can be responsible for additional heat transfer enhancement. It should be noted that study of the film condensation of nanofluids is important due to the wide range of applications in nanoparticle synthesis, thermosyphons cooling, and heat transfer in nano and micro heat exchangers [16]. 1.2. Migration of nanoparticles There are several papers in literature deal with the steady or unsteady motion of particles in regular fluids [17,18]. However, for nanofluids, nanoparticles migrate because of nano-scale slip mechanisms that intensify the thermal conductivity and heat transfer rate of nanofluids. According to Buongiorno [19], Brownian
/
c g l q s
nanoparticle volume fraction normal temperature difference, c = ðT sat T w Þ=T w transverse direction dynamic viscosity (kg/m s) density (kg/m3) shear stress (Pa)
Subscripts B Brownian index BT ratio of Brownian to thermophoretic index bf base fluid k thermal conductivity index p nanoparticle sat saturation condition t time index T thermophoresis index w condition at the wall l dynamic viscosity index q density index Superscripts t transpose of a matrix ⁄ dimensionless variable
diffusion and thermophoresis are merely two important slip mechanisms in nanofluids. The effects of Brownian motion and thermophoresis on the nanoparticle migration of nanofluids have been investigated by several researchers. For example, Yang et al. [20,21] consider the effects of nanoparticle migration on forced convective heat transfer of alumina/water and titania/water nanofluids in circular, parallel plate, and tube-in-tube channels. Malvandi et al. [22] conducted a numerical analysis on mixed convection of nanofluids in a vertical concentric annulus. In another study, Malvandi and Ganji [23] investigated the effects of the nanoparticle migration on alumina/water nanofluids in a parallelplate channel. They demonstrated that nanoparticles move from the adiabatic wall (nanoparticles depletion) toward the cold wall (nanoparticles accumulation) and construct a non-uniform nanoparticle distribution. In addition, the anomalous heat transfer rate occurs when the Brownian motion takes control of the nanoparticle migration. Hedayati and Domairry [24,25] studied the effects of nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. They indicated that nanoparticles migration has significant effects on heat transfer characteristics of nanofluids. More details can be found in different scientific researches, for example [26–33]. 1.3. Motivation Recently, the thermophysical behaviors of film condensation of ferrofluids (magnetic nanofluids) as a class of metallic nanofluids in the presence of variable-directional magnetic field have been considered theoretically [34,35]. Their results indicated that anisotropic behaviors of thermophysical properties of ferrofluids in the presence of magnetic field are able to control the heat transfer behavior. However, film condensation of non-magnetic (regular) nanofluids (simply nanofluids) over a vertical cylinder has not been a subject of a study yet. Consequently, there is a need to consider the different behavior of heat transfer rate at film condensation
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
of isotropic nanofluids. Also, the effects of nanoparticle migration on heat transfer enhancement at film condensation of nanofluids are completely a novel subject. Thus, in this paper, the falling film condensation of nanofluids over a vertical cylinder has been investigated theoretically considering the effects of nanoparticles migration. Modified Buongiorno’s model is employed for the nanofluids taking into account the dependency of Brownian motion and thermophoresis to the temperature and the nanoparticle concentration, respectively. The main aim of the present study is to develop a formula for the nanoparticle migration inside the film condensation over a vertical cylinder and consider how the nanoparticle migration affects the heat transfer rate. The results of this study can be applied to enhance the thermal performance on various fields such as air cooled power station condensers [36], biotechnology [37], food processing [38] and industrial machines [39]. 2. Model formulation Fig. 1 depicts the physical geometry of a continuous condensate film of nanofluids falls along the outer surface of a vertical cylinder in contact with a vapor, when the surface temperature (Tw) is cooled below the local vapor saturation temperature (Tsat). A two-dimensional coordinate system is considered where the xaxis is aligned vertically and the r-axis is normal to the walls. Because of the gravity, the formed condensate film at the top of the cylinder wall flows downward. Flow is assumed to be laminar and the density of vapor is negligible. The vapor temperature is constant in a way that the condensation is the merely heat transfer mechanism between the liquid and vapor interface. Further, the shear stress at the liquid-vapor interface is assumed to be negligible. The film thickness d is very small relative to the cylinder length (boundary layer assumptions) and heat and mass transfer across the condensate film is much larger than those in the streamwise direction.
the base fluid, as Buongiorno [19] asserted. Thermophoresis moves the nanoparticles from warmer to colder region, inducing a nonuniform nanoparticle volume fraction distribution. Brownian diffusion is directly proportional to gradient of the nanoparticle volume fraction and originate owing to random drifting of nanoparticle within the suspension. From [19], nanoparticle distribution (/) can be written as
@ t ð/Þ þ r ðu/Þ ¼
Because of the thermophoresis and Brownian diffusion in nanoscale flows, nanoparticles will have a slip velocity relative to
rðJp Þ
ð1Þ
where Jp ¼ qp ðDB r/ þ DT rTT Þ and the Brownian diffusion coefficient, DB, and thermophoresis diffusion coefficient, DT, can be written as
DB ¼
kB T ; 3plbf dp
DT ¼ b
lbf / qbf
ð2Þ
where kB is the Boltzmann’s constant, lbf is the dynamic viscosity of the base fluid, T is the local temperature, dp is the nanoparticle diameter, and b ¼ 0:26 2k
kbf
bf þkp
.
2.2. Transport equations The governing equations are the conservation equations of the mass, momentum, and thermal energy in general coordinate system which can be written as:
@ t ðqÞ þ r ðquÞ ¼ 0
ð3Þ
@ t ðquÞ þ r ðquuÞ ¼ rp þ r s
ð4Þ
@ t ðqcp TÞ þ r ðqcp uTÞ ¼ r q þ hp r Jp
ð5Þ
where
hp
is
the
s ¼ lðru þ ðruÞt Þ q¼
rfflT} k |fflfflffl{zfflffl
conduction heat flux
2.1. Modeling the nanoparticle migration
1
qp
specific is
þ
the hp Jp |{z}
enthalpy
of
nanoparticles,
shear stress, and is the energy flux relative
nanoparticle diffusion heat flux
to the nanofluid velocity. Further, q; l; k; c are the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluids respectively which can be written as:
q ¼ F q ð/Þqbf ; qbf ¼ 998:2 kg=m3 l ¼ F l ð/Þlbf ; lbf ¼ 9:93 104 kg=ðm sÞ k ¼ F k ð/Þkbf ;
kbf ¼ 0:597 W=ðm KÞ ( qp ¼ 3880 kg=m3 ; alumina qp /; F q ð/Þ ¼ 1 / þ qbf qp ¼ 4175 kg=m3 ; titania al ¼ 39:11; bl ¼ 533:9; alumina ; F l ð/Þ ¼ ð1 þ al / þ bl /2 Þ al ¼ 5:45; bl ¼ 108:2; titania ak ¼ 7:47; bk ¼ 0; alumina F k ð/Þ ¼ ð1 þ ak / þ bk /2 Þ ak ¼ 1:292; bk ¼ 11:99; titania ð6Þ For the problem illustrated in Fig. 1, using the cylindrical coordinate systems, and equating rhp ¼ cp rT, one may simply obtain governing equations for steady, laminar, and incompressible flow as follows:
Fig. 1. The geometry of physical model and coordinate system.
1 d du r lð/Þ þ qð/Þg ¼ 0 r dr dr
ð7Þ
1 d dT rkð/Þ ¼0 r dr dr
ð8Þ
@ @/ DT @T ¼0 DB þ @r @r T @r
ð9Þ
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Fig. 2. Contour lines of nanoparticle volume fraction for different values of NBT and T⁄.
It should be stated that, as Buongiorno [19] stated, the heat transfer associated with the nanoparticle diffusion can be neglected with respect to the other terms and the governing equations. In addition, the boundary conditions can be written as:
(
r¼R:
u ¼ 0; T ¼ T w ; DB @/ þ DTT @r
r ¼Rþd:
du dr
@T @r
¼ 0:
ð10Þ
¼ 0; T ¼ T sat ; / ¼ /sat :
where R is the cylinder radius. The governing equations of Eqs. (7)– (9) can reduced further with introducing the following nondimensional parameters:
rR ; g¼ d ¼
d e¼ ; R
T sat T w ; Tw
u ¼ qbf
NBT ¼
u
lbf
gd2
;
T Tw T ¼ ; T sat T w
c
CBT w CT c
ð11Þ
2
d u þ dg2
Fk
d T þ dg2 2
eF l dF l d/ du þ Fq ¼ 0 þ eg þ 1 d/ dg dg
@/ / @T þ ¼0 @ g NBT ð1 þ cT Þ2 @ g
ð13Þ
/ðgÞ ¼ /sat e
T ¼ 1;
ð14Þ
/ ¼ /sat
ð15Þ
The heat transfer rate can be formulated as
kbf ðT sat T w Þ dT dT q00w ¼ kð/Þ ¼ F k ð0Þ dr y¼R dg g¼0 d
ð16Þ
Thus, heat transfer enhancement (HTE) can be given by
HTE ¼
q00w d dT ¼ F k ð0Þ kbf ðT sat T w Þ dg g¼0
N
T ðgÞ1 BT ð1þcÞð1þcT ðgÞÞ
ð18Þ
⁄
g ¼ 0 : u ¼ 0; T ¼ 0 du ¼ 0; dg
In this section, the governing equations are solved and the effects of pertinent parameters are discussed. It should be stated that due to non-linearity of governing equations, they cannot be solved analytically and they should be solved with semianalytical methods [40–42] or numerical ones as presented here.
Eq. (14) with boundary conditions of Eq. (15) can be solved analytically. With a simple integration from Eq. (14), the nanoparticle volume fraction distribution can be written as
Substituting Eq. (11) into Eq. (10), the boundary conditions can be expressed as
g¼1:
3. Results and discussion
ð12Þ
eF k dF k d/ dT ¼0 þ eg þ 1 d/ dg dg
HTE is very useful to describe the physics of the problem since it is constant and for regular fluids becomes HTE = 1. As a result, they show the enhancement in heat transfer rate.
3.1. Distribution of nanoparticles
where C B ¼ DTB , C T ¼ D/T . Thus, Eqs. (7)–(9) can be reduced as
Fl
Fig. 3. Algorithm of the numerical method.
ð17Þ
In Eq. (18), T would change from 0 at the wall to 1 at the border of condensate film. In addition, with the nanoparticle diameter around and less than 100 nm, the ratio of Brownian motion to thermophoretic force N BT can be changed over a wide range of 0.2–10. Fig. 2 demonstrates the contour of / via T⁄ and NBT. It can be observed that for NBT > 1, nanoparticle distribution is uniform (/ /sat) and there is no significant migration among nanparticles. We call this the region of uniform nanoparticle distribution. In this region, nanoparticles are homogeneously distributed in the film which diminishes the nanoparticle concentration gradient (Brownian diffusion is dominant). For NBT < 1, however, nanoparticles migrate from the wamer region (T⁄ = 1) toward the colder region (T⁄ = 0) and form a non-uniform nanoparticle distribution (thermophoresis is dominant). The migration of nanoparticles toward the cold wall increases the nanoparticles concentration there which, in turn, intensifies the viscosity and thermal conductivity of film. It should be stated that NBT is inversely proportional to nanoparticle diameter; thus, larger nanoparticles (lower NBT) boosts the nanoparticle concentration inside the film.
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Fig. 4. Comparison of the obtained results with the reported ones of Nusselt [1] for film condensation over a vertical plate for (a) velocity and (b) temperature profiles when e = 106, /sat = 0.
5
Fig. 5. The effects of e on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when NBT = 0.5, c = 0.2, and /sat = 0.06.
3.2. Accuracy of the results
In this section, the results for the flow and the thermal fields for justified values of parameters are obtained and discussed. All w results have been carried out for 0:1 < c ffi T satTT < 0:3 and to prow
The effect of e on the nanoparticle volume fraction (/), velocity (u⁄) and temperature (T⁄) is shown in Fig. 5. It can be seen that the effect of e on the profiles is insignificant, particularly for the temperature and the nanoparticle volume fraction profiles. But, it can be observed that increasing e slightly increases the velocity of the film. From Fig. 6, it can be realized that as NBT decreases, nanoparticles significantly migrate from the warmer region (vapor-liquid interface) toward the colder one (cold cylinder wall). Thus, the nanoparticle volume fraction through the condensate film increases which pushes the nanoparticles toward at the cold wall. This dynamic leads to an increase in the viscosity of the nanofluids, thereby reducing the velocities of the film and the temperature gradient near the cold wall. Fig. 7 illustrates that increasing /sat enhances the nanoparticle volume fraction inside the condensate film. Therefore, it reduces the velocity inside the condensate film as well as the wall temperature gradient. Reducing the velocity gradient is a useful attribute since it decreases the friction at the surface. However, reducing the temperature gradient at the wall has a negative effect on the heat transfer rate. Fig. 8 shows w that rising c = T satTT reduces the migration of nanoparticles toward w
vide sufficient information, three different values of 0.2, 0.5, and 1 for NBT have been considered.
the condensate film. As a result, the viscosity and thermal conductivity decrease with increasing c and enrich the momentum
Eq. (18) can be coupled with a system of Eqs. (12) and (13) to solve the flow and thermal fields inside the film condensation. Runge-Kutta-Fehlberg method as an efficient algorithm is utilized to solve the governing equations, whose accuracy is shown elsewhere [22]. Runge-Kutta-Fehlberg is an adaptive method with the order O(h4) with an error estimator of order O(h5). It can adapts the number and the position of the grid points during each iteration to improve the accuracy. To clarify, the numerical algorithm is shown in Fig. 3. The numerical results for velocity and temperature profiles for regular fluids (/sat = 0) and e = 106 is compared with the analytical results of Nusselt [1] for film condensation of regular fluid over a vertical plate in Fig. 4. As is obvious from Fig. 4, the results are in the best agreement.
3.3. Flow and thermal fields
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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Fig. 6. The effects of NBT on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, c = 0.2, and /sat = 0.06.
Fig. 7. The effects of /sat on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, c = 0.2, and NBT = 0.5.
through the film. It should be mentioned that with a constant Tsat, increasing c originates from the reduction of wall temperature Tw. Thus, as the wall temperature Tw becomes smaller, migration of nanoparticles as well as the concentration of nanoparticle volume fraction through the condensate film reduce.
negative effects on the heat transfer rate at greater nanoparticle volume fractions. Moreover, despite the titania-water nanofluid, increasing c reduces the heat transfer rate for alumina-water nanofluid. This comes from the fact that increasing c decreases the nanoparticle migration inside a condensate film and increases the temperature gradient at the wall. However, thermal conductivity of nanofluids significantly reduces at the wall. Therefore, mutual effects of thermal conductivity reduction and the wall temperature gradient increment (see Eq. (17)) determines the net enhancement/deterioration of the heat transfer rate, as c increases. Fig. 9 illustrates the different impacts of titania and alumina nanoparticles on heat transfer enhancement. The figure is divided to two sections. For NBT < 1, the heat transfer enhancement is significantly changed while for NBT > 1, the heat transfer enhancements for both titania-water and alumina water nanofluids reach a plateau. As it is obvious, the alumina and titania nanoparticles have different trends via NBT. Alumina-water nanofluid expresses a higher enhancement in the heat transfer rate relative to titaniawater nanofluid, especially for the lower values of NBT. In other words, with respect to the alumina-water nanofluid, the titaniawater nanofluid does not represent a significant enhancement in heat transfer rate. In addition, the most enhancement of the heat
3.4. Heat transfer enhancement The heat transfer enhancement (HTE) for different values of e, /sat, and c for alumina-water and titania-water nanofluids has been shown in Table 1. Clearly, increasing e improves the heat transferate for both alumina and titania nanoparticles. Physically, increasing e corresponds to reducing the cylinder radius. Thus, smaller cylinder radius results in a greater heat transfer enhancement. In addition, it can be observed that heat transfer enhancement at titania-water nanofluids is much lower than such enhancement for alumina-water nanofluid. Therefore, aluminawater nanofluid exhibits a better performance. Notably, increasing /sat intensifies the heat transfer rate for the alumina-water nanofluid while has a reverse effect for the titania-water nanofluid. Thus, it can be deduced that adding titania nanoparticles has
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
A. Malvandi et al. / Advanced Powder Technology xxx (2016) xxx–xxx
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Fig. 9. The effects of NBT on heat transfer enhancement (HTE) for alumina/water and titania/water nanofluids when e = 0.01, c = 0.2, and /sat = 0.06.
4. Conclusions
Fig. 8. The effects of c on nanoparticle distribution (/), velocity (u⁄) and temperature (T⁄) profiles for alumina-water nanofluid when e = 0.01, NBT = 0.5, and /sat = 0.06.
Table 1 Heat transfer enhancement (HTE) for different values of c, /sat, e, and different nanoparticles type when NBT = 1. /sat
c
Alumina-water nanofluid
Titania-water nanofluid
e = 0.1
e = 0.01
e = 0.001
e = 0.1
e = 0.01
e = 0.001
0.02
0.1 0.2 0.3
1.301 1.288 1.277
1.246 1.234 1.224
1.241 1.228 1.218
1.079 1.078 1.077
1.033 1.033 1.032
1.029 1.028 1.027
0.06
0.1 0.2 0.3
1.805 1.765 1.734
1.729 1.691 1.661
1.721 1.683 1.653
1.055 1.062 1.067
1.010 1.017 1.022
1.006 1.013 1.017
0.1
0.1 0.2 0.3
2.309 2.242 2.190
2.212 2.148 2.098
2.202 2.138 2.089
0.920 0.947 0.966
0.881 0.907 0.926
0.877 0.903 0.922
transfer rate for alumina-water nanofluid is corresponded to larger nanoparticles (lower NBT), whereas smaller nanoparticles leads to a better enhancement for titania-water nanofluid. Thus, it can be concluded that the heat transfer behavior of the nanofluids is extremely depended to thermophysical properties of nanofluids.
Condensate falling film of nanofluids over a vertical cylinder is theoretically investigated considering the effects of nanoparticles migration. The Brownian motion and thermophoretic diffusivities have been considered via the modified Buongiorno model which takes into account the effects of nanoparticles slip velocity relative to the base fluid. Different nanoparticle types namely titania and alumina nanoparticles are considered and their thermophysical behavior on the enhancement of the heat transfer rate is studied. The results have been obtained for different values of parameters including Brownian motion to thermophoretic diffusivities NBT, saturation nanoparticle volume fraction /sat, and normal temperaw . The major findings of this paper can be ture difference c ¼ T satTT w expressed as: The migration of nanoparticles toward the cold wall increases the nanoparticle concentration there which intensify the local viscosity and thermal conductivity of the film. Migration of nanoparticles is significantly enhanced as NBT decreases. Thus, larger nanoparticles (lower NBT) boosts the nanoparticle concentration inside the film. For NBT < 1, the region of significant nanoparticle migration, heat transfer rate is significantly changes while for NBT > 1 nanoparticle distribution becomes uniform and heat transfer rate does not change considerably. Decreasing the wall temperature Tw, reduces the migration of nanoparticles as well as the concentration of nanoparticle volume fraction through the condensate film. Temperature gradient at the wall is reduced as the migration of nanoparticles increases which has a negative effect on the heat transfer rate. However, increment of thermal conductivity at the cold wall because of nanoparticles accumulation plays a positive role in the heat transfer enhancement. Thus, there is a tradeoff between thermal conductivity enhancement and a reduction in temperature gradient at the walls which determines the net enhancement/deterioration of heat transfer rate. Reducing the cylinder radius improves the heat transfer rate for both alumina and titania nanoparticles
Please cite this article in press as: A. Malvandi et al., Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.06.025
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