PERFORMANCE OF NANOFLUID IN FREE CONVECTIVE HEAT TRANSFER INSIDE A CAVITY WITH NON-ISOTHERMAL BOUNDARY CONDITIONS

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1 Proceedings o the International Conerence on Mechanical Engineering and Renewable Energy 2015 (ICMERE2015) November, 2015, Chittagong, Bangladesh ICMERE2015-PI-058 PERFORMANCE OF NANOFUID IN FREE CONVECTIVE HEAT TRANSFER INSIDE A CAVITY WITH NON-ISOTHERMA BOUNDARY CONDITIONS Salma Parvin 1, *, Raju Chowdhury 2, M.A.H Khan 3 and M.A. Alim Department o Mathematics, Bangladesh University o Engineering & Technology, Dhaka-1000, Bangladesh 2 Department o Natural Science, Stamord University Bangladesh, Dhaka-1209, Bangladesh. 1,* salpar@math.buet.ac.bd, 2 raju_chy_23@yahoo.com, 3 mahkhan@math.buet.ac.bd, 4 maalim@math.buet.ac.bd Abstract-The ree convective low and heat transer characteristics o nanoluid inside a prismatic enclosure with non-uniorm temperature distribution maintained at the bottom wall is analyzed numerically. The water based nanoluid with alumina nanoparticle is used as the operational luid through the enclosure. The governing partial dierential equations with proper boundary conditions are solved by Finite Element Method using Galerkin s weighted residual scheme. Calculations are perormed or dierent solid volume raction o nanoparticles 0 χ 0.1 with Rayleigh number An enhancement in heat transer rate is observed with the increase o nanoparticles volume raction. Keywords: Natural Convection; inite element method; Nanoluid; Solid Volume Fraction; Prismatic enclosure 1. INTRODUCTION Many engineering applications o natural convection in encloseres including geophysics, geothermal reservoirs, insulation o building, heat exchanger design, building structure etc. motivate many researchers to perorm numerical simulation to investigate the low pattern, temperature distribution, and heat low. The low thermal conductivity o conventional heat transer luids, commonly water, has restricted designers. Fluids containing nanosized solid particles oer a possible solution to conquest this problem. The nanoluid has greater eective thermal conductivity than pure base luid. Most o the available literature on this topic concerns regular geometries such as rectangular or square enclosures, while actual applications demand the consideration o irregular shapes. iterature reviews on natural convection inside triangular, trapezoidal and rhombic enclosures are available in [1-5]. In view o various applications o thermal processes, a comprehensive understanding o heat transer and low circulations within prismatic cavities is very much essential or industrial development. Current work attempts to analyze heat transer and energy distributions using heatline concept inside a nanoluid illed prisamtic cavity. The heat recovery system or true path o convective heat transer can be visualized by heatline method. Kimura and Bejan [6] and Bejan [7] irst introduced the concept o heatline. Heatlines represent heatlux lines which represent the trajectory o heat low in the system and they are normal to the isotherms or conductive heat transer. Various applications using heatlines were studied by Bello-Ochende [8], Costa [9-11], Mukhopadhyay et al. [12,13] and Deng and Tang [14]. Dalal and Das [15] have used heatline method or the visualization o low in a complicated cavity. Recently, Yaseen [16] has studied and analyzed numerically the steady natural convection low in a prismatic enclosure with strip heater on bottom wall. For this enclosure, top inclined walls are considered at low temperature, two vertical walls were adiabatic and strip heater was at constant high temperature mounted on the bottom enclosure while the reminder bottom wall was kept at low temperature. Recently Ahmed et.al [17] perormed numerical analysis or natural convection lows within prismatic enclosures based on heatline approach. Heat transer in cavities illed with nanoluid is the research interest o many researchers. Parvin et al. [18] studied the natural convection heat transer in an enclosure with a heated body illed with nanoluid. Nasrin and Parvin [19] investigated numerically the buoyancy-driven low and heat transer in a trapezoidal cavity illed with water Cu nanoluid. In their work, a correlation is developed graphically or the average Nusselt number as a unction o the Prandtl number as well as the cavity aspect ratio. Abu- Nada and Chamkha [20] perormed a numerical study o natural convection heat transer in a dierentially heated enclosure illed with CuO-EG-Water nanoluid. The results were compared with Brinkman model and MG models or nanoluid viscosity and thermal conductivity. Either enhancement or decline was reported or the average Nusselt number as the volume raction o nanoparticles

2 increased. Heat Transer and Entropy Generation in an Odd-shaped Cavity illed with Nanoluid is analyzed by Parvin and Chamkha [21]. It must be noticed that, adding nanoparticles into the base luid does not always increase its thermal conductivity [22]. Teja et al. [24] have reported that the thermal conductivity o 2 nm titania nanoparticles is smaller than the thermal conductivity o the base luid at the same temperature. Parvin et al. [23] numerically investigated the natural convection heat transer rom a heated cylinder contained in a square enclosure illed with water Cu nanoluid. Their results indicated that heat transer augmentation is possible using highly viscous nanoluid. From the above discussion, the literature contains many investigations into the heat transer perormance o natural convection o nanoluid in regular shaped cavities. However, a comprehensive analysis on heat low during natural convection o nanoluid in irregular enclosure with the heatline approach is yet to appear in the literature. Accordingly, the present study perorms a numerical investigation into the natural convection heat low within a prism shaped cavity containing nanoluid with non isothermal bottom wall. The simulations ocus speciically on the eects o the dierent volume raction o nanoluid on the streamlines, isotherm distribution, heatlines and average Nusselt number. Average temperature and velocity are also presented. 2. MATHEMATICA FORMUATION 2.1 Momentum and energy ormulation The physical domain is shown in Fig. 1. The luid properties are assumed to be constant except the density variation which is determined according to the Boussinesq approximation. Under these assumptions, the governing equations or steady two dimensional laminar incompressible lows can be written in dimensionless orm as: U V 0 (1) X Y U U P n U U U V Pr (2) X Y n X X Y V V P n V V U V Pr X Y n Y X Y (3) 1 ss Ra Pr n U V X Y X Y 1 is the density, where, n s Cp 1 Cp Cp is the heat n s capacitance, 1 n is the thermal expansion n s coeicient, k C n n p n is the thermal diusivity, (4) n is dynamic viscosity and k n k k 2k 2 k k s s k 2k k k s s conductivity o the nanoluid and x X, y Y, u U, v V, is the thermal T T T h c T 2 p g ( T 3 h Tc ) Pr P, Pr, Ra (5) 2 2 The boundary conditions can be summarized by the ollowing equations: At the bottom wall: sin W For the side walls: For the inclined walls: Y θ = 0 θ x = 0 V U 0,, U 0, θ = sin (π) V 0 V 0, U 0,, c, V 0 θ = 0 0 X 0 Fig.1: Schematic diagram o the physical system 2.2 Streamunction and heatunction The relationships between streamunction and velocity components U, V or two-dimensional lows are U and V (6) Y X which give a single equation U V (7) X Y Y X The no-slip condition is valid at all boundaries as there is no cross-low. Hence 0 is used or boundaries. The heat low within the enclosure is displayed using the heatunction Π obtained rom conductive heat luxes, as well as convective heat luxes (Uθ, X Y Vθ). The heatunction satisies the steady energy balance equation (Eq. (4) ) such that U and V (8) Y X X Y which yield a single equation U θ x = 0 X

3 U V X Y Y X 2.3 Average Nusselt number The average Nusselt number, average temperature and average velocity may be expressed as 1 S k n Nu dn S k 0 N av dv / V and Vav V dv / V where S is the non-dimensional length o the surace and V is the volume to be accounted. (9) 3. NUMERICA IMPEMENTATION The Galerkin inite element method is used to solve the non-dimensional governing equations along with boundary conditions or the considered problem. The equation o continuity has been used as a constraint due to mass conservation and this restriction may be used to ind the pressure distribution. The inite element method o Reddy [24] is used to solve the Eqns. (6) - (9) where the pressure P is eliminated by a constraint. The continuity equation is automatically ulilled or large values o this penalty constraint. Then the velocity components (U, V) and temperature (θ) are expanded using a basis set. The Galerkin inite element technique yields the subsequent nonlinear residual equations. The non-linear residual equations are solved using Newton Raphson method to determine the coeicients o the expansions. The convergence o solutions is assumed when the relative error or each variable between consecutive iterations is recorded below the n1 n 4 convergence criterion ε such that 10, where n is the number o iteration and is a unction o U, V and θ. little dierence with the results obtained or the other elements. Hence, considering the non-uniorm grid system o 2270 elements is preerred or the computation. χ = 0 χ = 0.05 χ = 0.1 Fig.3. Eect o χ on Isothems Fig.2: Grid independent test 3.1 Grid Independent Test An extensive mesh testing procedure is conducted to guarantee a grid-independent solution or Ra = 10 5 and Pr = 6.2, χ = 0 in a prismatic enclosure. Five dierent non-uniorm grid systems with the ollowing number o elements within the resolution ield: 454, 926, 1504, 2270 and 9948 are exami8ned. The numerical scheme is carried out or highly precise key in the average Nusselt (Nu) number to understand the grid ineness as shown in Fig. 2. The scale o the average Nusselt numbers or 2270 elements shows a 4. RESUTS AND DISCUSSION The numerical computation has been carried out through the inite element method to analyze natural convection within a nanoluid illed prismatic enclosure based on heatline concept. Results o isotherms, streamlines and heatlines or various values o nanoparticle volume raction χ (=0, 0.05, 0.1) with Rayleigh number Ra = 10 5 and Prandtl number Pr = 6.2 in the prismatic enclosure are displayed. In addition, the values o average Nusselt number at the top inclined walls and the bottom wall as well as average temperature and average velocitiy in the cavity have been calculated or the mentioned parameter. 4.1 Eect on Isotherms The inluence o particle volume raction χ on isotherms or the present coniguration has been demonstrated in Fig. 3. The pattern o isotherms are

4 smooth and are symmetric to the central vertical line or all the considered values o χ. A thermal plume rises rom the middle o the bottom wall because o sinusoidal boundary temperature. At χ = 0, that is in the case o base luid, the higher temperature lines remain near the hot bottom wall. These lines move toward the cold top inclined walls due to the increasing values o χ. This happens due to the presence o nanoparticle in the luid and buoyancy eect which causes higher temperature gradient and thus the isothermal lines move rom hot walls to the cold wall. χ = 0 χ = 0.05 χ = 0.1 χ = 0 χ = 0.1 χ = 0.05 Fig.5. Eect o χ on heatlines Fig.4. Eect o χ on streamlines 4.2 Eect on Streamlines Streamlines corresponding to dierent nanoluid solid volume raction χ are shown in Fig. 4. It is seen rom the igure that the trend o streamlines are similar or all cases. There are two symmetric circulation cells ormed inside the enclosure. With the increasing value o the χ the streamlines are less dense near the central vertical line. It is noteworthy to mention that the central core o the circulatory cell is reduded in size or higher nanoparticle concentration. Higher low intensity is seen or the base luid. 4.3 Eect on Heatlines The heatlines are constructed based on the thermal boundary conditions. As seen rom the Fig. 5, the heatlines are similar to streamlines at the core signiying convective heat low and a large amount o heat low occurs rom the middle portion o the bottom wall as seen rom dense heatlines. The two heat circulations in the system are observed and a very intense heat low occurs across the middle o the cavity represented by dense heatline or pure water. It is interesting to observe that heat transport in a large regime at the core is due to convection. The large regime o convection is due to the large amount o heat transport rom the bottom wall associated with large intensity o circulations. As the value o χ increases the convective heat transport has been suppressed. 4.4 Average Nusselt number Fig. 6 shows the distribution o the average Nusselt number o the top inclined walls and bottom wall versus the solid volume raction o nanoluid. General observation is that the average Nusselt number increases with nanoparticle concentration and it is utilized to represent the overall heat transer rate within the domain. Similar trend o heat transer is seen or

5 both the walls. That is adding nanoparticles increases the thermal conductivity which leads to enhanced heat transer. Fig.6. Eect o χ on heat transer the increment o χ. The movement o the luid becomes slower due to higher values o χ. 5. CONCUSION The prime objective o the current investigation is to analyze a physical as well as computational insight due to heatlow or natural convection within a prismatic enclosure illed with nanoluid. The key controlling parameter or this analysis is which govern the overall heat transer rate, i.e. Nusselt number. The heat transer rates have analyzed with average Nusselt number or the top inclined walls and the bottom wall o the enclosure. The motivation is to understand the eect o the parameter on the heat low process. The visualization o heat low inside any cavity is incomplete unless we know about the heat low and hence we have introduced the heatline concept in the prismatic enclosure which enables us to understand the heat low trajectory. During conduction dominant heat transer, it is observed that isotherms, streamlines and heatlines are ound to be smooth. Also, the heatlines are seen normal to the isotherm during the heat transer. However, as χ increases, convection is weaken. But heat transer is increased due to conduction. 6. ACKOWEDGEMENT The present work is done in the department o mathematics BUET. Fig.7. Eect o χ on average temperature Fig.8. Eect o χ on average 4.5 Average temperature and average velocity Fig. 7 demonstrates the distribution o average temperature with the nanoluid volume raction. It is seen that the average temperature decreases or increasing values o χ. The average velocity in the domain is depicted in Fig.8 which is also decreased by 7. REFERENCES [1] T. Basak, G. Arvind and S. Roy, Visualization o heat low due to natural convection within triangular cavities using Bejan s heatline concept, Int. J. Heat Mass Transer, vol. 52 (11-12), pp , [2] T. Basak, S. Roy, D. Ramakrishna and I. Pop, Visualization o Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach, Numerical Heat Transer, Part A, vol. 57 (6), pp , [3] T. Basak, S. Roy and I. Pop, Heat low analysis or natural convection within trapezoidal enclosures based on heatline concept, Int. J. Heat Mass Transer, vol. 52 (11-12), pp , [4] R. Anandalakshmi and T. Basak, Heatline based thermal management or natural convection in porous rhombic enclosures with isothermal hot side or bottom wall, Energy Conversion and Management, vol. 67, pp , [5] T. Basak, S. Roy and I. Pop, Heat low analysis or natural convection within trapezoidal enclosures based on heatline concept, Int. J. Heat Mass Transer, vol. 52, pp , [6] S. Kimura and A. Bejan, The heatline visualization o convective heat-transer, ASME J. Heat Transer, vol. 105 (4), pp , [7] Bejan, Convection Heat Transer, third ed., Wiley, Hoboken, NJU, [8] F.. Bello-Ochende, A heat unction ormulation or thermal convection in a square cavity, Int. Commun. Heat Mass Transer, vol. 15, pp , 1988.

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