Natural Convection and Entropy Generation in Partially Heated Porous Wavy Cavity Saturated by a Nanofluid

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1 Proceedings o the 5 th International Conerence o Fluid Flow, Heat and Mass Transer (FFHMT'18) Niagara Falls, Canada June 7 9, 2018 Paer No. 123 DOI: /hmt Natural Convection and Entroy Generation in Partially Heated Porous Wavy Cavity Saturated by a Nanoluid Ali J. Chamkha 1, Ammar I. Alsabery 2,3, Nader Nader 1, Ali Al-Mudha 4 1 Mechanical Engineering Deartment, Prince Mohammad Bin Fahd University Al-Khobar 31952, Saudi Arabia achamkha@mu.edu.sa 2 Rerigeration & Air-conditioning Technical Engineering Deartment, the Islamic University Naja, Iraq 3 School o Mathematical Sciences, Universiti Kebangsaan Malaysia UKM Bangi, Selangor, Malaysia 4 Manuacturing Engineering Deartment, Public Authority or Alied Education and Training Shuweikh 70654, Kuwait Abstract Steady laminar natural convection and entroy generation in a orous wavy cavity saturated by a nanoluid with artial bottom temerature is studied numerically using the inite element method. An isothermal heater is laced on the bottom horizontal wall o the cavity with length h while the wavy vertical walls are maintained at the same constant cold temerature T c. The remainder o the bottom wall and the to wall are ket adiabatic. The boundaries o the domain are assumed to be imermeable and the luid within the cavity is a water-based nanoluid having Cu nanoarticles. The Boussinesq aroximation is alicable and the Tiwari and Das' nanoluid model is considered. The Forchheimer-Brinkman-extended Darcy model is assumed to hold. The numerical comutations are obtained or various values o Darcy number, nanoarticle volume raction and the amlitude o the wave. Based on the obtained results, it is ound that the strength o the streamlines increases with the increment o the Darcy number. Keywords: Natural Convection, Entroy Generation, Wavy Porous Cavity, Non-Darcy Model, Partial Heating. 1. Introduction Convection heat transer in orous medium a signiicant henomenon in science and engineering systems due to wide range o industrial alications such as geothermal reservoirs, waste nuclear rocessing, matrix heat exchangers, otimization o solidiication rocesses o metals and alloys, loat glass roduction, low and heat transer in solar onds, air conditioning in rooms, otimization o solidiication rocesses o metals and alloys, dissemination control o chemical waste and ollutants, electronic ackages, grain storage systems and many others [1]. Natural convection and heat transer in closed cavities with dierent shaes takes a large art o the heat transer literature. Square, rectangle, triangle, cylindrical, ellitical and sherical geometries have been studied by many researchers. Comlex geometries cover dierent tyes o geometrical conigurations, namely wavy walls cavities, concave and convex curved walls cavities, etc. A very recent comrehensive literature survey concerning convection heat transer in wavy orous cavities is given by Shenoy et al. [2]. The authors o this book gave an excellent background in the ield o convective heat transer in wavy cavities illed with viscous luids, orous media, and nanoluids. Alsabery et al. [3] numerically studied the Eects o nonuniorm heating and wall conduction on natural convection heat transer in a square orous cavity using local thermal nonequilibrium (LTNE) model. The study o convective heat transer roblems within comlex geometries have received a considerable attention in the literature due to its wide alications in engineering roblems such as; solar collectors, micro-electronic devices, electrical and nuclear comonents, etc. The roblem o natural convection heat transer in a wavy cavity was considered by Das and Mahmud [4]. They ound that the number o undulations o the wavy wall is clearly aected the heat transer characteristics within the cavity. Rostami [5] numerically studied the unsteady luid low and heat transer roblem in a cavity with vertical wavy walls and horizontal straight walls. Natural convection in wavy orous cavities under the inluence o thermal radiation and with the use o thermal non-equilibrium model is studied by Mansour et al. [6]

2 A nanoluid is deined as a smart luid with susended nanoarticles o average sizes below 100 nm in conventional heat transer luids such as water, oil, and ethylene glycol. A nanoluid as a working medium has been considered by many researchers or the simle reason that it has the resence o nanoarticles resulting in higher thermal conductivity o medium and the heat transer becoming enhanced. There has been a surge in research activities concerning natural convective heat transer in nanoluid-saturated orous media. The work o Sun and Po [7] considered the natural convection heat transer in a triangular cavity heated by a wall heater and illed with a orous medium and saturated with dierent nanoluids tyes. Chamkha and Ismael [8] studied the conjugate natural convective heat transer in a orous cavity illed with nanoluids and heated by a triangular thick wall. Abu-Nada and Chamkha [9] numerically studied the eects o lid-driven wall on mixed convection low in a lid-driven cavity with a wavy wall illed with a nanoluid. They ound that heat transer rate increases with the increment o the volume raction o nanoarticles or all values o Richardson numbers. Recently, the roblem o natural convection in a artially heated wavy orous cavity illed with a nanoluid under the eects o Brownian diusion and thermohoresis was investigated by Sheremet et al. [10]. They ound that the local heat source has an eicient inluence o the nanoluid low and heat transer rate. Nevertheless, the study o natural convection and entroy generation in artially heated orous wavy cavity saturated by a nanoluid. Thereore, the aim o this study is to investigate the natural convection heat transer and entroy generation in a orous wavy cavity saturated by a nanoluid with artial bottom temerature. 2. Mathematical Formulation The steady two-dimensional natural convection roblem in a wavy cavity with a length L, is illustrated in Fig. 1. An isothermal heater is laced on the bottom horizontal wall o the cavity with length h while the wavy vertical walls are maintained at the same constant cold temerature T c. The remainder o the bottom wall and the to wall are ket adiabatic. The boundaries o the domain are assumed to be imermeable, the luid within the cavity is a water-based nanoluid having Cu nanoarticles. The Boussinesq aroximation is alicable and the Tiwari and Das' with Maxwell-Garnett and Brinkman models are used or modelling the nanoluid heat transer. The Forchheimer-Brinkman-extended Darcy model is assumed to hold. The governing equations or natural convection low using conservation o mass, momentum and energy equations can be written in dimensionless orm as ollows: Fig. 1: Physical model o convection in a wavy orous cavity together with the coordinate system. U X V Y 0, (1) 123-2

3 C Pr ( / ) U V U U V P Prb ( n / ) U, X Da[ ( / ) (1 )] U U F n U U 3/2 X Y Da ( / ) (1 ) X Y (2) C Pr ( / ) U V V U V Pr( n / ) ( / ) (1 ) V RaPr, Da[ ( / ) (1 )] V V F n V V P 3/2 X Y Da ( / ) (1 ) X Y Y n (3) k ( C ) U e V X Y k ( C ) n X Y, (4) The governing equations o Navier Stokes equations (1) (3), the energy equation (4) are transormed into dimensionless orms using the ollowing dimensionless variables: x y ul vl Tn Tc s X, Y, U, V,, S, k k (1 ) k L L T T L h c 3 2 g T T L K L Pr. h c, Ra, Da, P L e n s (5) The dimensionless boundary conditions o Eqs. (1)-(4) are as ollows: U V 0, 1 on (1 H) / 2 X (1 H) / 2, Y 0, (6) U V 0, 0 on 0 X (1 H ) / 2 and (1 H) / 2 X 1, Y 0, Y (7) U V 0, 0 on 1 (0.5 A(1 cos(2 NX ))1 A/ 2), 0 Y 1, (8) U V 0, 0 on (0.5 A(1 cos(2 NX))1 A/ 2), 0 Y 1, (9) U V 0, 0 on 0 X 1, Y 1. Y (10) The local Nusselt number evaluated at the bottom horizontal wall, which is deined by: ke n Nu k Y Y 0. (11) Finally, the average Nusselt number evaluated at the bottom horizontal wall o the cavity which is given by: 123-3

4 1 H n 2 1H n 2 Nu Nu d X. (12) In dimensionless orm, local entroy generation can be exressed as: S GEN 2 k e n U V U V ( U V ) N 2 k X Y X Y Y X Da (13) where, N T0 k L( T ) 2 is the irreversibility distribution ratio and S T L 0 GEN Sgen 2 k ( T). The terms o Eq. 19 can be searated to the ollowing orm: S S S (14) GEN where S θ and S Ψ are the entroy generation due to heat transer irreversibility (HTI) and luid riction irreversibility (FFI), resectively. S k e k X Y (15) S 2 n U V U V ( U V ) N 2 X Y Y X Da (16) By Integrating Eq. 16 over the domain, the global entroy generation (GEG) or the resent two-dimensional study is obtained SGEN X Y S X Y S X Y (17) GEG d d d d d d. It is aroriate to mention Bejan number in order to determine which one is the dominant, heat transer or luid riction irreversibility. Bejan number is deined as: S dxdy Be. S dxdy GEN (18) When Be > 0.5, the HTI is the dominant, while when Be < 0.5, the FFI is the dominant. The governing dimensionless equations (1) (4) subject to the boundary conditions (6) (10) are solved with Galerkin weighted residual inite element method [11]. The comutational domain is discretised into triangular elements. Triangular Lagrange inite elements o dierent orders are used or each o the low variables within the comutational domain. Residuals or each conservation equation are obtained by substituting the aroximations into the governing equations. To simliy the nonlinear terms in the momentum equations, a Newton-Rahson iteration algorithm was used [11]

5 3. Results and Discussion In this section, we resent numerical results or the streamlines, isotherms and isentroic with various values o Darcy number (10 6 Da 10 2 ), nanoarticle volume raction (0 φ 0.05) and amlitude o the wave (0.05 A 0.3), where the values o Rayleigh number, orosity o the media, number o oscillations and Prandtl number are ixed at Ra = 10 6, ε = 0.5, N = 3 and Pr = 6.2. The values o the average Nusselt number are calculated or various values o Da and φ. The thermohysical roerties o the base luid (water) and solid Cu hases are tabulated in Table 1. Table 1: Thermo-hysical roerties o water and Cu nanoarticles. Physical roerties Fluid hase (water) Cu C ρ k μ Figure 2 illustrates the eects o various Darcy number (Da) on the streamlines, isotherms and isentroic mas or φ = 0.03 and A = At low Darcy number (Da = 10 5 ), the low within the cavity is characterized by two streamlines cells at the lower art o the cavity aected by the low velocities that revail in the cavity. The intensity o the isotherm atterns decreases next to the cold wavy walls while in the middle o the cavity the atterns tend to take carved lines. The entroy generation in the cavity is rimarily due to heat transer irreversibility and it is almost negligible near the uer ortion o the cavity due to insigniicant heat low in that region because o the slow velocity. With the increasing o Da the ermeability o the orous medium is increased and thus the luid low is intense resulting in enhanced convective heat rom the hot bottom wall. At this stage, due to stronger luid low, the entroy generation is clearly higher comared to that or low Darcy number and the uer o the cavity articiates in the entroy generation. This is because as the luid circulation is stronger or higher Da which leads to a high velocity gradients and thereore a more entroy generation due to luid riction. However, the density o the entroy generation tends to decrease within the centre o the cavity due to the act that most o the energy moves toward the wavy vertical walls

6 Fig. 2: Variation o the streamlines (let), isotherms (middle), and isentroic (right) evolution by Darcy number (Da) or φ = 0.03 and A = Figure 3 illustrates the eects o various amlitude o the wave (A) on the streamlines, isotherms and isentroic mas or Da = 10 3 and φ = It is clearly observed rom this igure that the amlitude o the wave has an inluence on the geometric shae o the low cell, as well as on the distribution o isotherms and isentroic lines. The low within the cavity aears with two streamlines cells at the lower art o the cavity. The isotherm atterns observe with high intensity next the wavy walls o the cavity. The isentroic distribution shows a weak behaviour within the cavity, isentroic lines aear with high density within the wavy cavity aected by the lower thermal gradient. The intensity o the streamlines increases with the increasing o the amlitude o the wave, and as a result, the strength o the low circulation increases. The intensity o the isotherm atterns decreases next the wavy wall with the increment o the amlitude due to the reduction o the gradient o the boundary layer. Increasing the amlitude o the wave u to the high values is clearly enhanced the entroy generation due to the strong luid circulation

7 Fig. 3: Variation o the streamlines (let), isotherms (middle), and isentroic (right) evolution by amlitude o the wave (A) or Da = 10 3 and φ = Figure 4(a) shows the eects o various Darcy numbers on the local Nusselt number and along the heated art o the bottom wall or φ = 0.03 and A = Increasing Darcy tends to increase the gradient o the boundary layer and as a result, the local heat transer enhances. Higher heat transer enhancement is obtained with higher Darcy number. Figure 4(b) resents the eects o various nanoarticle volume raction on the local Nusselt number and along the heated art o the bottom wall or Da = 10 3 and A = The local heat transer increases by the augmentation o nanoarticle volume raction due to the higher thermal conductivity o the nanoarticles

8 Fig. 4: Variations o local Nusselt numbers interace with X or dierent (a) Da and (b) φ at A = Figure 5(a) shows the eects o nanoarticle volume raction on the average Nusselt number with Darcy number or A = The heat transer rate is clearly enhanced with increasing nanoarticle volume raction. The values o average Nusselt number denote that the heat transer rate increases and then decreases and increases ater a while with the increment o nanoarticle volume raction. Due to the various low velocity seeds with dierent amount o nanoarticles. Figure 5(b) illustrates the eects o various values o the nanoarticle volume raction on Bejan number with Darcy number. The average Bejan number tends to decrease as the Darcy number increases which clariy the dominance o the irreversibility due to the luid riction with the high Darcy numbers. Fig. 5: Variation o (a) average Nusselt number and (b) Bejan number (Be) with Da or dierent φ at A = Conclusion The resent study considered the natural convection and entroy generation in a orous wavy cavity saturated by a nanoluid with artial bottom temerature. The governing equations with the boundary conditions are solved numerically using the inite element method. The Forchheimer-Brinkman-extended Darcy and Tiwari and Das' nanoluid models are assumed to hold. Based on the obtained results, it is ound that the strength o the streamlines increases with the increment o the Darcy number. The isotherms aear with high density close to the wavy walls. The entroy generation looks very strong with high values o the Darcy number. However, the density o the entroy generation tends to decrease within the centre o the cavity due to the act that most o the energy moves toward the wavy vertical walls. The heat transer rate is clearly enhanced with the increasing o the nanoarticle volume raction and the Darcy number due to the increasing o the 123-8

9 thermal conductivity. On the other hand, Bejan number reduces by the increment o the Darcy number when the luid riction irreversibility (FFI) is the dominant. Acknowledgements The irst and third authors thank Prince Mohammad Bin Fahd University or the inancial suort o this work under an internal grant roject. Reerences [1] A. Narasimhan, Essentials o heat and luid low in orous media. CRC Press, Boca Raton, [2] A. Shenoy, M. Sheremet, I. Po, Convective low and heat transer rom wavy suraces: Viscous luids, orous media, and nanoluids. CRC Press, Taylor & Francis Grou, Boca Raton,, [3] A. I. Alsabery, A. J. Chamkha, I. Hashim and P. G. Siddheshwar, Eects o nonuniorm heating and wall conduction on natural convection in a square orous cavity using LTNE model, Journal o Heat Transer, vol. 139, no. 12, , [4] P. K. Das and S. Mahmud, Numerical investigation o natural convection inside a wavy enclosure, International Journal o Thermal Sciences, vol. 42, , [5] J. Rostami, Unsteady natural convection in an enclosure with vertical wavy walls, Heat and Mass Transer, vol. 44, , [6] M. Mansour, M. A. El-Aziz, R. Mohamed and S. E. Ahmed, Numerical simulation o natural convection in wavy orous cavities under the inluence o thermal radiation using a thermal non-equilibrium model, Transort in Porous Media, vol. 86, , [7] S. Qiang, I. Po, Free convection in a triangle cavity illed with a orous medium saturated with nanoluids with lush mounted heater on the wall, International Journal o Thermal Sciences, vol. 50, no. 11, , [8] A. J. Chamkha and M. A. Ismael, Conjugate heat transer in a orous cavity illed with nanoluids and heated by a triangular thick wall, International Journal o Thermal Sciences, vol. 67, , [9] E. Abu-Nada, A. J. Chamkha, Mixed convection low o a nanoluid in a lid-driven cavity with a wavy wall, International Communications in Heat and Mass Transer, vol. 57, , [10] M. Sheremet, D. Cimean and I. Po, Free convection in a artially heated wavy orous cavity illed with a nanoluid under the eects o brownian diusion and thermohoresis, Alied Thermal Engineering, vol. 113, , [11] A. I. Alsabery, M. A. Ismael, A. J. Chamkha and I. Hashim, Mixed convection o Al 2 O 3 -water nanoluid in a double lid-driven square cavity with a solid inner insert using Buongiorno s two-hase model, International Journal o Heat and Mass Transer, vol. 119, ,