ISSN 0018-151X, High Temperature Pleiades Publishing, Ltd., 2017

Numerical Investigation of Heat Transfer Enhancement in a Square Ventilated Cavity with Discrete Heat Sources Using Nanofluid1 H. Moumni*, H. Welhezi, and E. Sediki Thermal Radiation Laboratory, Fac. Sci. of Tunis, University of Tunis El Manar, Tunisia *e-mail: [email protected] Received March 14, 2016

Abstract⎯A numerical study of a laminar mixed convection problem in a ventilated square cavity partially heated from bellow is carried out. The fluid in the cavity is a water-based nanofluid containing Cu nanoparticles. The effects of monitoring parameters, namely, Richardson number, Reynolds number, and solid volume fraction on the streamline and isotherm contours as well as average Nusselt number along the two heat sources are analyzed. The computation is performed for Richardson number ranging from 0.1 to 10, Reynolds number from 10 to 500, and the solid volume fraction from 0 to 0.1. The results show that by adding nanoparticles to the base fluid and increasing both Reynolds and Richardson numbers the heat transfer rate is enhanced. It is also found, regardless of the Richardson and Reynolds numbers, and the volume fraction of nanoparticles, the highest heat transfer enhancement occurs at the left heat source surface. DOI: 10.1134/S0018151X17030166

Mixed convection problem in lid-driven cavities using nanofluids has been the subject of many studies [16–18]. For instance, in [19] authors used the finite volume method to investigate flow and heat transfer in a square cavity with a heat source on the bottom wall using Al2O3–water nanofluid. They observed that when the heat source was located in the middle of the bottom wall and the Ra was kept constant, the effect of nanoparticles on heat transfer enhancement increased with an increase in Re. Tiwari et al. [20] conducted a numerical investigation of mixed convection in twosided lid-driven differentially heated square cavity filled with copper–water nanofluid. In order to study the effect of the wall movement in different directions, three different directions were considered. Their results showed that both Ri and the direction of the moving walls affect the fluid flow and heat transfer in the cavity. An experimental investigation of the effects of nanoparticles concentration and size on nanofluid viscosity in a wide temperature range was carried out in [21, 22]. It was shown that viscosity decrease sharply with temperature particularly for high concentration of nanoparticles. Sourtiji et al. [23] performed a numerical study of the mixed convection flows through an alumina– water nanofluid inside a square ventilated cavity due to incoming flow oscillation. It was found that after a certain time duration, a periodic variation in the fluid flow and temperature field in the cavity is created due to the oscillating velocity at the inlet port. Mahmoudi et al. [24] reported results of a numerical simulation of mixed convection flow of copper–water nanofluid in

INTRODUCTION Mixed convection fluid flow and heat transfer in a ventilated cavities has received considerable attention over the past several years due to their increasing number of applications in many industrial transport processes and in the engineering devices, such as the nuclear reactors, design of solar collectors, and the cooling of microprocessors and electronic components. But low thermal conductivity of conventional fluids, such as water, oil, and ethylene glycol is a major limitation of heat transfer enhancement in places where there is a need of high heat flux (for example, ventilated cavity and heat exchangers). To overcome this drawback, an innovative technique of heat transfer enhancement can be designed by using nanometersized particles (<100 nm) dispersed in conventional fluids. The nanofluids, which consist of suspended metallic nanoparticles, have proven to have high thermal conductivity and to be very stable [1–5]. Such fluid might be able to meet the rising demand of an efficient heat transfer agent. The presence of nanoparticles in nanofluid increases appreciably the effective thermal conductivity of the fluid and consequently enhances the heat transfer characteristics [6–12]. Over the past several years, many researches have been conducted in order to simulate the effect of suspended nanoparticles on the convective heat transfer process within the cavity. The analysis of natural convection heat transfer and fluid flow in enclosures filled with nanofluid has been reported by many authors [13–15]. 1 The article is published in the original.

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the effect of the presence of nanoparticles on the flow and heat transfer characteristics for the mixed convection problem in a vented square cavity filled with Cu– water nanofluid and heated locally from below.

L

Nanofluid

Tc

g Tc

δ2

d1 y

uin Tc

0

x

δ1

L

d2 eS1

In this study the effective thermal conductivity of nanofluids calculated with a model proposed by Maxwell [28]. To determine the viscosity of nanofluid, a model given in [29] is used. The consequence of varying the Reynolds, Richardson numbers, and the nanoparticles volume fraction on the hydrodynamic and thermal characteristics are investigated and discussed.

eS2

PROBLEM DEFINITION AND MATHEMATICAL FORMULATION Fig. 1. Schematic of the problem.

a vented square cavity with adiabatic side and top walls and constant heat flux at the bottom. Results showed that both Re and Ri and also the solid concentration largely influence thermal characteristics and the flow field. In [25], authors found that an increase in the copper volume fraction increases the rate of heat transfer for all values of Ri. Recently Khorasanizadeh et al. [26] extended the work in [25] and studied mixed convection flow of Cu–water nanofluid in a square ventilated cavity at different inclination angles. It was found that the heat transfer across the cavity could be either enhanced or mitigated depending on the inclination angle and Ri. In many studies of mixed convection phenomenon in a ventilated cavity, the wall of cavity is fully or partially heated from below with single constant temperature or constant heat flux source. While in many applied cases, such as cooling of electronic components, electronic devices are treated as heat sources which are incorporated on the surface [27]. To the best of knowledge of the authors, the effect of nanoparticles volume fraction on mixed convection in a ventilated square cavity with two discrete heat sources on the bottom has not been largely investigated. Therefore, the purpose of this work is to study Table 1. Thermophysical properties of water and copper [30] Property

Water

Copper

ρ

997.1

8933

Cp

4179

385

k

0.613

401

β

21 × 10–5

1.67 × 10–5

Figure 1 shows the physical model of the problem under investigation. It consists of a two-dimensional square enclosure with inlet and outlet ports. The bottom wall is discretely heated at constant temperature Th. The two heat sources are of the length es = L/5 and are located at the distance d = L/5 from the side cold walls (Tc ).The remaining parts of the bottom wall and the ceiling are considered adiabatic. Width of the inlet and outlet ports is identical δ = L/10. The flow enters into the cavity at a constant temperature Tc and uniform velocity. The fluid in the cavity is a water-based nanofluid containing Cu nanoparticles. It is assumed that both the fluid phase and nanoparticles are in thermal equilibrium and no slip occurs between them. Also it is assumed that the nanofluid is Newtonian and incompressible and the flow is laminar. The thermophysical properties of the nanofluid are listed in Table 1. The thermophysical properties of nanofluid used in this study are assumed to be constant except for the density varying in the buoyancy force terms which is determined using the Boussinesq approximation. In this section, the continuity, momentum, and energy equations for the steady mixed convection flow are presented in a dimensionless form. For this purpose the following dimensionless variables are introduced:

y X = x, Y = , U = u , V = v , L L uin uin T − Tc tuin p , τ= , and P = . θ= T h − Tc L ρ nf uin2 Based on the assumptions above, the dimensionless governing equations can be expressed in the incompressible unsteady form as follows: HIGH TEMPERATURE

NUMERICAL INVESTIGATION OF HEAT TRANSFER ENHANCEMENT

∂ U + ∂ V = 0, ∂X ∂Y ∂ U + ∂ (UU ) + ∂ (UV ) ∂τ ∂X ∂Y ρ ⎛ ∂ 2U ∂ 2U ⎞ 1 = − ∂P + 1 f + ⎟, 2.5 ⎜ ∂ X Re ρ nf (1 − ϕ) ⎝ ∂ X 2 ∂ Y 2 ⎠

(1)

and Y = 0;

⎧0 ≤ X ≤ D ⎪ ∂θ U = V = 0, = 0 for ⎨D + ε ≤ X ≤ 1 − ( D + ε ) ∂Y ⎪1 − D2 ≤ X ≤ 1 ⎩ and Y = 0. The heat transfer rate is evaluated by computing the average Nusselt number Nu through the two heat surfaces. The local Nusselt number on each heat source surface can be defined as:

hnf L , kf

where hnf is the heat transfer coefficient HIGH TEMPERATURE

(4)

(Th − Tc ) ∂θ (5) . ∂ Y Y =0 L After substituting Eqs. (5) and (4) into Eq. (3), the local Nusselt number can be written as: q = −k nf

∂ V + ∂ (UV ) + ∂ (VV ) ∂τ ∂X ∂Y ρ ⎛ ∂ 2V ∂ 2V ⎞ 1 = − ∂P + 1 f + ⎜ ⎟ ∂ Y Re ρ nf (1 − ϕ) 2.5 ⎝ ∂ X 2 ∂ Y 2 ⎠ (2) ρ ⎛ ρβ ⎞ + Ri f ⎜1 − ϕ + s s ϕ ⎟ θ, ρ nf ⎝ ρ fβ f ⎠ ∂θ + ∂ (U θ) + ∂ (V θ) = α nf 1 ⎛ ∂ 2θ + ∂ 2θ ⎞ . ⎜ ⎟ ∂τ ∂X ∂Y α f Re Pr ⎝ ∂ X 2 ∂ Y 2 ⎠ The boundary conditions associated with the problem are as follows: U = 1, V = 0, θ = 0 for X = 0 and 0 ≤ Y ≤ H ; U = V = 0, θ = 0 for X = 0 and H ≤ Y ≤ 1; U = 0, V = 0, θ = 0 for X = 1 and 0 ≤Y ≤ 1−H; 2 2 ∂ U = ∂θ = 0, V = 0 for X = 1 and ∂X ∂X 1−H ≤Y ≤ 1+H; 2 2 2 2 U = 0, V = 0, θ = 0 for X = 1 and 1 + H ≤ Y ≤ 1; 2 2 U = V = 0, ∂θ = 0 for 0 ≤ X ≤ 1 and Y = 1; ∂Y ⎧D ≤ X ≤ D + ε U = V = 0, θ = 1 for ⎨ ⎩1 − ( D + ε ) ≤ X ≤ 1 − D

Nu =

q . T h − Tc Here, q is the wall heat flux per unit area: hnf =

3

(3)

Nu = −

( )

k nf ∂θ . k f ∂Y

The average Nusselt numbers along the heat sources S1 and S2 are obtained by integrating the local Nusselt number along each heat source [31]: D +ε

Nu S1 =

∫

Nu dX ,

D 1− D

Nu S 2 =

∫

(10) Nu dX .

1−(D +ε)

NUMERICAL METHOD, GRID INDEPENDENCE TEST, AND CODE VALIDATION The unsteady Navier–Stokes and energy equations are discretized on a staggered, non-uniform Cartesian grid using a finite volume approach. The procedure adopted here deserves a detailed explanation: the nonlinear terms in Eq. (1) are treated explicitly with a second-order Adams–Bashforth scheme. The convective terms in Eq. (2) are treated semi-implicitly and the diffusion terms in Eqs. (1) and (2) are treated implicitly. The time derivatives in momentum and energy equations are performed by an Euler backward second-order implicit scheme. In order to minimize the numerical diffusion errors, Quadratic Upstream Interpolation for Convective Kinetics (QUICK) [32] and second-order central differencing schemes (CDS) are used to approximate the advection and diffusion terms, respectively. To overcome the pressure–velocity coupling, we use the projection method [33, 34] which is one of the fractional step methods or pressure correction class methods. Further details on projection methods are provided by Chorin [35, 36]. The set of resulting algebraic equations are solved iteratively and are performed by the Red and Black Successive over Relaxation method (RBSOR) [37]. Such a procedure is often employed to avoid divergence in the iterative solution of algebraic equations and to accelerate the convergence. The convergence of iteration for the solution is obtained until the values of the variables cease to change from previous iteration to the next by less than the prescribed value. To examine whether the solution can be or not considered as steady, the following criterion is adopted:

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Table 2. Result of grid independence examination (Cu– water, Ri = 1, Re = 100, ϕ = 0.1) Number of grids in X–Y 020 × 20 040 × 40 060 × 60 080 × 80 100 × 100

Nu S1

Nu S 2

5.8975 5.7487 (2.58%) 5.6565 (1.60%) 5.6404 (0.28%) 5.6415 (0.01%)

2.1579 2.3587 (9.30%) 2.3714 (0.53%) 2.3803 (0.37%) 2.3826 (0.09%)

Table 3. Comparison of Nu at different Ra

Ra

103

10 4

105

106

107

Present work [38] [39] [40]

1.117 1.118 1.117 1.121

2.244 2.243 2.246 2.286

4.521 4.519 4.518 4.546

8.824 8.800 8.792 8.652

16.526 – 16.408 16.790

Table 4. Comparisons of Nu at Ra = 103,ϕ = 0.1, B = 0.4

Nanoparticles material Present work [41] [42]

Cu

Al2O3

Ag

TiO4

5.462 5.462 5.402 5.200 5.459 5.459 5.398 5.197 (0.054%) (0.054%) (0.074%) (0.057%) 5.451 5.451 5.391 5.189 (0.201%) (0.201%) (0.204%) (0.211%)

∑Φ

m +1 i, j

−6

− Φ i, j ≤ 10 . m

i, j

Here, the generic variable Φ represents the set of three variables U ,V , or θ, the superscript m indicates the iteration number and the subscript sequence (i, j ) represents the space coordinates X and Y. Grid Independence Test In order to take account of the presence of large gradients near the walls, we generate a non-uniform grid mesh which is finer near the boundaries using following grid point distribution:

(

)

tanh ⎡ 2i − 1 arctanh(α x )⎤ ⎢⎣ N ⎥⎦ x (i ) = 1 + , 2 2α x where α x = 0.9 , the variable N represents the grid point, and 1 ≤ i ≤ N refers to the index node. Similar

grid point distribution is used in the second direction of the cavity. For the purpose of obtaining a grid independent solution, a grid sensitivity analysis is performed for the mixed convection heat transfer in the cavity (Fig. 1). Five grids were tested ranging from a coarse grid (20 × 20) to a much finer one (100 × 100). Table 2 depicts the corresponding average Nusselt numbers (relative difference between results is given in brackets). It clearly shows that the difference between the results obtained for the 80 × 80 and 100 × 100 grids is less than 0.01% for Nu along the left heat source and 0.09% for the right one. For accuracy, required in reasonable computation time, the grid of 80 × 80 nodes is adopted for all computations presented in this study. Note that computations are carried out with a time step close to 10–3.

Code Validation In order to establish the code credibility, we validated the developed code based on some published results in the literature. Case of the natural convection flow in an enclosed cavity filled with pure fluid is tested first; results are gathered in Table 3 for average Nusselt number along the heated wall of the cavity. An excellent agreement is shown with former results of different approaches. Results for different nanoparticles materials are gathered in Table 4. The computed Nu along the heat source surface exhibit excellent agreement with the previous studies. RESULTS AND DISCUSSION In this study, the mixed convection flows through a copper–water nanofluid in a vented square cavity is numerically investigated. The computational results are obtained for 10 ≤ Re ≤ 500 , 0.1 ≤ Ri ≤ 10 , and 0 ≤ ϕ ≤ 0.1. The Prandtl number of the base fluid (water) is equal to 6.2. The streamlines contours corresponding to steady state cases at Re = 10, Ri = 0.1, 1, 10 for both pure fluid and ϕ = 0.1 are shown in Fig. 2. At Re = 10 and low value of Richardson 0.1, the flow nearly occupies the whole cavity and three small rotating vortices are formed in the corners of the cavity. These are two counter clockwise (CCW) rotating vortices located in the two top corners and one clockwise (CW) rotating vortex in the bottom right corner. When Ri rises from 0.1 to 1 the CCW rotating cell becomes larger and squeezes the induced flow path, while the recirculation cell located at the right top corner of the cavity vanishes. Furthermore, as Ri increases to 10, the size of the rotating vortices is sharply increased, what leads to a large change in the streamlines structure. The results show that regardless of the value of Ri, the addition of nanoparticles leads to a minor difference in HIGH TEMPERATURE

NUMERICAL INVESTIGATION OF HEAT TRANSFER ENHANCEMENT

1.0

ϕ=0

Ri = 0.1

ϕ = 0.1

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 1.0

Ri = 1

0 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 1.0

Ri = 10

0 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

0

ϕ=0

Ri = 0.1

5

ϕ = 0.1

Ri = 1

Ri = 10

0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

Fig. 2. Streamlines contours in the cavity at different Ri and Re = 10.

Fig. 3. Streamlines contours in the cavity at different Ri and Re = 500.

the streamlines contour compared to the case of ϕ = 0 (pure water).

The influence of Ri and solid volume fraction on heat transfer in an enclosure for various Re is displayed in Figs. 4 and 5. At Re = 10 and Ri = 0.1 (Fig. 4), the isotherms are distributed uniformly and nearly parallel to the two heat sources for both ϕ = 0 and 0.1, indicating that conduction is the dominant mechanism for heat transfer in the cavity. In this mode of heat transfer in the range of Ri, the higher thermal conductivity of particles has no significant influence on thermal field in the cavity in comparison with pure water. As Ri increase to 10, the flow intensity augments, and isotherms become nonlinear pointing to the enhancement of buoyancy effect, so that the convection mode is pronounced. Further increase in the solid concentration leads to an increase in the heat transfer between heat sources and working fluid.

Figure 3 illustrates the streamlines contours at Re = 500 and various Ri for pure water and nanofluid with ϕ = 0.1. It is obvious that the increase in Re leads to a great change in the flow pattern. The analysis of streamline structure at Re = 500 reveals the existence of three recirculation cells due to the higher energy of the induced flow. In this case the fluid flow is characterized by two small CW rotating vortices located at the top left corner and at the bottom right corner. A remarkable strong CCW rotating cell is developed above the open lines of the inlet flow. On the other hand, it is clearly discernible that an increase in Ri produces a change in the streamlines structure. As Ri rises from 0.1 to 10, the CCW rotating cell enlarges and compresses the induced flow more, indicating the domination of natural convection in the upper part of the cavity. A comparison between the streamlines corresponding to the pure fluid and nanofluid shows that the size of the rotating cell is slightly increased with increase in solid volume fraction. HIGH TEMPERATURE

Figure 5 shows the isotherms contour plots as a function of Ri and solid fraction at Re = 500. In this case the isotherms become more tightened at the vicinity of the two heat sources. From this figure, it is noticed that at Ri = 0.1, the isotherms are nonlinear and bunched near the heat sources. As Ri increases, the nonlinearity of the isotherms become higher, indi-

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MOUMNI et al.

1.0

ϕ=0

Ri = 0.1

ϕ = 0.1

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 1.0

Ri = 1

0 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

Ri = 10

1.0

0 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

0

ϕ=0

Ri = 0.1

ϕ = 0.1

Ri = 1

Ri = 10

0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

Fig. 4. Isotherms contours in the cavity at different Ri and Re = 10.

Fig. 5. Isotherms contours in the cavity at different Ri and Re = 500.

cating that the natural convection heat transfer is well established. The isotherms are clustered heavily near the left and the bottom surface of the cavity, which indicates that the temperature gradients in this region are steeper. As for the effect of the nanoparticle volume fraction ϕ, a comparison between the isotherms corresponding to the pure fluid and nanofluid shows that by increasing ϕ from 0 to 0.1, the nanofluid does not have a considerable effect on thermal field in the cavity. The results clearly show that the presence of nanoparticles in the base fluid, have no significant effect on the patterns of isotherms. Variation of the heat transfer rate quantified by the average Nusselt number along two heat sources versus volume fraction of nanoparticles and Richardson numbers at different values of Re is shown in Fig. 6. For all range of Ri considered and for each Re value, the addition of nanoparticles suspension to pure water has a significant effect on the heat transfer characteristics. In fact, increasing ϕ at constant Re and Ri enhances considerably the heat transfer rate from the

heat sources. This is due to the higher thermal conductivity of nanofluid. In addition, when keeping Re constant, the average Nusselt number along the two heat sources increases sharply with increasing Ri for pure water as well as for nanofluid. It can also be seen that Nu at the heated sources increases as Re increases with fixed Ri. However, the effect of addition of nanoparticles on Nu is more significant for higher Re. So that at Ri = 10 the increase in ϕ up to 0.1 leads to an increase in the heat transfer rate for the left heat source by about 17% for Re = 10 and by 25% for Re = 500. Therefore, it can be concluded that the highest heat transfer enhancement occurs in the case of high Re, Ri, and ϕ. The same phenomena can be observed for the right heat source (S2), but a precise observation can be achieved by examining the values of Nu along each heat source. The highest heat transfer enhancement occurs along the left heat source, while the right source heat transfer is lower. This is because the induced cold fluid flows over the left heated surface. HIGH TEMPERATURE

NUMERICAL INVESTIGATION OF HEAT TRANSFER ENHANCEMENT

Re = 10 NuS1 2.4 2.2

Ri = 0.1 Ri = 1 Ri = 10

Re = 500

NuS2 1.5

NuS1 14

1.4

13

1.3

12

2.0

1.2

1.8

1.1

1.0 0 0.02 0.04 0.06 0.08 0.10 0 0.02 0.04 0.06 0.08 0.10 ϕ ϕ

7

NuS2 8 6

11

4

10 2 0 0.02 0.04 0.06 0.08 0.10 0 0.02 0.04 0.06 0.08 0.10 ϕ ϕ

Fig. 6. Variation of Nu along the two heat sources (S1, S2) with respect to ϕ and Ri at different values of Re.

CONCLUSIONS

REFERENCES

In this paper, a numerical study was performed in order to investigate the effect of copper–water nanofluid on mixed convection heat transfer in a vented square cavity with two heat sources in the bottom wall using the finite volume method. A parametric study was undertaken and effects of solid volume fraction, Reynolds and Richardson numbers were investigated. Results clearly indicate that the addition of Cu nanoparticles in a base fluid produces a considerable enhancement of heat transfer with respect to that of the pure fluid. As Reynolds number increases, the effect is more pronounced. The results also indicate that the Nusselt number is an increasing function of the Richardson and Reynolds numbers and the solid volume fraction. The highest heat transfer enhancement occurs at the left heat source surface.

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Nomenclature: Cp—specific heat, J kg−1 K–1, d— distance of the heat source from the wall, m, D— dimensionless distance of the heat source from the wall d/L = 0.2, es—length of the heat source, m, H— dimensionless width of port δ L, G—gravitational acceleration, Gr = gβΔ TL3 ν 2f —Grashof number, k—thermal conductivity, W m–1 K–1, L—enclosure width, m, p—pressure, Pa, Pr = ν f α f —Prandtl number, Re = uin L ν f —Reynolds number, Ri =

Gr Re 2—Richardson number, t—time, s, T—temperature, K, u, v—velocity components in x and y direction, m s–1, uin—velocity at the inlet of the cavity, m s–1, x, y—Cartesian coordinates, m. Greek symbols: α—thermal diffusivity, m2 s–1, β—thermal expansion coefficient, K–1, δ —width of inlet and outlet ports, m, μ—dynamic viscosity, kg m–1 s–1, ρ—density, kg m–3, ν —kinematic viscosity, m2 s–1, ϕ—volume fraction of the nanoparticles, ε—dimensionless length of the heater es/L = 0.2. Subscripts: c—cold, f—fluid, h— hot, nf—nanofluid, s—solid particles, S1—the left heater, S2—the right heater. HIGH TEMPERATURE

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