A Numerical Study on Natural Convection in a Square Inclined Porous Enclosure Filled with an Ag Nanoliquids

Transcription

1 Applied Mathematical Sciences, Vol. 11, 2017, no. 34, HIKARI Ltd, A Numerical Study on Natural Convection in a Square Inclined Porous Enclosure Filled with an Ag Nanoliquids Rado Yendra Department of Mathematics Faculty of Science Technology Universitas Islam Sultan Syarif Kasim (UIN Suska) Pekanbaru, Riau, Indonesia Habibis Saleh Department of Mathematics University Riau Pekanbaru, Riau, Indonesia Copyright c 2017 Rado Yendra and Habibis Saleh. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Finite difference technique is used to understand fluid flow in an inclined porous enclosure filled with water based Ag nanoliquids. The enclosure has square cross-section and it is constantly heated from a wall and cooled from opposite wall while other walls are adiabatic. The oscillating flow development was found at ω = 15 before reaching their steady condition. The steady heat transfer enhancement by increasing the Ag concentration are linearly dependent with increasing the orientation angles. Keywords: natural convection, finite difference method, nanoliquids 1 Introduction Nanoliquids refer to nanometer-sized particles dispersed in a base liquid having relatively low thermal conductivity like water in order to obtain a liquid

2 1696 Rado Yendra and Habibis Saleh (l,l) aaa aaaaaa aaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaa y aaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa T c Nanoliquids T h aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaa ω aaaaaaaaa aaaaaa aaa x Figure 1: Schematic representation of the model with improved thermo-physical properties. Convective flow in enclosures filled with dispersed nanoparticles in water were studied by [15] using finite difference method. [3], [14] and [6] studied conjugate heat transfer a in an enclosure filled with different nanoparticles. Heated partially porous layered enclosure filled with nanoliquids was studied by [4]. [8] investigated an interaction between nanoparticles and magnetic field in a porous enclosure. [17] analyzed the convective flow inside nanoliquids saturated porous enclosure via heatline concept. Comprehensive review of convection heat transfer and fluid flow in porous media with nanoliquids was conducted by [9]. Recently, [5] included the viscous dissipation and radiation effects. The flow and heat transfer characteristics of the nanoliquids for the case of unsteady and transient flow situations has not been previously considered. Therefore, the present paper investigates unsteady natural convection in a porous square enclosure filled with argentum nanoparticles with various concentration dispersed in water. Effects of inclination angle on unsteady heat transfer and fluid flow are also studied. 2 Mathematical Formulation A schematic diagram of an inclined porous enclosure with sides of length l is shown in Figure 1. The flow inside the porous medium is assumed to obey Darcy law. The liquids in the enclosure is a water-based nanoliquids containing Ag, nanoparticles. It is also assumed that nanoparticles are suspended in the nanoliquids using either surfactant or surface charge technology. As a result, in keeping with the Darcys law, the basic continuity, momentum, and energy

3 Numerical study on natural convection 1697 equations can be written as: u x + v y = 0 (1) µ nl K u = p x + [φρ spβ sp + (1 φ)ρ bl β bl ] g(t T c ) cos ω (2) µ nl K v = p y + [φρ spβ sp + (1 φ)ρ bl β bl ] g(t T c ) sin ω (3) σ T t + u T x + v T y = α nl ( 2 T x + 2 T 2 y 2 The equations (1) (4) can be written in terms of the stream function ψ defined as u = ψ/ y and v = ψ/ x. The resulting non-dimensional forms of the governing Eqs. (1)-(4) are: ( 1 2 ) Ψ (1 φ) 2.5 X + 2 Ψ = Ra [(1 φ) + φ(ρ 2 Y 2 sp /ρ bl )(β sp /β bl )] ( Θnl Y cos ω Θ ) nl X sin ω Θ nl + Ψ Θ nl τ Y X Ψ Θ nl X Y = α ( nl 2 ) Θ nl α bl X + 2 Θ nl 2 Y 2 3 Numerical Method The Finite Difference (FD) scheme consisting of the Alternating Direction Implicit (ADI) method and the Tri-Diagonal Matrix Algorithm (TDMA) is employed to solve the governing equations. The finite difference equation (FDE) of (5) in the ADI scheme are: ) (4) (5) (6) Ψ n i,j ( τ/2) Ψ n+ 1 2 i,j 1 = Ψn+ 2 i+1,j 2Ψ n+ 1 2 i,j + Ψ n+ 1 2 i 1,j ( X) 2 + Ψn i,j+1 2Ψ n i,j + Ψ n i,j 1 ( Y ) 2 + (S Ψ ) n i,j (7) and Ψ n+1 i,j Ψ n+ 1 2 i,j ( τ/2) 1 = Ψn+ 2 i+1,j 2Ψ n+ 1 2 i,j + Ψ n+ 1 2 i 1,j ( X) 2 + Ψn+1 i,j+1 2Ψ n+1 i,j + Ψ n+1 i,j 1 ( Y ) 2 + (S Ψ ) n i,j (8)

4 1698 Rado Yendra and Habibis Saleh Nu Grid size in the X- and Y -directions 140 Figure 2: Grid sensitivity check at φ = 0.0 and Ra = 100 in a steady state Eqs. (7) and (8) are written in tridiagonal forms as: d x Ψ n+ 1 2 i 1,j + (1 + 2d x )Ψ n+ 1 2 i,j d x Ψ n+ 1 2 i+1,j = d y Ψ n i,j+1 + (1 2d y )Ψ n i,j + d y Ψ n i,j 1 ( τ/2)(s Ψ ) n i,j (9) with d y Ψ n+1 i,j 1 + (1 + 2d y )Ψ n+1 i,j d y Ψ n+1 i,j+1 = d x Ψ n+ 1 2 i+1,j + (1 2d x )Ψ n+ 1 2 i,j + d x Ψ n+ 1 2 i 1,j ( τ/2)(s Ψ ) n i,j (10) [ (S Ψ ) i,j = (1 φ) 2.5 Ra (1 φ) + ρ ] ( sp β sp (Θnl ) i,j+1 (Θ nl ) i,j 1 ρ bl β bl 2 Y (Θ ) nl) i+1,j (Θ nl ) i 1,j sin ω 2 X cos ω (11) The FDE of (6) could be treated in the similar way. The steady Ψ and Θ are achieved with the following criteria is fulfilled: Θ k+1 i,j Θ k i,j max 10 5 (12) Θ k i,j Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results are grid independent. A grid test was performed using sets of grids in the range to for Ra = 100 as presented in Fig. 2. The result showed insignificant differences for the grids and above. Therefore, for the all computations in this thesis for similar problems to this subsection, the uniform grid was employed.

5 Numerical study on natural convection Results and Discussion Figure 3 illustrates the time history of streamlines of water and nanoliquids for ω = 90 at τ = , 0.001, 0.003, 0.008, 0.02, 0.05, 0.2, 0.4. The time history of the flow field is described as follows. Initially, at τ = 0, left and right walls are cold and there is no fluid motion inside the cavity. As heating started, τ = , the fluid temperatures adjoining the hot-right wall rise. The fluid moves due to buoyancy force from the right region of the cavity to the left region. This movement creates the cells in the left portion of enclosure. The cells are elongated vertically. At τ = 0.001, the cells get bigger and stronger, occupies more portion of the enclosure. As time increases, the clockwise circulation cells occupies the left-half of enclosure and its strength increases significantly. The core of the cells starts to move to the center of the enclosure. Later, the cells occupy most area of the enclosure and reach steady state before τ = 0.2. We observe that the strength of the flow circulations of the nanoliquids always weaker than the base liquid at all time. Fig. 4 shows the variation of average Nusselt number for various values of ω at Ra = 500 and φ = At early stages of the flow growing, higher angle gives higher Nu values. Later, the average Nusselt number are identic for some ω, such as at τ = 0.02 for ω = 15, 30 and at τ = 0.03 for ω = 45, 60. We also observe the Nu oscillations at ω = 15 for τ < 0.1. This attributes to the periodical convective flow development before reaching their steady condition. The numerical values of Nu achieve their respective steady conditions are later for lower ω. Figure 5 shows the steady average Nusselt number against solid volume fraction for different orientation angle at Ra = 500. Increasing the Ag concentration significantly enhances the heat transfer rate for any configurations. Lower orientation angle has lower average Nusselt number for the fixed nanoparticles concentration. These result signifies that the steady heat transfer enhancement by increasing the Ag concentration are linearly dependent with increasing the orientation angles. 5 Conclusion The present numerical simulation study the effects of the Ag concentration on unsteady natural convection in a tilted square porous enclosure. The dimensionless forms of the governing equations are solved using the FD scheme consisting of ADI method and TDMA. The nanoliquids circulations are weaker than the water circulations. The oscillating flow development was found at ω = 15 before reaching their steady condition. The steady heat transfer enhancement by increasing the Ag concentration are linearly dependent with increasing the orientation angles.

6 Rado Yendra and Habibis Saleh =-1.98, (Ψ min ) nf =-1.86 =-5.45, (Ψ min ) nf = τ = τ = =-9.34, (Ψ min ) nf =-8.75 =-13.5, (Ψ min ) nf = τ = τ = =-14.1, (Ψ min ) nf =-12.9 =-13.5, (Ψ min ) nf = τ = 0.02 τ = =-13.4, (Ψ min ) nf =-12.3 =-13.4, (Ψ min ) nf =-12.3 τ = 0.2 τ = 0.4 Figure 3: Time history of streamlines [nanoliquids (solid lines) with φ = 0.05 and pure liquid (dashed lines)] for Ra = 500 and ω = 90 at τ = , 0.001, 0.003, 0.008, 0.02, 0.05, 0.2, 0.4.

7 Numerical study on natural convection ω =15 ω =30 ω =45 ω =60 12 Nu τ Figure 4: Variation of Nu with time, τ, for different ω at Ra = 500 and φ = Nu ω =15 ω =30 ω =45 ω = φ Figure 5: Variation of Nu with φ for different ω at Ra = 500 in steady state.

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