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Heat transfer at film condensation of moving vapor with nanoparticles over a flat surface
International Journal of Heat and Mass Transfer 82 (2015) 316–324
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Heat transfer at film condensation of moving vapor with nanoparticles over a flat surface Andriy A. Avramenko a, Igor V. Shevchuk b,⇑, Andrii I. Tyrinov a, Dmitry G. Blinov a a b
Institute of Engineering Thermophysics, National Academy of Sciences, Kiev 03057, Ukraine MBtech Group GmbH & Co., KGaA, 70736 Fellbach-Schmiden, Germany
a r t i c l e
i n f o
Article history: Received 14 April 2014 Received in revised form 13 September 2014 Accepted 18 November 2014
Keywords: Heat transfer Nanofluid Condensation Liquid film
a b s t r a c t Fluid mechanics, heat and mass transfer taking place at condensation of moving vapor with nanoparticles near a flat plate were simulated analytically. An approximate analytical model was employed for simulation of the transport phenomena in the film of condensate, which takes into consideration mechanisms of the Brownian and thermophoretic diffusion. An important novelty of this model is that it suggested five major dimensionless parameters, which are included in the functional dependence of the heat transfer and fluid flow parameters in the condensate film of the nanoparticle concentration and physical properties: (i) parameter A, i.e. the relation between the thermophoretic and Brownian diffusion; (ii) the nanoparticle concentration in the vapor u1; (iii) the density R of the nanoparticles normalized by that of the fluid; (iv) the thermal conductivity of nanoparticles K normalized by that of the fluid; (v) and the parameter m describing the properties of the nanofluid viscosity. Consequently, novel analytical solutions were deduced for the velocity profiles, the mass flow rate, the thickness of the condensate film and the Nusselt number was obtained as a function of the aforementioned dimensionless parameters. It can be concluded that an increase in the nanoparticles concentration favors augmentation of the processes of momentum and heat transfer. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Condensation phenomena take place quite often in biotechnology [1–3] and in food processing [4–6]. A request for augmentation of the heat and mass transfer processes and the overall efficiency of the technological processes is an actual task of researcher and practical engineers dealing with such processes. The number of investigations into of heat transfer during condensation of the vapor with nanoparticles is rather limited. Authors of the work [7] conducted experiments on heat transfer at condensation of nanofluid vapor with iron oxide nanoparticles in a thermosyphon. They found out that an increase in the mean HTC is a function of the inclination angle c of the condenser. At c = 30°, the HTC increased by 9% at the nanoparticle concentration u = 2%, and 19% at u = 5.3%. For c = 45°, the HTC increased by 6% at u = 2%, and by 14% at u = 5.3%. For c = 45° and 60°, the HTC grew by 8% at u = 2%, and by 15% at u = 5.3%. The angle c = 90° yields in the HTC enhancement of 7% at u = 2%, and 13% at u = 5.3%. Thus, ⇑ Corresponding author at: MBtech Group GmbH & Co. KGaA, Salierstr. 38, 70736 Fellbach-Schmiden, Germany. Tel.: +49 (176) 483 07019. E-mail addresses: [email protected], [email protected] (I.V. Shevchuk). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.059 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
heat transfer enhancement at due to the nanoparticles is independent of the inclination angle. Experiments on heat transfer at the vapor condensation in the presence of copper oxide nanoparticles with the concentration u = 1.0% at the pressure of 7.45 kPa in inclined and horizontal grooved heat pipes were performed in the study [8]. For a pure fluid, the HTC in inclined tubes went up by about 60–80% over the horizontal tube heat transfer rate and reached a maximum at the inclination angle of 75°. In the nanofluid, the HTC increased by approximately 60–100%, which means that the maximal HTC growth at the expense of nanoparticles did not exceed 20%. Authors of the work [8] stated that heat transfer enhancement occurred because on the wall a thin porous layer formed with high thermal conductivity. Authors of the work [9] carried out 3D numerical simulations of heat transfer at condensation of the nanofluid vapor in thermosiphons. They demonstrated that the maximum HTC value attained 1740 W/ m2 K for the vapor with iron oxide nanoparticles at u = 5.3%. The minimum HTC value was 1450 W/m2 K at u = 2%. The simulations agreed well with experiments performed in the work [7]. Nusselt [10] has analytically investigated the heat transfer and fluid flow at the film condensation of the moving vapor. His model enabled finding out expression for the velocity and temperature
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317
Nomenclature A cp DB DT G k K qw r R T u U x, y
parameter A, the relation between the mechanisms of the thermophoretic and Brownian diffusion, Eq. (15) specific heat capacity of the nanoparticles Brownian diffusion coefficient thermophoretic diffusion coefficient mass flow rate thermal conductivity normalized thermal conductivity of the nanoparticles, Eq. (17) heat flux density at the wall latent heat of vaporization ratio of the densities of the nanoparticles and the fluid, Eq. (36) temperature streamwise velocity component (x-component) dimensionless velocity, Eq. (29) Cartesian coordinates
DT
g H
l q s u
temperature difference across the film dimensionless coordinate dimensionless temperature, Eq. (13) dynamic viscosity density shear stress, Eq. (8) nanoparticle concentration (volume fraction)
Subscripts f properties of fluid p properties of the nanoparticles w wall 1 outer boundary of the condensation film Acronyms HTC heat transfer coefficient
Greek symbols d condensate film thickness
profiles in the film of the condensate, flow rate through the film and the Nusselt number. The model of Nusselt [10] has proved to be an adequate and physically justifiable mathematical description of the film condensation of the moving vapor. Nanofluids possess unique properties, which motivate scientific community and industry to keep on intensive research of their fundamental aspects and practical applications. The most important properties of nanofluids are high thermal conductivity and low susceptibility of to sedimentation, fouling, erosion and clogging as compared to ordinary fluids with microparticles [11]. This inspired many promising applications of nanofluids, like those in nuclear energy, thermal management of systems with high dissipation rates of energy, cooling systems of electronic and optical devices, heat pipes and thermosyphons, as well as nanostructured materials and complex fluids [11–14]. It has been revealed that there is a slip between nanoparticles and the base fluid; however, in many cases nanofluids are modeled as uniform homogenous mixtures [14–25]. The works [17,26,27] demonstrated however that models employing the homogeneous (single-phase) flow approaches lead to underprediction of heat transfer rates between the nanofluids and solid bodies. Non-uniformity in nanofluids arises due to nanoparticle migration at the expense of the temperature and velocity gradients. As confirmed by experiments [18], the concentration gradient is also present in nanofluids. Works [19,20] provide more insight into theoretical aspects of non-uniformity in nanofluids. Self-similar analysis known as a powerful tool of theoretical investigations was used to model laminar and turbulent boundary layer flows of nanofluids in the works [26–30]. Properties of a nanofluid as functions of the nanoparticle concentration and temperature were modeled in the works [27,28] in order to investigate fluid flow, heat and mass transfer in a boundary layer over a flat plate. Symmetry analysis (Lie groups) enabled finding proper self-similar variables and functions specific for the problems studied in [26,27]. Work [28] represents a study of a natural convection flow of a nanofluid over a vertical plate. The authors [29] theoretically studied onset of convection in a horizontal nanofluid layer of finite depth. The laminar boundary layer, with the surface velocity set to be a function of the streamwise coordinate and all fluid properties being independent of the concentration of nanoparticles,
was investigated in the work [30]. As demonstrated in these theoretical works, an increase in the nanoparticle concentration causes enhancement of heat and mass transfer processes in nanofluids. As mentioned above, the number of the works devoted to the experimental study of the condensation heat transfer in the presence of nanoparticles is limited by the publications [7–9]. A theoretical insight into this problem is given by the authors in the recent investigation [31]. However, these studies examined the condensation of the stationary vapor. The present paper focuses on convective heat transfer in laminar flow of a film of a nanofluid over a flat surface under complicated velocity condition at the outer boundary of the film. The flow of the nanofluid results from condensation of a vapor of the same nanofluid that moves along the plate in the same direction as the nanofluid does. The plate is colder than the saturation temperature of the nanofluid vapor. To our knowledge, the stated problem has never been previously studied theoretically or experimentally. This paper deals with an investigation of the effect of nanoparticles on the velocity profiles and HTC in the nanofluid. While considering heat transfer problems, effect of the nanoparticle concentration on all thermophysical properties must be taken into account, since the HTC is affected not only by thermal conductivity, but also by flow structure, thickness of the condensation layer and other flow characteristics. We employ here the known relations between thermophysical properties and the nanoparticle concentration. The model proposed in the present work develops classical model of Nusselt [10] characterized above. The novelty of the present model lies in the inclusion of an equation for the nanoparticle concentration, which simulates heat transfer enhancement due to the addition of the nanoparticles and mass transfer effects. This novel model enabled deriving dimensionless parameters characterizing the effects of nanoparticles on fluid flow, heat and mass transfer processes. The model resulted also in the relations for the velocity profiles, condensate film thickness, mass flow rate through the film, as well as for the Nusselt number. The major influence of nanoparticles is incorporated via an additional equation for the nanoparticle concentration and functional dependence of the nanofluid viscosity, thermal conductivity and density on the nanoparticle concentration like in the work [32].
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where m = 2.5 [33]. The following boundary conditions serve as a closure for Eqs. (1)–(3)
u ¼ 0;
l
Fig. 1. Schematic layout of the problem.
2. Mathematical model Let us consider the laminar film condensation of a moving vapor near a flat wall, whose temperature is equal to Tw (Fig. 1). Vapor moves with a constant velocity u1 and contains nanoparticles with the concentration u1 . The film of condensate flows steadily under condition of zero pressure gradient along the x-coordinate. The thickness d of the film of condensate is small in comparison with the length of the plate, which enables us to use a boundary layer approximation. The vapor and condensate temperature at the outer border of the film y = d is equal to the saturation temperature T1 > Tw at a given pressure. The mass flow rate of the condensate in the film at its flow along the wall increases due to continuous vapor condensation. This phenomenon is taken into account in the mathematical model by the balance equation of the increase in the mass flow rate of the condensate. The condensation heat is withdrawn by the cold wall. The following assumptions were made in order to solve the stated problem: the inertia forces in the fluid are negligibly small in comparison with the viscosity forces; heat transfer across the film (along the y-coordinate) is much larger than convective heat transfer and thermal conductivity in the streamwise direction (along the x coordinate). The same is valid as applied to the mass transfer processes. The stress friction at the boundary of the fluid and vapor phases is equal to s1. The vapor density is much smaller than that of the condensate. Than the generalized mathematical model, which describes fluid flow, heat and mass transfer of a nanofluid in the condensate film can be written as follows:
d dy
l
du ¼0 dy
ð1Þ
0¼
d dT du dT DT dT dT þ qp cp DB k þ dy dy dy dy T 1 dy dy
ð2Þ
0¼
d du DT dT : DB þ dy dy T 1 dy
ð3Þ
In order to solve system (1)–(3), one needs to involve the following complementary equations
l¼
lf ð1 uÞm
;
q ¼ ð1 uÞqf þ uqp ; k ¼ kf
kp þ 2kf þ 2uðkp kf Þ ; kp þ 2kf uðkp kf Þ
ð4Þ ð5Þ ð6Þ
T ¼ Tw;
du ¼ s1 ; dy y¼d
du DT dT DB ¼ dy y¼0 T 1 dy y¼0 T ¼ T1;
at y ¼ 0;
u ¼ u1 at y ¼ d:
ð7Þ
ð8Þ
The last condition in Eq. (7) describes mathematically the fact that the total flux of nanoparticles on the wall at y = 0 (Stefan’s flow) due to the concentration gradient (the left-hand side) is equal to the total flux of nanoparticles due to the temperature gradient (the right-hand side) [34,35]. The mass flow rate through the film caused by the vapor condensation increases as the nanofluid flows downstream along the plate in the x-direction. This phenomenon is taken into consideration by the equation of the mass balance
dG ¼
qw dx; r
ð9Þ
Thus, we can say that, based on the model of Nusselt [10] and using the approach Buongiorno [24], we proposed a model for the film condensation of the moving vapor in the presence of nanoparticles. Following [10], the model does not take into account the inertial and convective terms. The main influence of the nanoparticles is incorporated via the equation for the concentration of nanoparticles and closing relations for the nanofluid properties (viscosity, density, thermal conductivity) in frames of the approach of Bonjurno [24]. In this approach, in contrast to the most of the models existing in the literature, we considered the effects of the nanoparticles through gradient terms and the non-linear characteristics of the nanofluid properties depending on the concentration, like e.g. in the work [32]. Consequently, the proposed model is a parabolic system of equations describing the hydrodynamic and thermal processes in frames of the boundary layer approximation. As it was shown in [26,27] on the basis of a strict analytical group analysis, the proposed approach describes correctly these processes in the parabolic approximation. The boundary condition for the concentration on the wall was proposed in the works [26,27]. It is set in such a way that it maintains an appropriate balance for the nanoparticle concentration. This condition regulates the relationship between the two diffusion mechanisms, i.e. thermophoretic and Brownian diffusion. At the outer boundary, a constant concentration of nanoparticles is defined, which initially corresponds to the concentration of the nanoparticles in the vapor, i.e. the ambience can be considered as a vapor–particle mixture with non-condensable particles. The boundary condition for the concentration on the outer boundary was not set in analogy with such conditions on the wall (Stefan or Ackermann correction), as it is usually done in the study of condensation of the vapor from the vapor–gas medium. Contrary to that, it was assumed that the concentration in the vapor remains unchanged over the entire ambience. Such a condition is adopted in the most of the studies on heat transfer in nanofluids. 3. Analytical soluton and analysis of results The integration of Eq. (3) results in
C ¼ DB
du DT dT þ ; dy T 1 dy
ð10Þ
where C is an integration constant; C = 0 with account for the boundary condition (7). Substituting Eq. (10) into Eq. (2), one can obtain
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d dT DT dT dT DT dT dT d dT þ qp cp ¼ : k k 0¼ þ dy dy dy dy T 1 dy dy T 1 dy dy
C1 ¼ 1 þ
3ðA þ 2u1 ÞðK 1Þ ; 2ðK þ 2Þ
319
ð19Þ
ð11Þ and the temperature distribution in the layer of condensate
H¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K þ 2 þ 3ðA þ u1 ÞðK 1Þ 3AðK 1Þð1 gÞð3AðK 1Þ þ 2ðK þ 2 þ 3ðK 1Þu1 ÞÞ þ ðK þ 2 þ 3ðK 1Þu1 Þ2 : 3AðK 1Þ
The integration of Eq. (11) and the reduction of the result to a dimensionless form yields
dH C1 ¼ hk þ2k þ2uðk k Þi ; p p f f dg
ð12Þ
ð20Þ
Using Eq. (19), one can finally transform the distribution of the nanoparticle concentration, Eq. (18), to the following form
kp þ2kf uðkp kf Þ
u¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3AðK 1Þð1 gÞð3AðK 1Þ þ 2ðK þ 2 þ 3ðK 1Þu1 ÞÞ þ ðK þ 2 þ 3ðK 1Þu1 Þ2 ðK þ 2Þ 3ðK 1Þ
where C1 is the integration constant, and
y g¼ ; d
T Tw T Tw H¼ ¼ ; T1 Tw DT
DT ¼ T 1 T w :
ð13Þ
Combining Eqs. (10) and (12), one can further obtain
du C1 þ A hk þ2k þ2uðk k Þi ¼ 0; p p f f dg
ð14Þ
kp þ2kf uðkp kf Þ
where
A¼
DT DT ; DB T 1
ð15Þ
is a dimensionless parameter expressing the relation between the mechanisms of the thermophoretic and Brownian diffusion. In the reality, the concentration of nanoparticles does not exceed 10%. Therefore, to simplify the integration, one can represent the denominator of the second term in Eq. (14) as the McLaren series
kp þ 2kf þ 2uðkp kf Þ 3ðK 1Þ u FKðK; uÞ ¼ 1 þ þ O½u2 ; K þ2 kp þ 2kf uðkp kf Þ ð16Þ
where
is normalized thermal conductivity of nanoparticles, i.e. the ratio of the thermal conductivities of nanoparticles kp and the fluid kf . The integration of Eq. (14) with allowance for Eq. (16) and the boundary condition (8) yields a profile of the concentration of nanoparticles in the film of condensate
u¼
lim u ¼ u1 :
ð22Þ
A!0
Fig. 2 demonstrates how the parameter A affects the profile of the nanoparticle concentration. It can be seen from Fig. 2 that an increase in the parameter A enhances the transport of nanoparticles towards the wall and increases nonlinearity of the nanoparticle concentration. At A = 0.1, the nanoparticle concentration profile is almost linear. For the case where the thermal conductivity of nanoparticles is close to that of fluid, i.e. K ? 1, distribution (21) exhibits a linear trend regardless of the value of the parameter A K!1
ð17Þ
Having obtained the profile of concentration (18), one can integrate Eq. (12). This integration in view of conditions (7) and (8) yields an expression for the integration constant
ð23Þ
Fig. 3 elucidates the effect of the parameter K on the nanoparticle concentration profile in the film. An increase in this parameter, as well as in the parameter A, leads to an increase in nonlinearity of the concentration profile. However, the effect of the parameter K is much weaker than the influence of the parameter A. It should be pointed out that the value of the concentration at the wall is not affected by the variation of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK þ 2ÞðK þ 2 þ 6AðK 1ÞC 1 Þ þ 6AðK 2 þ K 2Þðu1 AC 1 gÞ þ 9ðK 2 1Þu21 ðK þ 2Þ 3ðK 1Þ
ð21Þ
As one can see from Eq. (21), the value u1 of the nanoparticle concentration at the outer boundary of the film of condensate is a point of minimum. The nanoparticle concentration at the wall demonstrates a point of maximum resulting from the Brownian and thermophoretic diffusion and normalized thermal conductivity of nanoparticles K. As a result, an extremely thin porous layer is formed over the wall. This porous layer reduces overall thermal resistance, which correlates with the data of the work [7]. If A ? 0, the thermal diffusion effects are negligible, and function (21) tends to a constant value independent of the parameter K
lim u ¼ Að1 gÞ þ u1 :
kp K¼ : kf
:
:
ð18Þ
parameter K. The temperature profile (20) in the film of condensate under condition A ? 0 exhibits a linear trend regardless of the values K and u1
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Fig. 2. Effect of the parameter A on the nanoparticle concentration profile, K = 2, u1 = 0.01.
Fig. 5. Effect of the parameter K on the temperature profile, A = 0.9, u1 = 0.1.
increase in the values of both parameters A and K causes a reduction of the temperature gradient near the wall. Such effect of the parameter K immediately follows from Eq. (12). Further integration of Eq. (1) under the boundary condition (8) yields
s1 d ¼ l
du dg
ð26Þ
To undertake the second integration of Eq. (26), let us transform Eq. (4) for the viscosity using the Padé approximation. For this particular purpose, it is more convenient than the Taylor series. Imposing the simplest Padé approximation to Eq. (4), one can obtain
l¼ Fig. 3. Effect of the parameter K on the nanoparticle concentration profile, A = 0.9, u1 = 0.01.
limH ¼ g: A!0
ð24Þ
The temperature profile takes the same shape under condition K ? 1 at any arbitrary values of A and u1
limH ¼ g: K!1
ð25Þ
Figs. 4 and 5 depict the effect of the parameters A and K on the shape of the temperature profile. As seen from Figs. 4 and 5, an
Fig. 4. Effect of the parameter A on the temperature profile, K = 2, u1 = 0.1.
lf ð1 uÞm
lf 1 mu
;
ð27Þ
As demonstrated in the review paper [33], the value of m can vary over a wide range, so that the coefficient m can be further used as a parameter. The integration of Eq. (26) under the boundary condition (7) on the wall yields
h U ¼ 9A2 ðK 1Þ2 ð2mðð1 gÞG BÞ þ LÞ þ 6AðK 1Þ i ð2mðð1 gÞG BÞ þ LÞM þ 2mðG BÞM 2 h i = 27AðK 1Þ2 ð3AðK 1Þ þ 2MÞ ;
ð28Þ
where
Fig. 6. Effect of the parameter A on the velocity profile, K = 2, u1 = 0.1, m = 2.5.
A.A. Avramenko et al. / International Journal of Heat and Mass Transfer 82 (2015) 316–324
321
The next step is finding a solution for the heat transfer coefficient. Following Nusselt [10], let us derive a solution for the mass flow rate through the film of condensate
G¼
Z
d
qudy ¼
0
s1 d2 qf Z 1 U ½ð1 uÞ þ uRdg; lf 0
R ¼ qp =qf :
ð35Þ ð36Þ
Here parameter R defines the ratio of the densities of the nanoparticles qp and the fluid qf. The integration of Eq. (35) with allowance for Eqs. (21) and (28) yields a very cumbersome mathematical expression. Therefore, let us represent Eq. (35) as follows
G¼
Z
d
qudy ¼
0
Fig. 7. Effect of the parameter u1 on the velocity profile, A = 0.1,K = 2, m = 2.5.
s1 d2 qf FGðA; K; u1 ; m; RÞ; 2lf
ð37Þ
where the function FG(A, K, u1, m, R) has a very complicated analytical form. In the limiting case of K ?1, this function looks as
lim FGðA; K; u1 ; m; RÞ ¼ K!1
A ðR 1 þ mð3u1 3Ru1 2ÞÞ 3 þ ð1 mu1 Þð1 þ ðR 1Þu1 Þ
mA2 ðR 1Þ: 4
ð38Þ
In the case where R ? 1, the nanoparticle density approaches to the fluid density, and the Eq. (38) reduces to
2 lim FGðA; K; u1 ; m; RÞ ¼ 1 A u1 m: 3 K!1 R!1
Fig. 8. Effect of the parameter m on the velocity profile, A = 0.1, K = 2, u1 = 0.1.
U¼
ulf ; s1 d
B ¼ 2 þ K þ 3ðK 1ÞðA þ u1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9A2 ðK 1Þ2 ð1 gÞ þ 6AðK 1Þð1 gÞM þ M2 ;
ð29Þ ð30Þ
ð39Þ
It follows from here that in the absence of nanoparticles, i.e. at A = u1 = m = 0, the function FG is also equal to unity, and Eq. (37) transforms into the solution of Nusselt [10]. Substituting further Eq. (36) into the balance Eq. (9), one can derive a differential equation for the thickness of the condensate film
d FGðA; K; u1 ; m; RÞ dx ¼
!
s1 d 2 q f kw dH ¼ DT 2lf dr dg g¼0
kf dH : FKðK; uð0ÞÞDT dr dg g¼0
ð40Þ
ð31Þ
Assuming that the film of condensate starts developing from the front edge of the plate x = 0, a solution of Eq. (40) with zero boundary conditions
L ¼ 3ð2m 3 þ Kð3 þ mÞÞg;
ð32Þ
d ¼ 0 at x ¼ 0
M ¼ 2 þ K þ 3ðK 1Þu1 :
ð33Þ
takes the following form
G¼
Eq. (28) describes mathematically the dimensionless velocity profile in the film of the condensate. In the limit at A ? 0, we have
lim U ¼ gð1 mu1 Þ: A!0
ð34Þ
Thus the velocity profile at absence of nanoparticles is described by linear Eq. (34), i.e. dependence of Nusselt [10]. Figs. 6–8 describe effects of different factors on the shape of the velocity profile in the layer of condensate. As follows from Figs. 6– 8, an increase in any of the parameters A, u1 and n entails a decrease in the fullness of the velocity profile in the film of condensate. An increase in any of these three parameters results in increased the nanoparticle concentration near the wall followed with the increased viscosity in accordance with Eq. (4) and reduced velocity gradient in the near-wall region. The variation of the parameter K practically does not influence on the shape of the velocity profile.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6kf lf DTx FKðK; uð0ÞÞ dH 3 d¼ cs q1 u21 r qf FGðA; K; u1 ; m; RÞ dg g¼0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FKðK; uð0ÞÞ dH 3 ¼ d0 ; FGðA; K; u1 ; m; RÞ dg g¼0
ð41Þ
ð42Þ
where
cs ¼
2s1 q1 u21
ð43Þ
is the friction coefficient introduced by Nusselt [10], q1 and u1 are the vapor density and velocity at the interface between the film and the vapor, respectively,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6kf lf DTx 3 d0 ¼ cs q1 u21 r qf
ð44Þ
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Fig. 9. Effect of the parameter u1 on the normalized Nusselt number Nu/Nu0 at A ? 0, R = 3, m = 2.5.
is the thickness of the film of condensate in the absence of nanoparticles [10]. Using the temperature profile (20) together with Eq. (42), one can derive a relation for the HTC
h¼
" #2=3 kf dH FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : d0 dg g¼0
ð45Þ
Eq. (45) can be re-written in the form of the Nusselt number
" #2=3 hd0 dH Nu ¼ ¼ FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : kf dg g¼0
ð46Þ
In the absence of nanoparticles [34]
Nu0 ¼ 1:
ð47Þ
Fig. 11. Effect of the parameter A on the normalized Nusselt number Nu/Nu0. a) u1 = 0.01, b) u1 = 0.1. K = 6, m = 2.5.
With account for Eq. (46), the normalized Nusselt number can be written as
" #2=3 Nu dH ¼ FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : Nu0 dg g¼0
ð48Þ
Eq. (48) enables analyzing effects of nanoparticles on the heat transfer rate characterized by five non-dimensional parameters A,
Fig. 10. Effect of the parameter u1 on the normalized Nusselt number Nu/Nu0 at A ? 0. K = 4, m = 2.5.
K, u1, m and R. The numerical values of these parameters were evaluated using the data from the studies [24,33]. Fig. 9 demonstrates effects of the nanoparticle concentration at the interface between the condensate and vapor u1 on the normalized Nusselt number Nu/Nu0 at different values of the parameter K. Here A ? 0, which represents the case where the Brownian diffusion by far exceeds the thermophoretic diffusion. As can be seen from Eq. (22), the nanoparticle concentration in this case is practically constant across the film of condensate and equal to u1. As evident from Fig. 9, an increase in the nanoparticle concentration at the interface between the condensate and vapor u1 leads to heat transfer enhancement, which amplifies with an increase in the difference between thermal conductivities of the particles and the fluid. Although, as Fig. 4 demonstrates, an increase in the parameter K causes a decrease in the temperature gradient at the wall, while the normalized Nusselt number increases. Therefore, in view of the form of Eq. (48), it can be concluded that the main factor that causes heat transfer enhancement in this case is the overall increase in the thermal conductivity of the nanofluid, but not the change of the temperature profile shape. Fig. 10 exhibits the effect of the parameter R on the normalized Nusselt number, which demonstrates a sustainable increasing trend with an increase in the relative density of nanoparticles. The effect of the parameter A, i.e. the relation the thermophoretic and Brownian diffusion, on the normalized Nusselt number is outlined in Fig. 11a (at low u1 = 0.01) and Fig. 11b (at high u1 = 0.1) for different values of the parameter R. Fig. 11
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in the condensate film. Obviously, the growth of these parameters leads to the heat transfer augmentation because of the increase in the nanoparticle concentration in near wall region. As expected, an increase in the parameters R and K also increases the HTC, because of the increased effective thermal conductivity of the nanofluids. A larger parameter R effectively means the larger total density of a nanofluid, whereas the effective thermal conductivity is proportional to the density of the medium. An increase in the parameter K entails an increase of thermal conductivity of the nanoparticles in comparison with baseline thermal conductivity of the pure fluid. Obviously, the growth of the parameter K causes an increase in the effective thermal conductivity of nanofluid. 4. Conclusions
Fig. 12. Effect of the parameter m on the normalized Nusselt number Nu/Nu0. K = 4, R = 3.
demonstrates that an increase in the parameter A enhances heat transfer. This means that heat transfer is enhanced following the amplification of the thermophoretic diffusion or the suppression of the Brownian diffusion. This trend is in line with Eq. (21), which testifies that the amplification of the thermophoretic diffusion leads to the increase in the nanoparticle concentration near the wall followed with the heat transfer enhancement. It can be also concluded that the heat transfer enhancement caused by the increase in the parameter A stands in relation to the absolute value of concentration u1. At low levels of u1, the heat transfer enhancement is more significant than that at high levels of u1, or, in other words, the rate of increase in Nu/Nu0 subsides for the higher values of u1. For higher nanoparticle concentrations, one can observe stronger effects of the parameter R. Effect of the exponent m in Eq. (4) for the dynamic viscosity of the nanofluid on the normalized Nusselt number Nu/Nu0 is shown in Fig. 12. Heat transfer diminishes with the increasing exponent m. It can be explained by the fact that the viscosity increases with the increasing exponent m, which is accompanied with a decrease in the velocity gradient near the wall (Fig. 8). This entails a reduction in the transport of momentum and, hence, in heat transfer. The magnitude of the predicted heat transfer augmentation is consistent with the experimental data of the works [7–9]. The analytical study performed above enabled revealing the mechanisms of hydrodynamic and thermal processes at the condensation of a moving vapor with nanoparticles. The vapor was considered as a vapor–particle mixture with non-condensable particles. Diffusion processes of these particles are controlled by two diffusion mechanisms, i.e. the thermophoretic and Brownian diffusion. As a result of the diffusion processes, an extremely thin porous layer formed over the wall. This porous layer reduces overall thermal resistance, which agrees with the data of the paper [7]. The level of concentration of the nanoparticles in this layer depends on the counteraction of the thermophoretic and Brownian diffusion. The amplification of the thermophoretic diffusion leads to the increased concentration of the nanoparticles on the wall, while Brownian diffusion augmentation entails the opposite effect. The relationship between these two diffusion mechanisms are taken into account by the dimensionless parameter A. Diffusion processes affect strongly the hydrodynamic and thermal processes in the condensate layer. Strengthening of the thermophoretic diffusion fosters an increase in the concentration of the nanoparticles on the wall and consequently a reduction of the velocity gradient in the near-wall region. Therefore, the increase in the parameters A and u1 results in the decreased fullness of the velocity profile
The paper represents an approximate model for fluid flow and heat transfer at film condensation of moving vapor with nanoparticles near a flat plate. This model involves the Brownian and thermophoretic diffusion mechanisms, together with the effect of the nanoparticle concentration on the fluid properties. The novelty of this mathematical model as compared with the classical Nusselt problem [10] is the addition of an equation for the nanoparticle concentration, which describes the heat and mass transfer augmentation because of the presence of nanoparticles. The model enabled obtaining expressions for the profiles of the nanoparticle concentration, temperature and velocity in the film of condensate, as well as for the mass flow rate in the film and the Nusselt number. An analysis revealed that fluid flow and heat transfer in the film are affected by the following non-dimensional parameters: parameter A that stands for the relation between the thermophoretic and Brownian diffusion; the nanoparticle concentration in the vapor u1; the density R of the nanoparticles normalized by that of the fluid; the thermal conductivity of nanoparticles K normalized by that of the fluid; and the parameter m, which characterizes properties of the nanofluid viscosity. With each of the parameters A, u1, R and K increasing, heat transfer also increases. As an analysis of the effect of the parameter A demonstrated, the Brownian diffusion deteriorates heat transfer, while the thermophoretic diffusion enhances heat transfer. An increase in the exponent m deteriorates heat transfer, which is caused by the decrease in the momentum transfer near the wall. Relative increase in heat transfer at u1 = 0.1 can reach 20%, which is somewhat lower than heat transfer enhancement at forced convection over a flat plate [26]. Conflict of interest None declared. References [1] B.M. Ennis, C.T. Marshall, I.S. Maddox, A.H.J. Paterson, Continuous product recovery by in-situ gas stripping/condensation during solvent production from whey permeate using Clostridium acetobutylicum, Biotechnol. Lett. 8 (10) (1986) 725–730. [2] G.A. Mansoori, K. Schulz, E.E. Martinelli, Bioseparation using supercritical fluid extraction/retrograde condensation, Bio/Technology 6 (1988) 393–396. [3] M. Capellas, G. Caminal, G. Gonzalez, J. López-Santín, Pere Clapés, Enzymatic condensation of cholecystokinin CCK-8 (4–6) and CCK-8 (7–8) peptide fragments in organic media, Biotechnol. Bioeng. 56 (4) (1997) 456–463. [4] D.-W. Sun, L. Zheng, Vacuum cooling technology for the agri-food industry: past, present and future, J. Food Eng. 77 (2006) 203–214. [5] V. Mavrov, E. Bélières, Reduction of water consumption and wastewater quantities in the food industry by water recycling using membrane processes, Desalination 131 (1–3) (2000) 75–86. [6] A. Baudot, M. Marin, Pervaporation of aroma compounds: comparison of membrane performances with vapour–liquid equilibria and engineering aspects of process improvement, Food Bioprod. Process. 75 (2) (1997) 117– 142.
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[7] G. Huminicaaand, A. Huminic, Heat transfer characteristics of a two-phase closed thermosyphons using nanofluids, Exp. Therm. Fluid Sci. 35 (3) (2011) 550–557. [8] Z.H. Liu, Y.-Y. Li, R. Bao, Thermal performance of inclined grooved heat pipes using nanofluids, Int. J. Therm. Sci. 49 (2010) 1680–1687. [9] G. Huminic, A. Huminic, Numerical study on heat transfer characteristics of thermosyphon heat pipes using nanofluids, Energy Convers. Manage. 76 (2013) 393–399. [10] W. Nusselt, Die Oberflächenkondensation des Wasserdampfes, Z. Ver. Deutsch. Ing. 60 (541–546) (1916) 569–575. [11] Y. Wenhua, D.M. France, J.L. Routbort, S.U.S. Choi, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transfer Eng. 29 (5) (2008) 432–460. [12] S.K. Das, S.U.S. Choi, H.E. Patel, Heat transfer in nanofluids: a review, Heat Transfer Eng. 27 (2006) 3–19. [13] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [14] M.H. Buschmann, Nanofluids in thermosyphons and heat pipes: overview of recent experiments and modelling approaches, Int. J. Therm. Sci. 72 (2013) 1– 17. [15] U. Rea, T. McKrell, L. Hu, J. Buongiorno, Laminar convective heat transfer and viscous pressure loss of alumina–water and zirconia–water nanofluids, Int. J. Heat Mass Transfer 52 (2009) 2042–2048. [16] H.C. Brinkmann, The viscosity of concentrated suspensions and solution, J. Chem. Phys. 20 (1952) 571–581. [17] S.E.B. Maiga, C.T. Nguyen, N. Galanis, G. Roy, Heat transfer behaviors of nanofluids in a uniformly heated tube, Superlattices Microstruct. 35 (2004) 543–557. [18] D. Merhi, E. Lemaire, G. Bossis, F. Moukalled, Particle migration in a concentrated suspension flowing between rotating parallel plates: investigation of diffusion flux coefficients, J. Rheol. 49 (2005) 1429–1448. [19] M. Frank, D. Anderson, E.R. Weeks, J.F. Morris, Particle migration in pressuredriven flow of a Brownian suspension, J. Fluid Mech. 493 (2003) 363–378. [20] Y.L. Ding, D. Wen, Particle migration in a flow of nanoparticle suspensions, Powder Technol. 149 (2005) 84–92. [21] B. Ghasemi, S.M. Aminossadati, Periodic natural convection in a nanofluidfilled enclosure with oscillating heat flux, Int. J. Therm. Sci. 49 (2010) 1–9. [22] E.B. Ogut, Natural convection of water-based nanofluids in an inclined enclosure with a heat source, Int. J. Therm. Sci. 48 (2009) 2063–2073. [23] A. Ehle, S. Feja, M.H. Buschmann, Temperature dependency of ceramic nanofluids shows classical behaviour, J. Thermophys. Heat Transfer 25 (2011) 3.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat transfer at film condensation of moving vapor with nanoparticles over a flat surface Andriy A. Avramenko a, Igor V. Shevchuk b,⇑, Andrii I. Tyrinov a, Dmitry G. Blinov a a b
Institute of Engineering Thermophysics, National Academy of Sciences, Kiev 03057, Ukraine MBtech Group GmbH & Co., KGaA, 70736 Fellbach-Schmiden, Germany
a r t i c l e
i n f o
Article history: Received 14 April 2014 Received in revised form 13 September 2014 Accepted 18 November 2014
Keywords: Heat transfer Nanofluid Condensation Liquid film
a b s t r a c t Fluid mechanics, heat and mass transfer taking place at condensation of moving vapor with nanoparticles near a flat plate were simulated analytically. An approximate analytical model was employed for simulation of the transport phenomena in the film of condensate, which takes into consideration mechanisms of the Brownian and thermophoretic diffusion. An important novelty of this model is that it suggested five major dimensionless parameters, which are included in the functional dependence of the heat transfer and fluid flow parameters in the condensate film of the nanoparticle concentration and physical properties: (i) parameter A, i.e. the relation between the thermophoretic and Brownian diffusion; (ii) the nanoparticle concentration in the vapor u1; (iii) the density R of the nanoparticles normalized by that of the fluid; (iv) the thermal conductivity of nanoparticles K normalized by that of the fluid; (v) and the parameter m describing the properties of the nanofluid viscosity. Consequently, novel analytical solutions were deduced for the velocity profiles, the mass flow rate, the thickness of the condensate film and the Nusselt number was obtained as a function of the aforementioned dimensionless parameters. It can be concluded that an increase in the nanoparticles concentration favors augmentation of the processes of momentum and heat transfer. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Condensation phenomena take place quite often in biotechnology [1–3] and in food processing [4–6]. A request for augmentation of the heat and mass transfer processes and the overall efficiency of the technological processes is an actual task of researcher and practical engineers dealing with such processes. The number of investigations into of heat transfer during condensation of the vapor with nanoparticles is rather limited. Authors of the work [7] conducted experiments on heat transfer at condensation of nanofluid vapor with iron oxide nanoparticles in a thermosyphon. They found out that an increase in the mean HTC is a function of the inclination angle c of the condenser. At c = 30°, the HTC increased by 9% at the nanoparticle concentration u = 2%, and 19% at u = 5.3%. For c = 45°, the HTC increased by 6% at u = 2%, and by 14% at u = 5.3%. For c = 45° and 60°, the HTC grew by 8% at u = 2%, and by 15% at u = 5.3%. The angle c = 90° yields in the HTC enhancement of 7% at u = 2%, and 13% at u = 5.3%. Thus, ⇑ Corresponding author at: MBtech Group GmbH & Co. KGaA, Salierstr. 38, 70736 Fellbach-Schmiden, Germany. Tel.: +49 (176) 483 07019. E-mail addresses: [email protected], [email protected] (I.V. Shevchuk). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.059 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
heat transfer enhancement at due to the nanoparticles is independent of the inclination angle. Experiments on heat transfer at the vapor condensation in the presence of copper oxide nanoparticles with the concentration u = 1.0% at the pressure of 7.45 kPa in inclined and horizontal grooved heat pipes were performed in the study [8]. For a pure fluid, the HTC in inclined tubes went up by about 60–80% over the horizontal tube heat transfer rate and reached a maximum at the inclination angle of 75°. In the nanofluid, the HTC increased by approximately 60–100%, which means that the maximal HTC growth at the expense of nanoparticles did not exceed 20%. Authors of the work [8] stated that heat transfer enhancement occurred because on the wall a thin porous layer formed with high thermal conductivity. Authors of the work [9] carried out 3D numerical simulations of heat transfer at condensation of the nanofluid vapor in thermosiphons. They demonstrated that the maximum HTC value attained 1740 W/ m2 K for the vapor with iron oxide nanoparticles at u = 5.3%. The minimum HTC value was 1450 W/m2 K at u = 2%. The simulations agreed well with experiments performed in the work [7]. Nusselt [10] has analytically investigated the heat transfer and fluid flow at the film condensation of the moving vapor. His model enabled finding out expression for the velocity and temperature
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317
Nomenclature A cp DB DT G k K qw r R T u U x, y
parameter A, the relation between the mechanisms of the thermophoretic and Brownian diffusion, Eq. (15) specific heat capacity of the nanoparticles Brownian diffusion coefficient thermophoretic diffusion coefficient mass flow rate thermal conductivity normalized thermal conductivity of the nanoparticles, Eq. (17) heat flux density at the wall latent heat of vaporization ratio of the densities of the nanoparticles and the fluid, Eq. (36) temperature streamwise velocity component (x-component) dimensionless velocity, Eq. (29) Cartesian coordinates
DT
g H
l q s u
temperature difference across the film dimensionless coordinate dimensionless temperature, Eq. (13) dynamic viscosity density shear stress, Eq. (8) nanoparticle concentration (volume fraction)
Subscripts f properties of fluid p properties of the nanoparticles w wall 1 outer boundary of the condensation film Acronyms HTC heat transfer coefficient
Greek symbols d condensate film thickness
profiles in the film of the condensate, flow rate through the film and the Nusselt number. The model of Nusselt [10] has proved to be an adequate and physically justifiable mathematical description of the film condensation of the moving vapor. Nanofluids possess unique properties, which motivate scientific community and industry to keep on intensive research of their fundamental aspects and practical applications. The most important properties of nanofluids are high thermal conductivity and low susceptibility of to sedimentation, fouling, erosion and clogging as compared to ordinary fluids with microparticles [11]. This inspired many promising applications of nanofluids, like those in nuclear energy, thermal management of systems with high dissipation rates of energy, cooling systems of electronic and optical devices, heat pipes and thermosyphons, as well as nanostructured materials and complex fluids [11–14]. It has been revealed that there is a slip between nanoparticles and the base fluid; however, in many cases nanofluids are modeled as uniform homogenous mixtures [14–25]. The works [17,26,27] demonstrated however that models employing the homogeneous (single-phase) flow approaches lead to underprediction of heat transfer rates between the nanofluids and solid bodies. Non-uniformity in nanofluids arises due to nanoparticle migration at the expense of the temperature and velocity gradients. As confirmed by experiments [18], the concentration gradient is also present in nanofluids. Works [19,20] provide more insight into theoretical aspects of non-uniformity in nanofluids. Self-similar analysis known as a powerful tool of theoretical investigations was used to model laminar and turbulent boundary layer flows of nanofluids in the works [26–30]. Properties of a nanofluid as functions of the nanoparticle concentration and temperature were modeled in the works [27,28] in order to investigate fluid flow, heat and mass transfer in a boundary layer over a flat plate. Symmetry analysis (Lie groups) enabled finding proper self-similar variables and functions specific for the problems studied in [26,27]. Work [28] represents a study of a natural convection flow of a nanofluid over a vertical plate. The authors [29] theoretically studied onset of convection in a horizontal nanofluid layer of finite depth. The laminar boundary layer, with the surface velocity set to be a function of the streamwise coordinate and all fluid properties being independent of the concentration of nanoparticles,
was investigated in the work [30]. As demonstrated in these theoretical works, an increase in the nanoparticle concentration causes enhancement of heat and mass transfer processes in nanofluids. As mentioned above, the number of the works devoted to the experimental study of the condensation heat transfer in the presence of nanoparticles is limited by the publications [7–9]. A theoretical insight into this problem is given by the authors in the recent investigation [31]. However, these studies examined the condensation of the stationary vapor. The present paper focuses on convective heat transfer in laminar flow of a film of a nanofluid over a flat surface under complicated velocity condition at the outer boundary of the film. The flow of the nanofluid results from condensation of a vapor of the same nanofluid that moves along the plate in the same direction as the nanofluid does. The plate is colder than the saturation temperature of the nanofluid vapor. To our knowledge, the stated problem has never been previously studied theoretically or experimentally. This paper deals with an investigation of the effect of nanoparticles on the velocity profiles and HTC in the nanofluid. While considering heat transfer problems, effect of the nanoparticle concentration on all thermophysical properties must be taken into account, since the HTC is affected not only by thermal conductivity, but also by flow structure, thickness of the condensation layer and other flow characteristics. We employ here the known relations between thermophysical properties and the nanoparticle concentration. The model proposed in the present work develops classical model of Nusselt [10] characterized above. The novelty of the present model lies in the inclusion of an equation for the nanoparticle concentration, which simulates heat transfer enhancement due to the addition of the nanoparticles and mass transfer effects. This novel model enabled deriving dimensionless parameters characterizing the effects of nanoparticles on fluid flow, heat and mass transfer processes. The model resulted also in the relations for the velocity profiles, condensate film thickness, mass flow rate through the film, as well as for the Nusselt number. The major influence of nanoparticles is incorporated via an additional equation for the nanoparticle concentration and functional dependence of the nanofluid viscosity, thermal conductivity and density on the nanoparticle concentration like in the work [32].
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where m = 2.5 [33]. The following boundary conditions serve as a closure for Eqs. (1)–(3)
u ¼ 0;
l
Fig. 1. Schematic layout of the problem.
2. Mathematical model Let us consider the laminar film condensation of a moving vapor near a flat wall, whose temperature is equal to Tw (Fig. 1). Vapor moves with a constant velocity u1 and contains nanoparticles with the concentration u1 . The film of condensate flows steadily under condition of zero pressure gradient along the x-coordinate. The thickness d of the film of condensate is small in comparison with the length of the plate, which enables us to use a boundary layer approximation. The vapor and condensate temperature at the outer border of the film y = d is equal to the saturation temperature T1 > Tw at a given pressure. The mass flow rate of the condensate in the film at its flow along the wall increases due to continuous vapor condensation. This phenomenon is taken into account in the mathematical model by the balance equation of the increase in the mass flow rate of the condensate. The condensation heat is withdrawn by the cold wall. The following assumptions were made in order to solve the stated problem: the inertia forces in the fluid are negligibly small in comparison with the viscosity forces; heat transfer across the film (along the y-coordinate) is much larger than convective heat transfer and thermal conductivity in the streamwise direction (along the x coordinate). The same is valid as applied to the mass transfer processes. The stress friction at the boundary of the fluid and vapor phases is equal to s1. The vapor density is much smaller than that of the condensate. Than the generalized mathematical model, which describes fluid flow, heat and mass transfer of a nanofluid in the condensate film can be written as follows:
d dy
l
du ¼0 dy
ð1Þ
0¼
d dT du dT DT dT dT þ qp cp DB k þ dy dy dy dy T 1 dy dy
ð2Þ
0¼
d du DT dT : DB þ dy dy T 1 dy
ð3Þ
In order to solve system (1)–(3), one needs to involve the following complementary equations
l¼
lf ð1 uÞm
;
q ¼ ð1 uÞqf þ uqp ; k ¼ kf
kp þ 2kf þ 2uðkp kf Þ ; kp þ 2kf uðkp kf Þ
ð4Þ ð5Þ ð6Þ
T ¼ Tw;
du ¼ s1 ; dy y¼d
du DT dT DB ¼ dy y¼0 T 1 dy y¼0 T ¼ T1;
at y ¼ 0;
u ¼ u1 at y ¼ d:
ð7Þ
ð8Þ
The last condition in Eq. (7) describes mathematically the fact that the total flux of nanoparticles on the wall at y = 0 (Stefan’s flow) due to the concentration gradient (the left-hand side) is equal to the total flux of nanoparticles due to the temperature gradient (the right-hand side) [34,35]. The mass flow rate through the film caused by the vapor condensation increases as the nanofluid flows downstream along the plate in the x-direction. This phenomenon is taken into consideration by the equation of the mass balance
dG ¼
qw dx; r
ð9Þ
Thus, we can say that, based on the model of Nusselt [10] and using the approach Buongiorno [24], we proposed a model for the film condensation of the moving vapor in the presence of nanoparticles. Following [10], the model does not take into account the inertial and convective terms. The main influence of the nanoparticles is incorporated via the equation for the concentration of nanoparticles and closing relations for the nanofluid properties (viscosity, density, thermal conductivity) in frames of the approach of Bonjurno [24]. In this approach, in contrast to the most of the models existing in the literature, we considered the effects of the nanoparticles through gradient terms and the non-linear characteristics of the nanofluid properties depending on the concentration, like e.g. in the work [32]. Consequently, the proposed model is a parabolic system of equations describing the hydrodynamic and thermal processes in frames of the boundary layer approximation. As it was shown in [26,27] on the basis of a strict analytical group analysis, the proposed approach describes correctly these processes in the parabolic approximation. The boundary condition for the concentration on the wall was proposed in the works [26,27]. It is set in such a way that it maintains an appropriate balance for the nanoparticle concentration. This condition regulates the relationship between the two diffusion mechanisms, i.e. thermophoretic and Brownian diffusion. At the outer boundary, a constant concentration of nanoparticles is defined, which initially corresponds to the concentration of the nanoparticles in the vapor, i.e. the ambience can be considered as a vapor–particle mixture with non-condensable particles. The boundary condition for the concentration on the outer boundary was not set in analogy with such conditions on the wall (Stefan or Ackermann correction), as it is usually done in the study of condensation of the vapor from the vapor–gas medium. Contrary to that, it was assumed that the concentration in the vapor remains unchanged over the entire ambience. Such a condition is adopted in the most of the studies on heat transfer in nanofluids. 3. Analytical soluton and analysis of results The integration of Eq. (3) results in
C ¼ DB
du DT dT þ ; dy T 1 dy
ð10Þ
where C is an integration constant; C = 0 with account for the boundary condition (7). Substituting Eq. (10) into Eq. (2), one can obtain
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d dT DT dT dT DT dT dT d dT þ qp cp ¼ : k k 0¼ þ dy dy dy dy T 1 dy dy T 1 dy dy
C1 ¼ 1 þ
3ðA þ 2u1 ÞðK 1Þ ; 2ðK þ 2Þ
319
ð19Þ
ð11Þ and the temperature distribution in the layer of condensate
H¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K þ 2 þ 3ðA þ u1 ÞðK 1Þ 3AðK 1Þð1 gÞð3AðK 1Þ þ 2ðK þ 2 þ 3ðK 1Þu1 ÞÞ þ ðK þ 2 þ 3ðK 1Þu1 Þ2 : 3AðK 1Þ
The integration of Eq. (11) and the reduction of the result to a dimensionless form yields
dH C1 ¼ hk þ2k þ2uðk k Þi ; p p f f dg
ð12Þ
ð20Þ
Using Eq. (19), one can finally transform the distribution of the nanoparticle concentration, Eq. (18), to the following form
kp þ2kf uðkp kf Þ
u¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3AðK 1Þð1 gÞð3AðK 1Þ þ 2ðK þ 2 þ 3ðK 1Þu1 ÞÞ þ ðK þ 2 þ 3ðK 1Þu1 Þ2 ðK þ 2Þ 3ðK 1Þ
where C1 is the integration constant, and
y g¼ ; d
T Tw T Tw H¼ ¼ ; T1 Tw DT
DT ¼ T 1 T w :
ð13Þ
Combining Eqs. (10) and (12), one can further obtain
du C1 þ A hk þ2k þ2uðk k Þi ¼ 0; p p f f dg
ð14Þ
kp þ2kf uðkp kf Þ
where
A¼
DT DT ; DB T 1
ð15Þ
is a dimensionless parameter expressing the relation between the mechanisms of the thermophoretic and Brownian diffusion. In the reality, the concentration of nanoparticles does not exceed 10%. Therefore, to simplify the integration, one can represent the denominator of the second term in Eq. (14) as the McLaren series
kp þ 2kf þ 2uðkp kf Þ 3ðK 1Þ u FKðK; uÞ ¼ 1 þ þ O½u2 ; K þ2 kp þ 2kf uðkp kf Þ ð16Þ
where
is normalized thermal conductivity of nanoparticles, i.e. the ratio of the thermal conductivities of nanoparticles kp and the fluid kf . The integration of Eq. (14) with allowance for Eq. (16) and the boundary condition (8) yields a profile of the concentration of nanoparticles in the film of condensate
u¼
lim u ¼ u1 :
ð22Þ
A!0
Fig. 2 demonstrates how the parameter A affects the profile of the nanoparticle concentration. It can be seen from Fig. 2 that an increase in the parameter A enhances the transport of nanoparticles towards the wall and increases nonlinearity of the nanoparticle concentration. At A = 0.1, the nanoparticle concentration profile is almost linear. For the case where the thermal conductivity of nanoparticles is close to that of fluid, i.e. K ? 1, distribution (21) exhibits a linear trend regardless of the value of the parameter A K!1
ð17Þ
Having obtained the profile of concentration (18), one can integrate Eq. (12). This integration in view of conditions (7) and (8) yields an expression for the integration constant
ð23Þ
Fig. 3 elucidates the effect of the parameter K on the nanoparticle concentration profile in the film. An increase in this parameter, as well as in the parameter A, leads to an increase in nonlinearity of the concentration profile. However, the effect of the parameter K is much weaker than the influence of the parameter A. It should be pointed out that the value of the concentration at the wall is not affected by the variation of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK þ 2ÞðK þ 2 þ 6AðK 1ÞC 1 Þ þ 6AðK 2 þ K 2Þðu1 AC 1 gÞ þ 9ðK 2 1Þu21 ðK þ 2Þ 3ðK 1Þ
ð21Þ
As one can see from Eq. (21), the value u1 of the nanoparticle concentration at the outer boundary of the film of condensate is a point of minimum. The nanoparticle concentration at the wall demonstrates a point of maximum resulting from the Brownian and thermophoretic diffusion and normalized thermal conductivity of nanoparticles K. As a result, an extremely thin porous layer is formed over the wall. This porous layer reduces overall thermal resistance, which correlates with the data of the work [7]. If A ? 0, the thermal diffusion effects are negligible, and function (21) tends to a constant value independent of the parameter K
lim u ¼ Að1 gÞ þ u1 :
kp K¼ : kf
:
:
ð18Þ
parameter K. The temperature profile (20) in the film of condensate under condition A ? 0 exhibits a linear trend regardless of the values K and u1
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Fig. 2. Effect of the parameter A on the nanoparticle concentration profile, K = 2, u1 = 0.01.
Fig. 5. Effect of the parameter K on the temperature profile, A = 0.9, u1 = 0.1.
increase in the values of both parameters A and K causes a reduction of the temperature gradient near the wall. Such effect of the parameter K immediately follows from Eq. (12). Further integration of Eq. (1) under the boundary condition (8) yields
s1 d ¼ l
du dg
ð26Þ
To undertake the second integration of Eq. (26), let us transform Eq. (4) for the viscosity using the Padé approximation. For this particular purpose, it is more convenient than the Taylor series. Imposing the simplest Padé approximation to Eq. (4), one can obtain
l¼ Fig. 3. Effect of the parameter K on the nanoparticle concentration profile, A = 0.9, u1 = 0.01.
limH ¼ g: A!0
ð24Þ
The temperature profile takes the same shape under condition K ? 1 at any arbitrary values of A and u1
limH ¼ g: K!1
ð25Þ
Figs. 4 and 5 depict the effect of the parameters A and K on the shape of the temperature profile. As seen from Figs. 4 and 5, an
Fig. 4. Effect of the parameter A on the temperature profile, K = 2, u1 = 0.1.
lf ð1 uÞm
lf 1 mu
;
ð27Þ
As demonstrated in the review paper [33], the value of m can vary over a wide range, so that the coefficient m can be further used as a parameter. The integration of Eq. (26) under the boundary condition (7) on the wall yields
h U ¼ 9A2 ðK 1Þ2 ð2mðð1 gÞG BÞ þ LÞ þ 6AðK 1Þ i ð2mðð1 gÞG BÞ þ LÞM þ 2mðG BÞM 2 h i = 27AðK 1Þ2 ð3AðK 1Þ þ 2MÞ ;
ð28Þ
where
Fig. 6. Effect of the parameter A on the velocity profile, K = 2, u1 = 0.1, m = 2.5.
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The next step is finding a solution for the heat transfer coefficient. Following Nusselt [10], let us derive a solution for the mass flow rate through the film of condensate
G¼
Z
d
qudy ¼
0
s1 d2 qf Z 1 U ½ð1 uÞ þ uRdg; lf 0
R ¼ qp =qf :
ð35Þ ð36Þ
Here parameter R defines the ratio of the densities of the nanoparticles qp and the fluid qf. The integration of Eq. (35) with allowance for Eqs. (21) and (28) yields a very cumbersome mathematical expression. Therefore, let us represent Eq. (35) as follows
G¼
Z
d
qudy ¼
0
Fig. 7. Effect of the parameter u1 on the velocity profile, A = 0.1,K = 2, m = 2.5.
s1 d2 qf FGðA; K; u1 ; m; RÞ; 2lf
ð37Þ
where the function FG(A, K, u1, m, R) has a very complicated analytical form. In the limiting case of K ?1, this function looks as
lim FGðA; K; u1 ; m; RÞ ¼ K!1
A ðR 1 þ mð3u1 3Ru1 2ÞÞ 3 þ ð1 mu1 Þð1 þ ðR 1Þu1 Þ
mA2 ðR 1Þ: 4
ð38Þ
In the case where R ? 1, the nanoparticle density approaches to the fluid density, and the Eq. (38) reduces to
2 lim FGðA; K; u1 ; m; RÞ ¼ 1 A u1 m: 3 K!1 R!1
Fig. 8. Effect of the parameter m on the velocity profile, A = 0.1, K = 2, u1 = 0.1.
U¼
ulf ; s1 d
B ¼ 2 þ K þ 3ðK 1ÞðA þ u1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9A2 ðK 1Þ2 ð1 gÞ þ 6AðK 1Þð1 gÞM þ M2 ;
ð29Þ ð30Þ
ð39Þ
It follows from here that in the absence of nanoparticles, i.e. at A = u1 = m = 0, the function FG is also equal to unity, and Eq. (37) transforms into the solution of Nusselt [10]. Substituting further Eq. (36) into the balance Eq. (9), one can derive a differential equation for the thickness of the condensate film
d FGðA; K; u1 ; m; RÞ dx ¼
!
s1 d 2 q f kw dH ¼ DT 2lf dr dg g¼0
kf dH : FKðK; uð0ÞÞDT dr dg g¼0
ð40Þ
ð31Þ
Assuming that the film of condensate starts developing from the front edge of the plate x = 0, a solution of Eq. (40) with zero boundary conditions
L ¼ 3ð2m 3 þ Kð3 þ mÞÞg;
ð32Þ
d ¼ 0 at x ¼ 0
M ¼ 2 þ K þ 3ðK 1Þu1 :
ð33Þ
takes the following form
G¼
Eq. (28) describes mathematically the dimensionless velocity profile in the film of the condensate. In the limit at A ? 0, we have
lim U ¼ gð1 mu1 Þ: A!0
ð34Þ
Thus the velocity profile at absence of nanoparticles is described by linear Eq. (34), i.e. dependence of Nusselt [10]. Figs. 6–8 describe effects of different factors on the shape of the velocity profile in the layer of condensate. As follows from Figs. 6– 8, an increase in any of the parameters A, u1 and n entails a decrease in the fullness of the velocity profile in the film of condensate. An increase in any of these three parameters results in increased the nanoparticle concentration near the wall followed with the increased viscosity in accordance with Eq. (4) and reduced velocity gradient in the near-wall region. The variation of the parameter K practically does not influence on the shape of the velocity profile.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6kf lf DTx FKðK; uð0ÞÞ dH 3 d¼ cs q1 u21 r qf FGðA; K; u1 ; m; RÞ dg g¼0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FKðK; uð0ÞÞ dH 3 ¼ d0 ; FGðA; K; u1 ; m; RÞ dg g¼0
ð41Þ
ð42Þ
where
cs ¼
2s1 q1 u21
ð43Þ
is the friction coefficient introduced by Nusselt [10], q1 and u1 are the vapor density and velocity at the interface between the film and the vapor, respectively,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6kf lf DTx 3 d0 ¼ cs q1 u21 r qf
ð44Þ
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Fig. 9. Effect of the parameter u1 on the normalized Nusselt number Nu/Nu0 at A ? 0, R = 3, m = 2.5.
is the thickness of the film of condensate in the absence of nanoparticles [10]. Using the temperature profile (20) together with Eq. (42), one can derive a relation for the HTC
h¼
" #2=3 kf dH FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : d0 dg g¼0
ð45Þ
Eq. (45) can be re-written in the form of the Nusselt number
" #2=3 hd0 dH Nu ¼ ¼ FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : kf dg g¼0
ð46Þ
In the absence of nanoparticles [34]
Nu0 ¼ 1:
ð47Þ
Fig. 11. Effect of the parameter A on the normalized Nusselt number Nu/Nu0. a) u1 = 0.01, b) u1 = 0.1. K = 6, m = 2.5.
With account for Eq. (46), the normalized Nusselt number can be written as
" #2=3 Nu dH ¼ FKðK; uð0ÞÞ ½FGðA; K; u1 ; m; RÞ1=3 : Nu0 dg g¼0
ð48Þ
Eq. (48) enables analyzing effects of nanoparticles on the heat transfer rate characterized by five non-dimensional parameters A,
Fig. 10. Effect of the parameter u1 on the normalized Nusselt number Nu/Nu0 at A ? 0. K = 4, m = 2.5.
K, u1, m and R. The numerical values of these parameters were evaluated using the data from the studies [24,33]. Fig. 9 demonstrates effects of the nanoparticle concentration at the interface between the condensate and vapor u1 on the normalized Nusselt number Nu/Nu0 at different values of the parameter K. Here A ? 0, which represents the case where the Brownian diffusion by far exceeds the thermophoretic diffusion. As can be seen from Eq. (22), the nanoparticle concentration in this case is practically constant across the film of condensate and equal to u1. As evident from Fig. 9, an increase in the nanoparticle concentration at the interface between the condensate and vapor u1 leads to heat transfer enhancement, which amplifies with an increase in the difference between thermal conductivities of the particles and the fluid. Although, as Fig. 4 demonstrates, an increase in the parameter K causes a decrease in the temperature gradient at the wall, while the normalized Nusselt number increases. Therefore, in view of the form of Eq. (48), it can be concluded that the main factor that causes heat transfer enhancement in this case is the overall increase in the thermal conductivity of the nanofluid, but not the change of the temperature profile shape. Fig. 10 exhibits the effect of the parameter R on the normalized Nusselt number, which demonstrates a sustainable increasing trend with an increase in the relative density of nanoparticles. The effect of the parameter A, i.e. the relation the thermophoretic and Brownian diffusion, on the normalized Nusselt number is outlined in Fig. 11a (at low u1 = 0.01) and Fig. 11b (at high u1 = 0.1) for different values of the parameter R. Fig. 11
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in the condensate film. Obviously, the growth of these parameters leads to the heat transfer augmentation because of the increase in the nanoparticle concentration in near wall region. As expected, an increase in the parameters R and K also increases the HTC, because of the increased effective thermal conductivity of the nanofluids. A larger parameter R effectively means the larger total density of a nanofluid, whereas the effective thermal conductivity is proportional to the density of the medium. An increase in the parameter K entails an increase of thermal conductivity of the nanoparticles in comparison with baseline thermal conductivity of the pure fluid. Obviously, the growth of the parameter K causes an increase in the effective thermal conductivity of nanofluid. 4. Conclusions
Fig. 12. Effect of the parameter m on the normalized Nusselt number Nu/Nu0. K = 4, R = 3.
demonstrates that an increase in the parameter A enhances heat transfer. This means that heat transfer is enhanced following the amplification of the thermophoretic diffusion or the suppression of the Brownian diffusion. This trend is in line with Eq. (21), which testifies that the amplification of the thermophoretic diffusion leads to the increase in the nanoparticle concentration near the wall followed with the heat transfer enhancement. It can be also concluded that the heat transfer enhancement caused by the increase in the parameter A stands in relation to the absolute value of concentration u1. At low levels of u1, the heat transfer enhancement is more significant than that at high levels of u1, or, in other words, the rate of increase in Nu/Nu0 subsides for the higher values of u1. For higher nanoparticle concentrations, one can observe stronger effects of the parameter R. Effect of the exponent m in Eq. (4) for the dynamic viscosity of the nanofluid on the normalized Nusselt number Nu/Nu0 is shown in Fig. 12. Heat transfer diminishes with the increasing exponent m. It can be explained by the fact that the viscosity increases with the increasing exponent m, which is accompanied with a decrease in the velocity gradient near the wall (Fig. 8). This entails a reduction in the transport of momentum and, hence, in heat transfer. The magnitude of the predicted heat transfer augmentation is consistent with the experimental data of the works [7–9]. The analytical study performed above enabled revealing the mechanisms of hydrodynamic and thermal processes at the condensation of a moving vapor with nanoparticles. The vapor was considered as a vapor–particle mixture with non-condensable particles. Diffusion processes of these particles are controlled by two diffusion mechanisms, i.e. the thermophoretic and Brownian diffusion. As a result of the diffusion processes, an extremely thin porous layer formed over the wall. This porous layer reduces overall thermal resistance, which agrees with the data of the paper [7]. The level of concentration of the nanoparticles in this layer depends on the counteraction of the thermophoretic and Brownian diffusion. The amplification of the thermophoretic diffusion leads to the increased concentration of the nanoparticles on the wall, while Brownian diffusion augmentation entails the opposite effect. The relationship between these two diffusion mechanisms are taken into account by the dimensionless parameter A. Diffusion processes affect strongly the hydrodynamic and thermal processes in the condensate layer. Strengthening of the thermophoretic diffusion fosters an increase in the concentration of the nanoparticles on the wall and consequently a reduction of the velocity gradient in the near-wall region. Therefore, the increase in the parameters A and u1 results in the decreased fullness of the velocity profile
The paper represents an approximate model for fluid flow and heat transfer at film condensation of moving vapor with nanoparticles near a flat plate. This model involves the Brownian and thermophoretic diffusion mechanisms, together with the effect of the nanoparticle concentration on the fluid properties. The novelty of this mathematical model as compared with the classical Nusselt problem [10] is the addition of an equation for the nanoparticle concentration, which describes the heat and mass transfer augmentation because of the presence of nanoparticles. The model enabled obtaining expressions for the profiles of the nanoparticle concentration, temperature and velocity in the film of condensate, as well as for the mass flow rate in the film and the Nusselt number. An analysis revealed that fluid flow and heat transfer in the film are affected by the following non-dimensional parameters: parameter A that stands for the relation between the thermophoretic and Brownian diffusion; the nanoparticle concentration in the vapor u1; the density R of the nanoparticles normalized by that of the fluid; the thermal conductivity of nanoparticles K normalized by that of the fluid; and the parameter m, which characterizes properties of the nanofluid viscosity. With each of the parameters A, u1, R and K increasing, heat transfer also increases. As an analysis of the effect of the parameter A demonstrated, the Brownian diffusion deteriorates heat transfer, while the thermophoretic diffusion enhances heat transfer. An increase in the exponent m deteriorates heat transfer, which is caused by the decrease in the momentum transfer near the wall. Relative increase in heat transfer at u1 = 0.1 can reach 20%, which is somewhat lower than heat transfer enhancement at forced convection over a flat plate [26]. Conflict of interest None declared. References [1] B.M. Ennis, C.T. Marshall, I.S. Maddox, A.H.J. Paterson, Continuous product recovery by in-situ gas stripping/condensation during solvent production from whey permeate using Clostridium acetobutylicum, Biotechnol. Lett. 8 (10) (1986) 725–730. [2] G.A. Mansoori, K. Schulz, E.E. Martinelli, Bioseparation using supercritical fluid extraction/retrograde condensation, Bio/Technology 6 (1988) 393–396. [3] M. Capellas, G. Caminal, G. Gonzalez, J. López-Santín, Pere Clapés, Enzymatic condensation of cholecystokinin CCK-8 (4–6) and CCK-8 (7–8) peptide fragments in organic media, Biotechnol. Bioeng. 56 (4) (1997) 456–463. [4] D.-W. Sun, L. Zheng, Vacuum cooling technology for the agri-food industry: past, present and future, J. Food Eng. 77 (2006) 203–214. [5] V. Mavrov, E. Bélières, Reduction of water consumption and wastewater quantities in the food industry by water recycling using membrane processes, Desalination 131 (1–3) (2000) 75–86. [6] A. Baudot, M. Marin, Pervaporation of aroma compounds: comparison of membrane performances with vapour–liquid equilibria and engineering aspects of process improvement, Food Bioprod. Process. 75 (2) (1997) 117– 142.
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