Fully developed mixed convection flow in a vertical channel filled with nanofluids

International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

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Fully developed mixed convection flow in a vertical channel filled with nanofluids☆ Hang Xu a,⁎, Ioan Pop b a b

State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP253, Romania

a r t i c l e

i n f o

Available online 17 June 2012 Keywords: Mixed convection Heat transfer Channel Nanoparticles Analytical solutions

a b s t r a c t In this paper, the fully developed mixed convection flow in a vertical channel filled with nanofluids is investigated. Analytical solutions for both the buoyancy-assisted and -opposed flow are obtained. Further analysis shows that the analytical solution for the opposing flow is only valid for a certain region of the Rayleigh number Ra in physical sense. Besides, the effects of the nanoparticle volume fraction φ on the temperature and the velocity distributions are then exhibited. It is confirmed that the nanoparticle volume fraction φ plays a key role for improving the heat and mass transfer characteristics of the fluids. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Vertical channel is a frequently encountered configuration in thermal engineering equipment, for example, collector of solar energy, cooling devices of electronic and micro-electronic equipments, etc. Due to its wide applications, numerous investigations have been done toward the understandings of fully developed mixed convection flow in a vertical channel. Reviews of the open literature reveal that there have been very much attention paid to fully developed mixed convection flow in vertical and inclined channels, such as Tao [1], Beckett [2] Beckett and Friend [3], Aung and Worku [4], Lavine [5], Cheng, Kou and Huang [6], Hamadah and Wirtz [7], Chen and Chung [8], Pan and Li [9], Barletta [10], Barletta, Magyari and Keller [11,12] and so on. Nanofluids (a term proposed by Choi [13]) are expected to have superior properties compared to conventional heat transfer fluids, as well as fluids containing micro-sized metallic particles. Wang and Mujumdar [14] pointed out that nanoparticles have the much larger relative surface area as compared to those of conventional particles, they therefore should not only significantly improve heat transfer capabilities, but also increase the stability of the suspensions. Choi et al. [15] showed that the addition of small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. On the other side, Buongiorno [16] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He has shown that in the absence of turbulent effects it is the Brownian diffusion and the thermophoresis that are important and he has written down conservation equations based on these two effects. However, several other ☆ Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. E-mail addresses: [email protected] (H. Xu), [email protected] (I. Pop). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2012.06.003

works (Keblinski et al. [17,18] and Prasher et al. [19]) proposed alternative mechanisms for the “abnormal increase” of the thermal conductivity and viscosity than those proposed by Buongiorno [16]. Further, it should be mentioned that Brinkman model and Maxwell model are correct especially for concentrations up to 0.8 (80%). For instance, by comparing the Brinkman model and experimental data which are supplied by Maïga et al. [20] and Polidori, Fohanno and Nguyen [21], one may find an augmentation of the dynamic viscosity of 30% at a 4% particle loading. More recently, Popa et al. [22] were making a comparison between the Maxwell model and the experimental data provided by Mintsa et al. [23] for thermal conductivity. Popa et al. [22] find that Maxwell's model strongly overestimates the thermal conductivity of the nanofluid. Some numerical and experimental studies on nanofluids can be found in Kang et al. [24], Abu-Nada [25], Khanafer et al. [26], Maïga et al. [27], Tiwari and Das [28], Oztop and Abu-Nada [29], Abu-Nada and Oztop [30], Ghasemi and Aminossadati [31], Mahmoudi et al. [32], Yacob et al. [33], Bachok, Ishak and Pop [34], Nield and Kuznetsov [35,36], Kuznetsov and Nield [37,38], etc. Daungthongsuk and Wongwises [39] studied the influence of thermophysical properties of nanofluids on the convective heat transfer and summarized various models used in literature for predicting the thermophysical properties of nanofluids. The comprehensive references on nanofluid can be found in the recent book by Das et al. [40] and in the review papers by Daungthongsuk and Wongwises [41], Trisaksri and Wongwises [42], Wang and Mujumdar [43,44], and Kakaç and Pramuanjaroenkij [45]. The aim of the present paper is the analysis of fully developed and laminar mixed convection flow in a vertical channel filled with nanofluids, the basic fluid being water. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. Both walls of the channel are kept at a temperature which increases or decreases linearly with the distance along the walls. The nanofluid model used is that proposed by Tiwari and Das [28] and the analysis is based on an analytical solution. The effects of buoyancy parameter and solid

H. Xu, I. Pop / International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

volume fraction are discussed about temperature and velocity distributions in the channel.

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subject to the boundary conditions uð−LÞ ¼ 0; uðþLÞ ¼ 0; T w ð−LÞ ¼ T 0 −C 1 L; T w ¼ T 0 þ C 1 L:

ð3Þ

2. Basic equations Consider the steady mixed convection flow, which is driven by an external pressure gradient and also by a buoyancy force, between two parallel long vertical plane walls filled with a nanofluid and separated by a distance 2 L. A coordinate system is chosen such that the x-axis is parallel to the gravitational acceleration vector g, but with the opposite direction. The y-axis is orthogonal to the channel walls, and the origin of the axes is such that the positions of the channel walls arey = − L and y = + L, respectively. A sketch of the system and of the coordinate axes is reported in Fig. 1. Following Tao [1], we assume that the temperature of the both walls is Tw(x) = T0 + C1x, where C1 is a constant, which is positive for buoyancy-assisted flow and negative for buoyancy-opposed flow, respectively, and T0 is the upstream reference wall temperature. The velocity field is given in this case by v(u, 0), so that the continuity equation reduces to ∂ u/∂ x = 0 and implies u = u(y). Also, the pressure gradient ∂ p/∂ y = 0 so that p = p(x) and dp/dx = constant. Using the nanofluid model proposed by Tiwari and Das [28], the momentum balance and energy equations according to the Boussinesq approximation can be written as

It is a common practice in channel flow studies to assume the mass flow rate as a prescribed quantity. Hence, the following average fluid velocity in the channel section will be considered as being prescribed 1 þL 1 L ∫ uðyÞdy ¼ ∫0 uðyÞdy: 2 L −L L

Um ¼

The physical quantities in Eqs. (1) and (2) are: φ is the nanoparticle volume fraction, βf and βs are the coefficients of thermal expansion of the fluid and of the solid, respectively, ρf and ρs are the densities of the fluid and of the solid fractions, respectively, μnf is the viscosity of the nanofluid and αnf is the thermal diffusivity of the nanofluid, which are given by μ nf ¼ 

ρ Cp

μf ð1−φÞ2:5



i d2 u h dp ; þ φ ρs βs þ ð1−φÞρf βf g ðT−T w Þ ¼ 2 dx dy

ð1Þ

αn f

∂2 T ∂T ¼u ; ∂x ∂y2

ð2Þ

knf  ; α nf ¼  ρ Cp

nf         k ks þ 2kf −2φ kf −ks nf   ; ¼ ð1−φÞ ρ C p þ φ ρ C p ; ¼  f s kf ks þ 2kf þ φ kf −ks

where μf is the dynamic viscosity of the base fluid and its expression has been proposed by Brinkman (1953), knf is the thermal conductivity of the nanofluid, kf and ks are the thermal conductivities of the base fluid and of the solid, respectively, and (ρ Cp)nf is the heat capacitance of the fluid nanofluid. It is worth mentioning that the expressions (5) are restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles. These equations were also used by Khanafer, Vafai and Lightstone [26], Oztop and Abu-Nada [29], and Abu-Nada and Oztop [30]. The thermophysical properties of fluid and nanoparticles are given in Table 1. We introduce now the following dimensionless variables y Y¼ ; L

x X¼ ; L

U ðY Þ ¼

u ; Um

θðY Þ ¼

T−T w ; C 1 L Re P r

P ðxÞ ¼

p ; ð6Þ ρf U 2m

where Re = Um L/νf is the Reynolds number and Pr = νf/αf is the Prandtl number. Substituting Eq. (6) into Eqs. (1) and (2), we get the following ordinary differential equations

x, u

y

y, v

2L

Tw(x) = T0 + C1 x

g

Tw(x) = T0 + C1 x

nf

;

ð5Þ

μ nf

o

ð4Þ

  i Ra 1 1 d2 U h dP ; þ ð1−φÞ þ φ ρs =ρf βs =βf θ¼ Re ð1−φÞ2:5 d Y 2 Re dX

ð7Þ

α nf d2 θ ¼ U; αf d Y 2

ð8Þ

subject to the following boundary conditions U ð−1Þ ¼ 0;

U ð1Þ ¼ 0;

θð−1Þ ¼ 0;

θð1Þ ¼ 0;

ð9Þ

Table 1 Thermophysical properties of fluid and nanoparticles given in [29].

Fig. 1. Schematic of the geometry and coordinate system.

Physical properties

Fluid phase (water)

Cu

Al2O3

TiO2

Cp (J/kg K) ρ (kg/m3) k (W/m K) α × 107 (m2/s) β × 105 (1/K)

4179 997.1 0.613 1.47 21

385 8933 400 11163.1 1.67

765 3970 40 131.7 0.85

686.2 4250 8.9538 30.7 0.9

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H. Xu, I. Pop / International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

along with the mass flux conservation relation (4), which becomes 1

∫0 U dY ¼ 1;

ð10Þ

where Ra = g βf C1 L 4/(νf αf) is the Rayleigh number. It should be mentioned that Ra > 0 for assisting flow and Ra b 0 for opposing flow, respectively. It should also be mentioned that for a regular fluid (φ = 0), Eqs. (7) and (8) reduce to those of Chen and Chung [8]. 3. Results analysis Differentiating U(Y) in Eq. (8) with respect to Y twice and then substituting it into Eq. (8), we obtain ‴′

θ þ α 1 ⋅θ ¼ α 2 ;

ð11Þ

where the unknown constants are determined using the boundary conditions (12) as  pffiffiffi i pffiffiffi 3 h pffiffiffi 2 βP cos 2 βP þ cosh 2 βP h pffiffiffi  pffiffiffi i ; α2 ¼ α r sin 2 βP −sinh 2 βP     pffiffiffi β β 2 cos pPffiffiffi cosh pPffiffiffi P P 2 2 h pffiffiffi  p i ; C1 ¼ C3 ¼ − ffiffiffi βP α r sin 2 βP −sinh 2 βP     pffiffiffi β β 2 sin pPffiffiffi sinh pPffiffiffi P P 2 2 h pffiffiffi  p i : C 2 ¼ −C 4 ¼ − ffiffiffi βP α r sin 2 βP −sinh 2 βP For Ra b 0, corresponding to α1 b 0. Similar to the procedures mentioned above, we obtain the analytical solution for Eq. (11) as

subject to the boundary conditions N

″

″

1

θð−1Þ ¼ θð1Þ ¼ θ ð1Þ ¼ θ ð−1Þ ¼ 0;

″

∫0 α r ⋅θ ðyÞ ¼ 1;

ð12Þ

   i 5 2 Re d P where α 1 ¼ ð1−φÞ ð1−φÞ þ ρρs ββs φ Ra α r , α 2 ¼ ð1−φÞ α r d X , and f f α nf αr ¼ α . f The analytical solutions for Eq. (11) satisfied the boundary conditions (12) which can be explicitly obtained for three different cases Ra = 0, Ra > 0 and Ra b 0, as illustrated hereinafter. For Ra = 0, we integrate Eq. (11) with respect to Y four times, obtaining 5 2

0

0

h

0

2

0

3

θðY Þ ¼ C 1 þ C 2 Y þ C 3 Y þ C 4 Y þ

α2 4 Y ; 24

ð13Þ

where the constants C10, C20, C30, and C40 and α2 are determined by the boundary conditions (12) as 0

C1 ¼ −

5 ; 8γ

0

C 2 ¼ 0;

0

C3 ¼

3 ; 4γ

0

C 4 ¼ 0;

3 α2 ¼ − : γ

ð14Þ

For Ra > 0, corresponding to α1 > 0, in this case, the homogeneous equation for Eq. (11) is ‴′

4

θ þ βP ⋅θ ¼ 0;

ð15Þ

where βP = α11/4. The characteristic equation for Eq. (15) can then be written as 4

4

r þ βP ¼ 0;

ð16Þ

which have four roots β r 1;2 ¼ pPffiffiffi ð1  iÞ; 2

β r 3;4 ¼ − pPffiffiffi ð1  iÞ: 2

ð17Þ

Thus the solution for Eq. (15) is given by    β β β P C P1 cos pPffiffiffi Y þ C 2 sin pPffiffiffi Y θ ðY Þ ¼ exp pPffiffiffi Y 2 2 2  β β β P P C 3 cos pPffiffiffi Y þ C 4 sin pPffiffiffi Y : þexp − pPffiffiffi Y 2 2 2

ð18Þ

It is obvious that θ(Y) = α2/βP4 is a particular solution of Eq. (11). Hence, the general solution of Eq. (11) reads    β β β P C P1 cos pPffiffiffi Y þ C 2 sin pPffiffiffi Y θðY Þ ¼ exp pPffiffiffi Y 2 2 2    β β β α P C P3 cos pPffiffiffi Y þ C 4 sin pPffiffiffi Y þ 42 ; þexp − pPffiffiffi Y βP 2 2 2

ð19Þ

N

N

θðY Þ ¼ C 1 expðβN Y Þ þ C 2 expð−βN Y Þ þ C 3 cosðβN Y Þ α N þ C 4 sinðβN Y Þ− 42 ; βN

ð20Þ

where 2 β3N ½1 þ expð2 βN Þ ; α r ½1−expð2 βN Þ þ tanðβN Þ þ tanðβN Þexpð2 βN Þ 1 C N1 ¼ C N2 ¼ ; 2 βN α r ½sinhðβN Þ−coshðβN ÞtanðβN Þ 1 N 1=4 C N3 ¼ ; C 4 ¼ 0; βN ¼ ð−α 1 Þ : βN α r ½cosðβN ÞtanhðβN Þ−sinðβN Þ α2 ¼ −

The temperature distribution of Al2O3–water fluid flow at the centerline of the channel as a function of the Rayleigh number Ra for some values of φ is plotted in Fig. 2. It is found from Fig. 2(a) that for the assisting flow case, as expected, the centerline temperature θ(0) enlarges gradually with increasing values of the Rayleigh number Ra. While for the opposing flow case, the centerline temperature θ(0) diminishes rapidly from its initial value to negative infinity as the Rayleigh number decreases from zero to a singular value Ras1. Beyond this value, θ(0) changes its sign and lessens continuously from positive infinity to negative infinity with the reduction of Ra, as shown in Fig. 2(b). Furthermore, due to Eq. (20), it is readily to know that there are multiple singularities, at those points θ(0) always approaches to infinity, as shown in Table 2. However the centerline temperature θ(0) cannot be infinite physically, we thereby are able to conclude that Eq. (20) is only valid for Ras1 b Ra b 0. Besides, the effect of φ on the centerline temperature distribution can be also seen in Fig. 2. From Fig. 2(a) and (b), we notice that the increasing values of φ result in the enhancement of the absolute values of θ(0) for any a prescribed value Ra. This provides a theoretical basis from mathematical point of view that the nanofluids can effectively improve on the heat transfer characteristics compared to the traditional fluids. The temperature distribution θ(Y) for various values of Ra with φ = 0.1 in Al2O3–water nanofluid is presented in Fig. 3. It is shown from Fig. 3(a), the temperature profiles θ(Y) grow monotonously with the increasing of the Rayleigh number Ra for the assisting flow case. Similar situation can be found for the opposing flow case, the temperature profiles θ(Y) increase consecutively as the Rayleigh number Ra evolves from Rac1 to zero, as shown in Fig. 3(b). Fig. 4 gives the velocity profiles for various values of Ra with φ = 0.1 in Al2O3–water nanofluid. As illustrated in Fig. 4(a), for the assisting flow case, the velocity profiles U(Y) decrease smoothly as the Rayleigh number Ra enlarges from 0 to a critical value Ra1(≈46.8602). While when Ra constantly enlarges, the centerline temperature U(0) diminishes unceasingly as before, synchronously, two crests alongside of the centerline come to appear and gradually enhance. As Ra approaches to another critical value Ra2(≈3795.6735), the

H. Xu, I. Pop / International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

(a) Ra 0

1089

centerline velocity U(0) reaches to its a minimum value. When Ra continuously evolves, the centerline temperature U(0) begins to increase again, while the amplitude of the existing two crests begins to decrease, at the same time two troughs beside the centerline start to appear. For other values φ in various nanofluids, such critical values of Ra can also be found as expected, as shown in Table 3. From the table, it is further found that both Ra1 and Ra2 increase with the increase of φ. For the opposing flow case, the centerline velocity U(0) reduces gradually with the increasing values of Ra from −400 to zero. Simultaneously, the amplitude of the two troughs alongside of the centerline decreases and finally is vanishing as Ra approaches to zero, as shown in Fig. 4(b). The effect of φ on the temperature profiles for Ra = ± 50 in Al2O3– water nanofluid is exhibited in Fig. 5. It is shown from Fig. 5(a) that the temperature profiles θ(Y) increases monotonously for the assisting flow case with the increasing values of φ. Similar trend can also be seen for the opposing flow case, the increasing values of φ

0

-0.1

-0.2

θ(0)

-0.3

-0.4

-0.5

-0.6

-0.7 0 10

10

1

10

2

10

3

10

4

(a) Ra 0

Ra

0

(b) Ra 0

-0.05

3

-0.1 10000

2

-0.15 2000

-0.2

θ(0)

θ(Y)

1

0

-0.25

800

-0.3

400

-0.35 -1

200

-0.4

100 50

-0.45

0

-2 -0.5 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

Y

-3 0 10

10

1

10

2

10

3

10

4

Ra

(b) Rac1 Ra 0 0

Fig. 2. The values of θ(0) for various values of Ra in Al2O3–water fluid. Dash–dot-dotted line: φ = 0.0; dashed line: φ = 0.1; solid line: φ = 0.2. 0

-0.5

-200

-1

Types of fluids

Singularities

φ = 0.0

φ = 0.1

φ = 0.2

Al2O3–water

Ras1 Ras2 Ras3 Ras4

− 237.903 − 2498.4 − 10875.9 − 31804.4 … − 237.903 − 2498.4 − 10875.9 − 31804.4 … − 237.903 − 2498.4 − 10875.9 − 31804.4 …

− 457.438 − 4803.9 − 20912.1 − 61153.2 … − 432.018 − 4536.94 − 19750. − 57754.9 … − 457.438 − 4803.9 − 20912.1 − 61153.2 …

− 902.185 − 9474.52 − 41244. − 120610. … − 797.644 − 8376.66 − 36464.8 − 106634. … − 839.29 − 8814.02 − 38368.7 − 112202. …

Cu–water

TiO2–water

… Ras1 Ras2 Ras3 Ras4 … Ras1 Ras2 Ras3 Ras4 …

θ(Y)

Table 2 The singular points given by Eq. (20).

-300

-350

-1.5 -380

-2

-400

-2.5

-3 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

Y Fig. 3. The temperature profiles for various values of Ra with φ = 0.1 in Al2O3–water fluid.

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H. Xu, I. Pop / International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

(a) Ra 0

4. Conclusion An investigation has been made of the flow subject to an external pressure gradient and a buoyancy force between two parallel vertical plane surfaces filled with a nanofluid oriented in the direction of the generating body force along which the temperature distribution is assumed to vary linearly. Analytical solutions for both the buoyancyassisted flow and the buoyancy-opposed flow have been obtained explicitly. An analysis shows that the analytical solution for the opposing flow case is not physically realistic when Ra b Ras1. While for the assisting flow case, we find that there are two critical values Ra1 and Ra2, that divide Ra into two regions Ra1 ≤ Ra b Ra2, and Ra ≥ Ra2. In each region, the velocity profiles remain similar shape for various Ra. While the shape of velocity profiles is different when Ra falls into the other regions. Besides, the nanoparticle volume fraction φ is found to be of significance in this problem, which has nonnegligible effects on the improvement of the heat and mass transfer characteristics of the fluids.

(a) Ra=50

(b) Rac1 Ra 0

Fig. 4. The velocity profiles for various values of Ra with φ = 0.1 in Al2O3–water fluid.

(b) Ra = -50

results in the enhancement of the temperature profiles, as shown in Fig. 5(b). The velocity distribution for various values of φ with Ra = ± 50 in Al2O3–water fluid is plotted in Fig. 6. It is shown from Fig. 6(a) that the centerline velocity U(0) increases consecutively as φ enlarges for the assisting flow case. While for the opposing flow case, the centerline velocity U(0) decreases continuously as φ evolves, as shown in Fig. 6(b).

Table 3 The critical Rayleigh numbers Ra1 and Ra2. Types of fluids Ra1 φ = 0.0 Al2O3–water Cu–water TiO2–water

Ra2 φ = 0.1

φ = 0.2

φ = 0.0

φ = 0.1

φ = 0.2

24.3709 46.8602 92.4203 1974.0428 3795.6735 7486.0445 24.3709 44.2561 81.7111 1974.0428 3584.7457 6618.5964 24.3709 45.1399 85.9773 1974.0428 3656.3292 6964.1645

Fig. 5. The temperature profiles for various values of φ in Al2O3–water fluid.

H. Xu, I. Pop / International Communications in Heat and Mass Transfer 39 (2012) 1086–1092

(a) Ra=50

(b) Ra=-50

Fig. 6. The velocity profiles for various values of φ in Al2O3–water fluid.

Acknowledgment We acknowledge Dr. Qiang Sun for fruitful discussions. H. Xu extends his sincere appreciation to the National Natural Science Foundation of China for the support through grant no. 10972136 and grant no. 50739004, and to the Higher Education of China for the support through grant no. 20090073120014.

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