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Heat transfer analysis in natural convection flow of nanofluid past a wavy cone
Heat transfer analysis in natural convection flow of nanofluid past a wavy cone Ahmer Mehmood, Muhammad Saleem Iqbal PII: DOI: Reference:
S0167-7322(16)32288-7 doi:10.1016/j.molliq.2016.09.029 MOLLIQ 6306
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 August 2016 3 September 2016 9 September 2016
Please cite this article as: Ahmer Mehmood, Muhammad Saleem Iqbal, Heat transfer analysis in natural convection flow of nanofluid past a wavy cone, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.09.029
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ACCEPTED MANUSCRIPT Heat transfer analysis in natural convection flow of nanofluid past a wavy
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cone
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Ahmer Mehmood, Muhammad Saleem Iqbal1
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Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad 44000, Pakistan.
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Abstract:
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The problem of natural convective heat transfer of water-based nanofluid along wavy cone surface is investigated numerically. Analysis is performed to study the heat transfer
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augmentation due to five types of nanoparticles, namely, alumina (𝐴𝑙2 𝑂3), copper (𝐶𝑢), silver
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(𝐴𝑔), copper oxide (𝐶𝑢𝑂) and titania (𝑇𝑖𝑂2). The flow has been assumed to be steady and fluid
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properties have been supposed constant except for the density depending upon temperature giving rise to the buoyancy force. Famous Tiwari and Das model of nanofluid has been utilized in this study. The effects of cone half angle 𝛾 and amplitude of the waviness 𝛼 on the Nusselt number (𝑁𝑢) and skin friction (𝐶𝑓 ) are studied. A comparison is made with the case of pure fluid flow past wavy cone at different values of 𝛼. It has been observed that the 𝑇𝑖𝑂2-nanoparticle shown to have the maximum cooling performance and 𝐶𝑢-nanoparticle appeared to have maximum heating performance for this study. The results shown in this research arrange for a significant source of reference for taming the natural convection heat transfer enactment along wavy cone. Present results for selected variables are matched with the already published work for pure fluid and are shown to be in good agreement. 1
Corresponding Author’s Email: [email protected] Cell No. +923335195130
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ACCEPTED MANUSCRIPT Keywords: Natural convection, heat transfer, Nanofluid, wavy cone surface
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1. Introduction Convective heat transfer phenomena over a cone is applicable in various designs of thermal
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equipment like heat exchangers, geothermal reservoirs, nuclear reactor cooling, solar energy
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plants, design of space crafts, drying dehydration process in chemical and food process and
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steam generators etc. The science and the art of heat and mass transfer augmentation has evolved into an important component of various aspects of thermal science and engineering. This poses a
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great challenge for choosing appropriate design and application information to achieve the industrial and technology goals. Regarding the cone geometry the phenomena of natural
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convection heat transfer over vertical cone have been studied by several researchers and
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scientists. Merk and Prins [1-2] discussed natural convection flow along a cone for pure fluid.
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Hering and Grosh [3-4] analyzed the free convection for low Prandtl number fluids over a nonisothermal cone. For high Prandtl number fluids Roy [5] investigated natural convection over an isothermal cone. Lin [6] discussed the uniform surface heat flux from a vertical cone. Alamgir [7] investigated free heat transfer characteristics from vertical cone using approximate technique. Pop et al. [8] discussed compressibility effects in natural convection from a vertical cone. Yih [9] analyzed under uniform mass flux free convection along a vertical cone in porous medium. Cheng [10] examined the effect of variable wall temperature in free convection flow of a micropolar fluid over a vertical permeable cone. Hossain and Paul [11-12] discussed the effect of uniform heat flux and non-uniform surface temperature in free convection from a vertical circular cone. Cheng [13] discussed the effect of variable wall temperature in natural convection flow of a micro polar fluid along a vertical cone. Pullepu [14] explained uniform heat flux case for unsteady natural convection from a vertical cone. 2
ACCEPTED MANUSCRIPT The above cited studies are restricted to the cases where the cone surface is supposed to be uniform. Attention have also been given to the cases where the cone surface is taken to be non-
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uniform such as the wavy one. Since the irregular surface shape changes the flow and hence the
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heat transfer rate. The wavy shaped surface occurs frequently in practice for example flat-plate solar collectors in refrigerators and flat-plate condensers. Pop and Na [15-17] using porous
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medium discussed the natural convection flow along a vertical and frustum of a wavy cone.
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Cheng [18] investigated heat and mass transfer phenomena in natural convection along wavy cone in porous media. Considering viscosity dependent temperature Hossain et al. [19] examined
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free convection flow over a vertical wavy cone. Taking viscosity as an exponential function of temperature Rahman et al. [20] discussed free convection flow beside the vertical wavy cone.
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Important contributions and preliminary work related to the problem of natural convection in a viscous fluid along wavy surfaces are briefly given here. Yao [21] discussed free convection
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along vertical wavy sheet. Moulic and Yao [22] studied same problem with uniform heat flux. Molla et al. [23] examined natural convection through vertical wavy plate with heat generation effects. Hossain and Rees [24] considering vertical wavy plate discussed heat and mass transfer effects. Rees and Pop [25] examined free convection phenomena beside vertical wavy sheet. Cheng [26-27] using porous medium analyzed heat and mass transfer effects and double diffusive natural convection through inclined wavy surface. Saddiqa et al. [28] explained influence of radiation in natural convection flow through vertical wavy sheet. In addition to the surface texture improving thermal properties of the working fluid is another strategy towards the expedition of heat transfer processes. With the advancement of modern nanotechnology it now became possible to prepare a mixture of base fluid and metallic particles of nano-size. Nanofluid is such type of fluid whose heat transfer capabilities can be reduced or
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ACCEPTED MANUSCRIPT increased as per desire. Nanofluid increases thermal conductivity of the conventional base fluid and they have no supplementary complications, like pressure drop, sedimentation erosion, and
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non-Newtonian behavior due to low concentration of nano elements and small size (1-100nm).
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Gorla and Kumari [29] investigated free convection flow through a vertical wavy sheet in nanofluid. Mehmood et al. [30] studied MHD effects along horizontal moving wavy surface in
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nanofluid. Mehmood and Iqbal [31] discussed wavy surface texture impact on natural convection
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boundary layer of nanofluid.
In light of above literature review, the aim of this research work is to investigate the effect of
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nanoparticle in natural convective heat transfer over wavy cone surface. Famous Tiwari and Das model [32] for nanofluid has been utilized in the development of governing equations. Using
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Keller-Box technique [33-35] the governing non similar transport equations have been solved. The results obtained are discussed through tables and graphs.
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1. Mathematical Formulation Consider the natural convection flow adjacent to a vertical cone with transversal wavy surface having constant temperature 𝑇𝑤 where the constant ambient temperature is denoted by 𝑇∞ . It is supposed that cone surface is hotter than the ambient fluid, i.e. (𝑇𝑤 > 𝑇∞ ). The coordinate system is selected in such a way that 𝑥 −axis runs from the apex to flat surface of the cone and the 𝑦 −axis is measured normally out word as described in Fig. 1. The density of fluid depends upon
temperature while other properties are supposed to be constant. Different empirical mathematical models have been devised for nanofluid based on the homogeneous distribution of the nanoparticle, see for instance [36]. The most popular models are the due to Buongiorno [37] and Tiwari and Das [32].Buongiorno discovered that seven slip mechanisms (inertia, Brownian diffusion, thermophoresis, diffusionphoresis, Magnus effect, fluid drainage and gravity settling)
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ACCEPTED MANUSCRIPT take place in convective transport in nanofluid. The Brownian diffusion and the thermophoresis are the most important factors in the Buongiorno model. Therefore, the Buongiorno model
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focuses on the Brownian motion and the thermophoresis effects in the transport phenomena of
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nanofluid. On the other hand, Tiwari and Das model [32], considers thermophysical properties of nanoparticle such as, thermal conductivity, density, specific heat capacity and concentration of
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nanoparticle in his model. Because of the advent of modern nanotechnology it has now been
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possible to creat nanoparticle of size 100 nm. Such a small sized particle mixup with the fluid and does not creat the problem of clogging and sedimentation. Due to this reason the Tiwari and
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Das model is preferred by most of the researchers dealing with the nanofluid. In this model only the volume fraction, the particle dimensions and material properties are important. According to
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the Tiwari and Das model [32] the nanofluid density is given by 𝜌𝑛𝑓 = (1 − 𝜙)𝜌𝑓 + 𝜙𝜌𝑝 , where
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𝜙 denotes the nanoparticle concentration and the subscripts, ‘𝑝’, ‘𝑓’ and ‘𝑛𝑓’ denote the
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nanoparticle, base fluid and the nanofluid respectively.
Figure 1: Flow geometry and coordinate system This improved density of the nanofluid plays important role in the free convection flow which is mainly established due to the gravitational body force. Because of the absence of circular component of velocity the flow is essentially two-dimensional. According to the Tiwari and Das
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ACCEPTED MANUSCRIPT Model [32] the mass, momentum and energy conservation laws after the consideration of above assumptions read as:
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(1)
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𝜕(𝑟̅ 𝑢̅) 𝜕(𝑟̅ 𝑣̅ ) + = 0, 𝜕𝑥̅ 𝜕𝑦̅
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Mass
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Momentum
𝜕𝑢̅ 𝜕𝑢̅ 1 𝜕𝑝̅ 1 + 𝑣̅ =− + 𝜈𝑛𝑓 ∇2 𝑢̅ + 𝑔(𝜌𝛽)𝑛𝑓 (𝑇 − 𝑇∞ )𝐶𝑜𝑠𝛾, 𝜕𝑥̅ 𝜕𝑦̅ 𝜌𝑛𝑓 𝜕𝑥̅ 𝜌𝑛𝑓
(2)
𝑢̅
𝜕𝑣̅ 𝜕𝑣̅ 1 𝜕𝑝̅ 1 + 𝑣̅ =− + 𝜈𝑛𝑓 ∇2 𝑣̅ − 𝑔(𝜌𝛽)𝑛𝑓 (𝑇 − 𝑇∞ )𝑆𝑖𝑛𝛾, 𝜕𝑥̅ 𝜕𝑦̅ 𝜌𝑛𝑓 𝜕𝑦̅ 𝜌𝑛𝑓
(3)
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𝑢̅
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Energy
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𝜕𝑇 𝜕𝑇 ∗ + 𝑣̅ = 𝛼𝑛𝑓 ∇2 𝑇, 𝜕𝑥̅ 𝜕𝑦̅
(4)
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𝑢̅
where (𝑢̅ , 𝑣̅) are constituents of velocity parallel to (𝑥̅, 𝑦̅ ), these are in the direction of longitudinal and transverse directions the conventional space coordinates, g is the acceleration due to gravity, 𝛾 is cone half angle, 𝑝̅ is the pressure of the fluid, 𝜇𝑛𝑓 = 𝜇𝑓 /(1 − 𝜙)2.5 is viscosity of nanofluid approximated by Brinkman model [38], heat capacitance of nanofluid is given by ∗ 𝜙 ((𝜌𝑐𝑝 )𝑝 ) + (1 − 𝜙)(𝜌𝑐𝑝 )𝑓 , thermal diffusivity of nanofluid is 𝛼𝑛𝑓 = κ𝑛𝑓 /(𝜌c𝑝 )𝑛𝑓 , thermal
conductivity of the nanofluid [39] is
κ𝑛𝑓 κ𝑓
=
(𝜅𝑝 +2𝜅𝑓 )−2𝜙(𝜅𝑓 −𝜅𝑝 ) (𝜅𝑝 +2𝜅𝑓 )+𝜙(𝜅𝑓 −𝜅𝑝 )
and (𝜌𝛽)𝑛𝑓 = 𝜙(𝜌𝛽)𝑝 +
(1 − 𝜙)(𝜌𝛽)𝑓 is the thermal expansion coefficient. The local radius 𝑟̅ of the corresponding cone
flat surface is described as 𝑟̅ (𝑥̅ ) = 𝑥̅ 𝑆𝑖𝑛𝛾 and the sinusoidal wavy surface is defined by 𝑦̅𝑤 = 𝜋𝑥̅
𝜎̅(𝑥̅ ) = 𝛼̅𝑆𝑖𝑛 ( 𝑙 ) as shown in Fig. 1. 6
ACCEPTED MANUSCRIPT The applicable boundary conditions in the perspective of the flow assumptions read as 𝑦̅ = 𝜎̅(𝑥̅ ): 𝑢̅ = 0, 𝑣̅ = 0, 𝑇 = 𝑇𝑤 ,
(5)
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𝑦̅ → ∞: 𝑢̅ = 0, 𝑝̅ = 𝑝∞ , 𝑇 = 𝑇∞ .
In accordance with the cone geometry we use stream function like 𝑟𝑣 = −𝜕𝜓/𝜕𝑥 , 𝑟𝑢 = 𝜕𝜓/𝜕𝑦,
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due to which the equation of continuity is satisfied identically. In order to normalize the system,
𝑣=
3 1 𝑔𝛽𝑓 (𝑇𝑤 − 𝑇∞ )𝑙 3 𝑇 − 𝑇∞ 𝑙2 , 𝜓(𝜉 , 𝜂) = 𝜉 4 𝑓(𝜉 , 𝜂), 𝑝 = 2 𝐺𝑟 −4 𝑝̅ , 𝐺 = , 𝑇𝑤 − 𝑇∞ 𝜈 𝜌𝑓 𝜈𝑓2
(6)
κ𝑛𝑓 𝜌 𝜌 , Ε1 = (1 − 𝜙)2.5 [1 − 𝜙 + 𝜙 ( 𝑝⁄𝜌𝑓 )] , Ε2 = [1 − 𝜙𝜙 ( 𝑝⁄𝜌𝑓 )], κ𝑓
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Ε=
𝜌𝑓 𝑙 −1 𝜈𝑓 𝜎̅(𝑥̅ ) 𝐺𝑟 4 (𝑣̅ − 𝜎𝜉 𝑢̅), 𝜎 = , 𝑟(𝜉) = 𝜉 𝑆𝑖𝑛𝛾, 𝑃𝑟 = , Η = √1 + 𝜎𝜉2 , 𝜇𝑓 𝑙 𝛼𝑓
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𝜃(𝜉 , 𝜂) =
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1 𝜌𝑓 𝑙 −1 𝑥̅ 𝑦̅ 𝑟̅ 𝑦̅ − 𝜎̅(𝑥̅ ) 1 , 𝑦 = , 𝑟 = , 𝜂 = 𝜉 −4 𝑦 , 𝑦 = 𝐺𝑟 4 , 𝑢 = 𝐺𝑟 2 𝑢̅ , 𝑙 𝑙 𝑙 𝑙 𝜇𝑓
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𝜉=𝑥=
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we utilize the suitable set of variables given by
Ε3 = [1 − 𝜙 + 𝜙 ((𝜌𝑐𝑝 ) /(𝜌𝑐𝑝 ) )] , Ε4 = [1 − 𝜙 + 𝜙 ((𝜌𝛽𝑝 ) /(𝜌𝛽𝑝 ) )],
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𝑝
𝑓
𝑝
𝑓
which transform the system (2) - (5) to the following form
Ε4 (1 − 𝜎𝜉 𝑇𝑎𝑛𝜉) Η2 7 1 Η𝜉 𝜕𝑓′ 𝜕𝑓 𝑓′′′ + 𝑓𝑓′′ − ( + 𝜉)𝑓′2 + 𝜃 = 𝜉 [𝑓 ′ − 𝑓′′ ], 2 Ε1 4 2 Η 𝜕𝜉 𝜕𝜉 Ε2 Η ΕΗ2 Ε3 𝑃𝑟
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𝜕𝜃
𝜕𝑓
𝜃′′ + 4 𝑓𝜃′ = 𝜉 [𝑓′ 𝜕𝜉 − 𝜃′ 𝜕𝜉 ],
(7) (8)
𝑓(𝜉 ,0) = 𝑓′(𝜉 ,0) = 𝜃(𝜉 ,0) − 1 = 0,
(9) 𝑓′(𝜉 , ∞) = 𝜃(𝜉 , ∞) = 0. Here Ε , Ε1 , Ε2 , Ε3 and Ε4 are the material parameters, 𝐺𝑟 is the Grashof number, Η is wavy contribution, 𝑃𝑟 is the Prandtl number, subscript ’ 𝜉 ’ represents derivative accordi---ng to 𝜉 and the ‘ ' ’ represents differentiation according to 𝜂. The wall shear stress 𝜏𝑤 = 𝜇𝑛𝑓 (∇𝑢̅ . 𝑛̂)𝑦=0 and
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number𝑁𝑢𝑥 = 𝑥̅ 𝑞𝑤 /𝜅𝑓 (𝑇𝑤 − 𝑇∞ ). In view of the non-dimensional transformations (6) their final
Η 𝑓′′ (𝜉 ,0), (1 − 𝜙)2.5 κ𝑛𝑓 κ𝑓
𝜃
′(𝜉 ,0)
(10)
,
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𝑁𝑢 = 𝑁𝑢𝑥 (𝐺𝑟𝑥 3 )−1/4 = Η
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𝐶𝑓 = 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4 =
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expressions read as:
2. Method of Solution
𝜎𝜉 1
, ).
Η Η
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where unit normal to the wavy surface is 𝑛̂ = (−
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Using implicit finite difference numerical scheme [33, 34, 35] the transport equations (7) and (8)
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together with the boundary conditions (9) have been solved. Any quantity 𝑓 in this scheme, at the point (𝜉𝑛 , 𝜂𝑗 ) is written as 𝑓𝑗𝑛 ; quantities at the midpoints of grid segments are assumed to second
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𝑛 𝑛 order as 𝑓𝑗𝑛−1/2 = 1/2(𝑓𝑗𝑛 + 𝑓𝑗𝑛−1 ), 𝑓𝑗−1/2 = 1/2(𝑓𝑗𝑛 + 𝑓𝑗−1 ), and the derivatives are assumed to 𝑛 second order as (𝜕𝑓/𝜕𝜉)𝑛−1/2 = 1/𝑘𝑛 (𝑓𝑗𝑛 + 𝑓𝑗𝑛−1 ), (𝜕𝑓/𝜕𝜂)𝑛𝑗−1/2 = 1/ℎ𝑗 (𝑓𝑗𝑛 + 𝑓𝑗−1 ) where 𝑓 is 𝑗
any dependent variable, 𝑛 and 𝑗 are the node locations along 𝜉 and 𝜂 directions correspondingly. By using the above central difference approximation in terms of the finite differences the differential equations are changed into the system of first order differential equations. According to Newton’s linearization process non-linear algebraic equations are linearized. Finally by tridiagonal block elimination method the system of linearized algebraic equations is solved. The solution was supposed to have converged and the iterative process was stopped if the comparative difference between the present and the previous iterations reached 10−5. In order to validate the precision of the current technique, we have matched our outcomes with those of Hearing et al. [3], Roy et al. [5], Singh et al. [8], Yih et al. [10] and Na et al. [16-17] for 8
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the skin friction coefficient 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4 and local Nusselt number 𝑁𝑢𝑥 (𝐺𝑟𝑥3 )−1/4 in Table 1. The comparisons are found to be superb match in all the above cases. After ensuring the accuracy of
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the current procedure we have employed it to the current problem.
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3. Results and Discussion
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In this exploring work we have shown the effects of amplitude of wavy surface, nanoparticle concentration, and cone half angle on the natural convection flow of nanofluid past a vertical
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wavy cone. The skin friction (𝐶𝑓 ) and the Nusselt number (𝑁𝑢) are the important physical
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quantities that are necessary to examine. The 𝐶𝑓 and 𝑁𝑢 plots are shown in Figs. 2-7 when 𝐶𝑢water based nanofluid is considered. Nanoparticle concentration 𝜙 varies from 0- 20 % and wavy
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amplitude 𝛼 varies from 0 to 0.2, the values of the cone half angle 𝛾 are taken 𝜋/12, 𝜋/6
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and 𝜋/4 for fixed Prandtl number (𝑃𝑟 = 7.0). The computation is performed for different values
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of the wavy amplitude 𝛼 , nanoparticle concentration parameter 𝜙 and the cone inclination half angle 𝛾. We used five nanoparticles viz. alumina (𝐴𝑙2 𝑂3), silver (𝐴𝑔), copper (𝐶𝑢), copper oxide (𝐶𝑢𝑂), and titania (𝑇𝑖𝑂2 ). Figures. 2 - 7 illustrate the change in 𝐶𝑓 and 𝑁𝑢 for increasing values of wavy amplitude 𝛼 , nanoparticle concentration parameter 𝜙 and the cone inclination half angle 𝛾 for Cu -water nanoparticle. These figures show that 𝐶𝑓 and 𝑁𝑢 enhance with the increase of nanoparticle concentration parameter 𝜙 and cone inclination angle 𝛾 whereas a decrease is depicted with an increase of wavy amplitude 𝛼. Figures 8 - 13 depict the variation of the 𝐶𝑓 and 𝑁𝑢 with the variation of wavy amplitude 𝛼 , nanoparticle volume fraction parameter 𝜙 and cone
inclination angle 𝛾 for the selected types of the nanoparticle. From these figures it is clear that the 𝑁𝑢 enhances with the increase of wavy amplitude 𝛼 , nanoparticle centration parameter 𝜙 and cone inclination angle 𝛾. The 𝐶𝑓 enhances with the increase of nanoparticle concentration parameter 𝜙 whereas decreases with the increase of wavy amplitude 𝛼 and cone inclination angle 9
ACCEPTED MANUSCRIPT 𝛾. Also it is noted that the 𝑁𝑢 is maximum in the case of 𝐶𝑢-nanoparticle whereas 𝐶𝑓 is maximum for 𝐴𝑔-nanoparticle; 𝐶𝑓 and 𝑁𝑢 are minimum for 𝑇𝑖𝑂2 -nanoparticle. Isotherms are
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plotted for some values of 𝛼 , 𝜙 (𝛼 = 𝜙 = 0.0, 0.1, 0.2) and 𝛾 (𝛾 = 𝜋/12, 𝜋/6, 𝜋/4) in
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Figures 14-16. The wavy pattern is prominent and the figures indicate that with the increase of
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these parameters the thermal boundary layer thickness increases.
Thermo-physical properties of the five nanoparticles 𝐶𝑢, 𝐶𝑢𝑂, 𝐴𝑙2 𝑂3, 𝐴𝑔 , 𝑇𝑖𝑂2 and water are
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shown in Table 2. The results of 𝐶𝑓 and 𝑁𝑢 are tabulated in Table 3. The values are reported for the selected values of 𝛼 , 𝜙 and 𝛾 when 𝑃𝑟 = 7.0 for copper (𝐶𝑢) nanoparticle along vertical
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wavy cone surface at 𝑥 = 0.5, 𝑥 = 1.0 and 𝑥 = 1.5. It is observed that the 𝑁𝑢 enhances with the
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increase of 𝛼 , 𝜙 and 𝛾 but 𝐶𝑓 decreases with the increase of 𝛼 and 𝛾 at crest and decreasing
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behavior at node and trough.
Fig. 2: Effect of 𝛼 on skin friction Fig. 3: Nusselt number plotted against 𝜉 for profile. variation of 𝛼 .
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Fig. 4: Influence of 𝜙 on skin friction.
Fig. 6: change of skin friction for different Fig. 7: Nusselt number graph for different 𝛾.
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𝛾.
Fig. 8: Effect of 𝜙 on Skin Friction for Fig. 9: Nusselt number profile against 𝜙 for different nanoparticles. different nanoparticles.
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Fig. 10: The effect of 𝛼 on Skin friction for Fig.11: Nusselt number against 𝛼 for different nanoparticles. different nanoparticles.
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Fig. 12: The effect of 𝛾 on 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4for Fig.13: Change in Nusselt number against 𝛾 for different nanoparticles. different nanoparticles.
Fig. 14: Isotherms plotted at 𝛼 (𝛼 = Fig. 15: Effect of 𝜙 (𝜙 = 0.0, 0.1, 0.2) on Isotherms. 0.0, 0.1, 0.2).
Fig. 16: Isotherms at various values of 𝛾 (𝛾 = 𝜋/12, 𝜋/6, 𝜋/4). 12
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Table 1: Comparison with the existing data for 𝑓 ′′ (0, 0) and −𝜃 ′ (0, 0) when 𝛼 = 𝜉 = 0, 𝛾 =
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0, 𝜑 = 0.
Hering [4] 1.5166 1.3550 1.0960 0.7694 -
Roy [5] 0.8600 0.4899 0.2897 0.1661 0.0940
Yih et al. [10] 1.6006 1.5135 1.3551 1.0960 0.7699 0.4877 0.2896 0.1661 0.0940
Present
0.033845 0.038402 0.075460 0.211345 0.451095 0.510399 1.033989 1.922854 3.470171 6.200679
Na et.al [16, 17] 0.07493 0.45101 0.51039 1.03397 1.92197 3.470882 6.204813
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1.452616 1.440436 1.348483 1.095916 0.819591 0.769428 0.487697 0.289635 0.166145 0.094042
Singh et al. [8] 0.81959 -
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0.0001 0.001 0.01 0.1 0.7 1 10 100 1000 10000
Na et.al [16, 17] 0.81959 -
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present
−𝜃 ′ (0, 0)
Singh et al. [8] 0.45109 -
Yih et al. [9] 0.0079 0.0246 0.0749 0.2116 0.5109 1.0339 1.9226 3.4696 6.1984
Hering [4] 0.0247 0.0748 0.2113 0.5104 -
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Pr
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𝑓 ′′ (0, 0)
𝐶𝑝 (𝐽/ 𝐾𝑔𝐾) 383.1 535.6 765 235 686.2 4179
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Properties
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Table 2: Thermo-physical properties of nanoparticles and water.
𝐶𝑢 𝐶𝑢𝑂 𝐴𝑙2 𝑂3 𝐴𝑔 𝑇𝑖𝑂2 Fluid(water)
𝜌(𝐾𝑔/ 𝑚3 ) 8954 6500 3970 10500 4250 997.1
𝜅(𝑊 /𝑚𝐾) 386 20 40 429 8.9538 0.613
𝛽 × 10−5 (1/𝐾) 1.67 1.80 0.85 1.89 0.90 21.0
Table 3: Skin friction (𝐶𝑓 ) and Nusselt number (𝑁𝑢) data for different 𝛼, 𝜙 𝑎𝑛𝑑 𝛾 when 𝑃𝑟 = 7.0 for cupper (𝐶𝑢) nanoparticle. 𝜉
0.5 (crest)
𝜙 0.2
𝛼 0.0 0.1
𝛾 𝜋/4
0.2
0 𝜋/12
0 0.1
13
𝐶𝑓 0.6401 0.6285 0.5100 0.5572 0.6452 0.6383
𝑁𝑢 1.1496 1.1542 0.9457 1.0529 1.1409 1.1442
Roy [5] 0.5275 1.0354 1.9229 3.4700 6.1998
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0.2 0.0 0.1
0 𝜋/12 𝜋/6 𝜋/4
0.2
0.0 0.1 0.0 0.1
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1.5 (trough)
𝜋/4
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0.2
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0 0.1
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1.0 (node)
𝜋/4
0 𝜋/12 𝜋/6 𝜋/4
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0.2
1.1492 1.1590 1.1496 1.2179 1.0007 1.1167 1.0717 1.1197 1.1697 1.2308 1.1496 1.1471 0.9156 1.0275 1.1298 1.1310 1.1332 1.1370
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0.2
0.6298 0.6170 0.6401 0.7183 0.5697 0.6233 0.4929 0.5490 0.6109 0.6914 0.6401 0.6621 0.5582 0.6160 0.6473 0.6584 0.6707 0.6869
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4. Concluding Remarks
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Heat transfer phenomena in natural convection flow has been investigated past a wavy cone for base fluid water and five nanoparticles, namely, alumina (𝐴𝑙2 𝑂3), copper (𝐶𝑢), copper oxide (𝐶𝑢𝑂), silver (𝐴𝑔), and titania (𝑇𝑖𝑂2). The non-linear transport equations have been solved numerically and the accuracy of the solution scheme has been proved by giving comparison with the already existing data. Results show that heat transfer rate is significantly increased by addition of nanoparticles with respect to the base liquid and heat transfer enhancement seems to be more prominent with the increase of the nanoparticle volume fraction. It is shown that Nusselt number is maximum in the case of 𝐶𝑢-nanoparticle and decreases successively for 𝐴𝑔nanoparticle, 𝐴𝑙2 𝑂3-nanoparticle, 𝐶𝑢𝑂-nanoparticle and 𝑇𝑖𝑂2 -nanoparticle. Furthermore, it is concluded that the nanofluid serves as very useful fluid in the expeditions of cooling and heating processes.
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References
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1. H.J. Merk, J.A. Prins, Thermal convection in laminar boundary layer-I. Int. Appl. Sci.
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Res. A4 (1953), pp. 11-24
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5. S. Roy, Free convection from a vertical cone at high Prandtl numbers. ASME, J. Heat Transf. 96 (1974), pp. 115-117
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6. F.N. Lin, Laminar convection from a vertical cone with uniform surface heat flux. Lett. Heat Mass Transf. 3 (1976), pp. 49-58 7. M. Alamgir, Overall heat transfer from vertical cones in laminar free convection an approximate method. ASME, J. Heat Transf. 101(1) (1979), pp. 174-176 8. P. Singh, V. Radhakrishnan, K.A. Narayan, Non similar solutions of free convection flow over a frustum of a cone. Ingen. Arch. 59 (1989), pp. 382-389 9. I. Pop, H.S. Takhar, Compressibility effects in laminar free convection from a vertical cone. Appl. Sci. Res. 48 (1991), pp. 71-82 10. K.Y. Yih, The effect of uniform lateral mass flux on free convection about a vertical cone embedded in porous media. Int. Commun. Heat Mass Transf. 24 (1997), pp. 1195-1205
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ACCEPTED MANUSCRIPT 11. M.A. Hossain, S.C. Paul Free convection from a vertical permeable cone with nonuniform surface temperature. Acta Mech. 151 (2001), pp. 103-114
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12. M.A. Hossain, S.C. Paul, Free convection from a vertical permeable circular cone with
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non-uniform surface heat flux. Heat Mass Transf. 37 (2001), pp. 167-173 13. C.Y. Cheng, Natural convection boundary layer flow of a micro polar fluid over a
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14. B. Pullepu, K. Ekambavanan, A.J. Chamkha, Unsteady laminar free convection from a
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vertical cone with uniform surface heat flux. Nonlinear Anal. Model. Control 13 (1) (2008), pp. 47–60
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15. I. Pop, T.Y. Na, Natural convection from a wavy cone. Appl. Sci. Res. 54 (1995), pp.
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16. I. Pop, T.Y. Na, Natural convection of a Darcian fluid about a wavy cone. Int. Comm. Heat Mass Transf. 21 (1994), pp. 891–899 17. I. Pop, T.Y. Na, Natural convection over a frustum of a wavy cone in a porous medium. Mech. Res. Comm. 22 (1995), 1pp. 81–190 18. C.Y. Cheng, Natural convection heat and mass transfer near a wavy cone with constant wall temperature and concentration in a porous medium. Mech. Res. Comm. 27 (2000), pp. 613–620 19. M.A. Hossain , M.S. Munir, I. Pop, Natural convection flow of a viscous fluid with viscosity inversely proportional to linear function of temperature from a vertical wavy cone. Int. J. Therm. Sci. 40 (2001), pp. 366–371
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ACCEPTED MANUSCRIPT 20. A. Rahman, M.M. Molla, M.M.A. Sarker, Natural convection flow along the vertical wavy cone in case of uniform surface heat flux where viscosity is an exponential function
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of temperature. Int. Comm. Heat Mass Transf. 38 (2011), pp. 774-780
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21. L.S. Yao, Natural convection along a vertical wavy surface. ASME J. Heat Transf. 105 (1983), pp. 465-468
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22. S.G. Moulic, L.S. Yao, Natural convection along a wavy surface with uniform heat flux.
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ASME J. Heat Transf. 111 (1989), pp. 1106-1108
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surface with heat generation/absorption. Int. J. Therm. Sci. 43 (2004), pp. 157-163 24. D.A.S. Rees, I. Pop, A note on a free convection along a vertical wavy surface in a
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porous medium. ASME J. Heat Transf. 115 (1994), pp. 505-508 25. M.A. Hossain, D.A.S. Rees, Combined heat and mass transfer in natural convection flow
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from a vertical wavy surface. Acta Mech. 136 (1999), pp. 133-141 26. C.Y. Cheng, Natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. Int. Commun. Heat Mass Transf. 27 (2000), pp. 1143-1154 27. C.Y. Cheng, Double diffusive natural convection along an inclined wavy surface in a porous medium. Int. Commun. Heat Mass Transf. 37 (2010), pp. 1471–1476 28. S. Siddiqa, M.A. Hossain, S.C. Saha, Natural convection flow with surface radiation along a vertical wavy surface. Num. Heat Transf. 64 (2013), pp. 400-415 29. R.S.R. Gorla, M. Kumari, Free convection along a vertical wavy surface in a nanofluid. J. Nano Engng. Nano Sys. 225(3) (2011), pp. 133-142
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ACCEPTED MANUSCRIPT 30. A. Mehmood, M. S. Iqbal, I. Mustafa, Cooling of moving wavy surface through MHD nanofluid. Zeitschrift für Naturforschung A, (2016) DOI: 10.1515/zna-2016-0044.
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31. A. Mehmood, M. S. Iqbal, Impact of surface texture on natural convection boundary
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layer of nanofluid. Therm. Sci., (2016), DOI: TSCI1151008122M.
32. R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven
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differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50 (200),
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pp. 2002-2018
33. T.Y. Na, Computational Methods in Engineering Boundary Value Problems. Academic
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Press New York 1979.
34. H.B. Keller, Numerical methods in boundary layer theory. Ann. Rev. Fluid Mech. 10
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(1988) 793-796.
35. T. Cebeci, P. Bradshaw, Physical and Computational Aspects of Convective Heat
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Transfer. Springer New York 1988. 36. W.J. Minkowycz, E.M. Sparrow, J.P. Abraham, Nanoparticles heat transfer and fluid flow. CRC Press, Taylor & Francis Group, Boca Raton, London, New York 2013. 37. J. Buongiorno, Convective transport in nanofluids. J. Heat Transf. 128 (2006), pp. 240250
38. H.C. Brinkman, The viscosity of concentrated suspensions and solution. J. Chem. Phy. 20 (1952), pp. 571–581 39. J.C. Maxwell Garnett, Colours in metal glasses and in metallic films. Philos. Trans. R. Soc. Lond. A 203 (1904), pp. 385 – 420
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ACCEPTED MANUSCRIPT Highlights
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3 4
Tiwari and Das model of nanofluid has been utilized for this research article. Heat transfer rate is significantly increased by addition of nanoparticles with respect to the base liquid. Copper has higher heat transfer as compared to other nanoparticles considered. 𝑇𝑖𝑂2 -nanoparticle shown to have the maximum cooling performance.
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S0167-7322(16)32288-7 doi:10.1016/j.molliq.2016.09.029 MOLLIQ 6306
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 August 2016 3 September 2016 9 September 2016
Please cite this article as: Ahmer Mehmood, Muhammad Saleem Iqbal, Heat transfer analysis in natural convection flow of nanofluid past a wavy cone, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.09.029
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ACCEPTED MANUSCRIPT Heat transfer analysis in natural convection flow of nanofluid past a wavy
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cone
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Ahmer Mehmood, Muhammad Saleem Iqbal1
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Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad 44000, Pakistan.
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Abstract:
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The problem of natural convective heat transfer of water-based nanofluid along wavy cone surface is investigated numerically. Analysis is performed to study the heat transfer
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augmentation due to five types of nanoparticles, namely, alumina (𝐴𝑙2 𝑂3), copper (𝐶𝑢), silver
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(𝐴𝑔), copper oxide (𝐶𝑢𝑂) and titania (𝑇𝑖𝑂2). The flow has been assumed to be steady and fluid
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properties have been supposed constant except for the density depending upon temperature giving rise to the buoyancy force. Famous Tiwari and Das model of nanofluid has been utilized in this study. The effects of cone half angle 𝛾 and amplitude of the waviness 𝛼 on the Nusselt number (𝑁𝑢) and skin friction (𝐶𝑓 ) are studied. A comparison is made with the case of pure fluid flow past wavy cone at different values of 𝛼. It has been observed that the 𝑇𝑖𝑂2-nanoparticle shown to have the maximum cooling performance and 𝐶𝑢-nanoparticle appeared to have maximum heating performance for this study. The results shown in this research arrange for a significant source of reference for taming the natural convection heat transfer enactment along wavy cone. Present results for selected variables are matched with the already published work for pure fluid and are shown to be in good agreement. 1
Corresponding Author’s Email: [email protected] Cell No. +923335195130
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ACCEPTED MANUSCRIPT Keywords: Natural convection, heat transfer, Nanofluid, wavy cone surface
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1. Introduction Convective heat transfer phenomena over a cone is applicable in various designs of thermal
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equipment like heat exchangers, geothermal reservoirs, nuclear reactor cooling, solar energy
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plants, design of space crafts, drying dehydration process in chemical and food process and
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steam generators etc. The science and the art of heat and mass transfer augmentation has evolved into an important component of various aspects of thermal science and engineering. This poses a
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great challenge for choosing appropriate design and application information to achieve the industrial and technology goals. Regarding the cone geometry the phenomena of natural
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convection heat transfer over vertical cone have been studied by several researchers and
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scientists. Merk and Prins [1-2] discussed natural convection flow along a cone for pure fluid.
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Hering and Grosh [3-4] analyzed the free convection for low Prandtl number fluids over a nonisothermal cone. For high Prandtl number fluids Roy [5] investigated natural convection over an isothermal cone. Lin [6] discussed the uniform surface heat flux from a vertical cone. Alamgir [7] investigated free heat transfer characteristics from vertical cone using approximate technique. Pop et al. [8] discussed compressibility effects in natural convection from a vertical cone. Yih [9] analyzed under uniform mass flux free convection along a vertical cone in porous medium. Cheng [10] examined the effect of variable wall temperature in free convection flow of a micropolar fluid over a vertical permeable cone. Hossain and Paul [11-12] discussed the effect of uniform heat flux and non-uniform surface temperature in free convection from a vertical circular cone. Cheng [13] discussed the effect of variable wall temperature in natural convection flow of a micro polar fluid along a vertical cone. Pullepu [14] explained uniform heat flux case for unsteady natural convection from a vertical cone. 2
ACCEPTED MANUSCRIPT The above cited studies are restricted to the cases where the cone surface is supposed to be uniform. Attention have also been given to the cases where the cone surface is taken to be non-
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uniform such as the wavy one. Since the irregular surface shape changes the flow and hence the
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heat transfer rate. The wavy shaped surface occurs frequently in practice for example flat-plate solar collectors in refrigerators and flat-plate condensers. Pop and Na [15-17] using porous
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medium discussed the natural convection flow along a vertical and frustum of a wavy cone.
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Cheng [18] investigated heat and mass transfer phenomena in natural convection along wavy cone in porous media. Considering viscosity dependent temperature Hossain et al. [19] examined
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free convection flow over a vertical wavy cone. Taking viscosity as an exponential function of temperature Rahman et al. [20] discussed free convection flow beside the vertical wavy cone.
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Important contributions and preliminary work related to the problem of natural convection in a viscous fluid along wavy surfaces are briefly given here. Yao [21] discussed free convection
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along vertical wavy sheet. Moulic and Yao [22] studied same problem with uniform heat flux. Molla et al. [23] examined natural convection through vertical wavy plate with heat generation effects. Hossain and Rees [24] considering vertical wavy plate discussed heat and mass transfer effects. Rees and Pop [25] examined free convection phenomena beside vertical wavy sheet. Cheng [26-27] using porous medium analyzed heat and mass transfer effects and double diffusive natural convection through inclined wavy surface. Saddiqa et al. [28] explained influence of radiation in natural convection flow through vertical wavy sheet. In addition to the surface texture improving thermal properties of the working fluid is another strategy towards the expedition of heat transfer processes. With the advancement of modern nanotechnology it now became possible to prepare a mixture of base fluid and metallic particles of nano-size. Nanofluid is such type of fluid whose heat transfer capabilities can be reduced or
3
ACCEPTED MANUSCRIPT increased as per desire. Nanofluid increases thermal conductivity of the conventional base fluid and they have no supplementary complications, like pressure drop, sedimentation erosion, and
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non-Newtonian behavior due to low concentration of nano elements and small size (1-100nm).
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Gorla and Kumari [29] investigated free convection flow through a vertical wavy sheet in nanofluid. Mehmood et al. [30] studied MHD effects along horizontal moving wavy surface in
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nanofluid. Mehmood and Iqbal [31] discussed wavy surface texture impact on natural convection
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boundary layer of nanofluid.
In light of above literature review, the aim of this research work is to investigate the effect of
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nanoparticle in natural convective heat transfer over wavy cone surface. Famous Tiwari and Das model [32] for nanofluid has been utilized in the development of governing equations. Using
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Keller-Box technique [33-35] the governing non similar transport equations have been solved. The results obtained are discussed through tables and graphs.
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1. Mathematical Formulation Consider the natural convection flow adjacent to a vertical cone with transversal wavy surface having constant temperature 𝑇𝑤 where the constant ambient temperature is denoted by 𝑇∞ . It is supposed that cone surface is hotter than the ambient fluid, i.e. (𝑇𝑤 > 𝑇∞ ). The coordinate system is selected in such a way that 𝑥 −axis runs from the apex to flat surface of the cone and the 𝑦 −axis is measured normally out word as described in Fig. 1. The density of fluid depends upon
temperature while other properties are supposed to be constant. Different empirical mathematical models have been devised for nanofluid based on the homogeneous distribution of the nanoparticle, see for instance [36]. The most popular models are the due to Buongiorno [37] and Tiwari and Das [32].Buongiorno discovered that seven slip mechanisms (inertia, Brownian diffusion, thermophoresis, diffusionphoresis, Magnus effect, fluid drainage and gravity settling)
4
ACCEPTED MANUSCRIPT take place in convective transport in nanofluid. The Brownian diffusion and the thermophoresis are the most important factors in the Buongiorno model. Therefore, the Buongiorno model
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focuses on the Brownian motion and the thermophoresis effects in the transport phenomena of
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nanofluid. On the other hand, Tiwari and Das model [32], considers thermophysical properties of nanoparticle such as, thermal conductivity, density, specific heat capacity and concentration of
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nanoparticle in his model. Because of the advent of modern nanotechnology it has now been
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possible to creat nanoparticle of size 100 nm. Such a small sized particle mixup with the fluid and does not creat the problem of clogging and sedimentation. Due to this reason the Tiwari and
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Das model is preferred by most of the researchers dealing with the nanofluid. In this model only the volume fraction, the particle dimensions and material properties are important. According to
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the Tiwari and Das model [32] the nanofluid density is given by 𝜌𝑛𝑓 = (1 − 𝜙)𝜌𝑓 + 𝜙𝜌𝑝 , where
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𝜙 denotes the nanoparticle concentration and the subscripts, ‘𝑝’, ‘𝑓’ and ‘𝑛𝑓’ denote the
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nanoparticle, base fluid and the nanofluid respectively.
Figure 1: Flow geometry and coordinate system This improved density of the nanofluid plays important role in the free convection flow which is mainly established due to the gravitational body force. Because of the absence of circular component of velocity the flow is essentially two-dimensional. According to the Tiwari and Das
5
ACCEPTED MANUSCRIPT Model [32] the mass, momentum and energy conservation laws after the consideration of above assumptions read as:
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(1)
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𝜕(𝑟̅ 𝑢̅) 𝜕(𝑟̅ 𝑣̅ ) + = 0, 𝜕𝑥̅ 𝜕𝑦̅
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Mass
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Momentum
𝜕𝑢̅ 𝜕𝑢̅ 1 𝜕𝑝̅ 1 + 𝑣̅ =− + 𝜈𝑛𝑓 ∇2 𝑢̅ + 𝑔(𝜌𝛽)𝑛𝑓 (𝑇 − 𝑇∞ )𝐶𝑜𝑠𝛾, 𝜕𝑥̅ 𝜕𝑦̅ 𝜌𝑛𝑓 𝜕𝑥̅ 𝜌𝑛𝑓
(2)
𝑢̅
𝜕𝑣̅ 𝜕𝑣̅ 1 𝜕𝑝̅ 1 + 𝑣̅ =− + 𝜈𝑛𝑓 ∇2 𝑣̅ − 𝑔(𝜌𝛽)𝑛𝑓 (𝑇 − 𝑇∞ )𝑆𝑖𝑛𝛾, 𝜕𝑥̅ 𝜕𝑦̅ 𝜌𝑛𝑓 𝜕𝑦̅ 𝜌𝑛𝑓
(3)
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𝑢̅
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Energy
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𝜕𝑇 𝜕𝑇 ∗ + 𝑣̅ = 𝛼𝑛𝑓 ∇2 𝑇, 𝜕𝑥̅ 𝜕𝑦̅
(4)
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𝑢̅
where (𝑢̅ , 𝑣̅) are constituents of velocity parallel to (𝑥̅, 𝑦̅ ), these are in the direction of longitudinal and transverse directions the conventional space coordinates, g is the acceleration due to gravity, 𝛾 is cone half angle, 𝑝̅ is the pressure of the fluid, 𝜇𝑛𝑓 = 𝜇𝑓 /(1 − 𝜙)2.5 is viscosity of nanofluid approximated by Brinkman model [38], heat capacitance of nanofluid is given by ∗ 𝜙 ((𝜌𝑐𝑝 )𝑝 ) + (1 − 𝜙)(𝜌𝑐𝑝 )𝑓 , thermal diffusivity of nanofluid is 𝛼𝑛𝑓 = κ𝑛𝑓 /(𝜌c𝑝 )𝑛𝑓 , thermal
conductivity of the nanofluid [39] is
κ𝑛𝑓 κ𝑓
=
(𝜅𝑝 +2𝜅𝑓 )−2𝜙(𝜅𝑓 −𝜅𝑝 ) (𝜅𝑝 +2𝜅𝑓 )+𝜙(𝜅𝑓 −𝜅𝑝 )
and (𝜌𝛽)𝑛𝑓 = 𝜙(𝜌𝛽)𝑝 +
(1 − 𝜙)(𝜌𝛽)𝑓 is the thermal expansion coefficient. The local radius 𝑟̅ of the corresponding cone
flat surface is described as 𝑟̅ (𝑥̅ ) = 𝑥̅ 𝑆𝑖𝑛𝛾 and the sinusoidal wavy surface is defined by 𝑦̅𝑤 = 𝜋𝑥̅
𝜎̅(𝑥̅ ) = 𝛼̅𝑆𝑖𝑛 ( 𝑙 ) as shown in Fig. 1. 6
ACCEPTED MANUSCRIPT The applicable boundary conditions in the perspective of the flow assumptions read as 𝑦̅ = 𝜎̅(𝑥̅ ): 𝑢̅ = 0, 𝑣̅ = 0, 𝑇 = 𝑇𝑤 ,
(5)
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𝑦̅ → ∞: 𝑢̅ = 0, 𝑝̅ = 𝑝∞ , 𝑇 = 𝑇∞ .
In accordance with the cone geometry we use stream function like 𝑟𝑣 = −𝜕𝜓/𝜕𝑥 , 𝑟𝑢 = 𝜕𝜓/𝜕𝑦,
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due to which the equation of continuity is satisfied identically. In order to normalize the system,
𝑣=
3 1 𝑔𝛽𝑓 (𝑇𝑤 − 𝑇∞ )𝑙 3 𝑇 − 𝑇∞ 𝑙2 , 𝜓(𝜉 , 𝜂) = 𝜉 4 𝑓(𝜉 , 𝜂), 𝑝 = 2 𝐺𝑟 −4 𝑝̅ , 𝐺 = , 𝑇𝑤 − 𝑇∞ 𝜈 𝜌𝑓 𝜈𝑓2
(6)
κ𝑛𝑓 𝜌 𝜌 , Ε1 = (1 − 𝜙)2.5 [1 − 𝜙 + 𝜙 ( 𝑝⁄𝜌𝑓 )] , Ε2 = [1 − 𝜙𝜙 ( 𝑝⁄𝜌𝑓 )], κ𝑓
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Ε=
𝜌𝑓 𝑙 −1 𝜈𝑓 𝜎̅(𝑥̅ ) 𝐺𝑟 4 (𝑣̅ − 𝜎𝜉 𝑢̅), 𝜎 = , 𝑟(𝜉) = 𝜉 𝑆𝑖𝑛𝛾, 𝑃𝑟 = , Η = √1 + 𝜎𝜉2 , 𝜇𝑓 𝑙 𝛼𝑓
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𝜃(𝜉 , 𝜂) =
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1 𝜌𝑓 𝑙 −1 𝑥̅ 𝑦̅ 𝑟̅ 𝑦̅ − 𝜎̅(𝑥̅ ) 1 , 𝑦 = , 𝑟 = , 𝜂 = 𝜉 −4 𝑦 , 𝑦 = 𝐺𝑟 4 , 𝑢 = 𝐺𝑟 2 𝑢̅ , 𝑙 𝑙 𝑙 𝑙 𝜇𝑓
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𝜉=𝑥=
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we utilize the suitable set of variables given by
Ε3 = [1 − 𝜙 + 𝜙 ((𝜌𝑐𝑝 ) /(𝜌𝑐𝑝 ) )] , Ε4 = [1 − 𝜙 + 𝜙 ((𝜌𝛽𝑝 ) /(𝜌𝛽𝑝 ) )],
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𝑝
𝑓
𝑝
𝑓
which transform the system (2) - (5) to the following form
Ε4 (1 − 𝜎𝜉 𝑇𝑎𝑛𝜉) Η2 7 1 Η𝜉 𝜕𝑓′ 𝜕𝑓 𝑓′′′ + 𝑓𝑓′′ − ( + 𝜉)𝑓′2 + 𝜃 = 𝜉 [𝑓 ′ − 𝑓′′ ], 2 Ε1 4 2 Η 𝜕𝜉 𝜕𝜉 Ε2 Η ΕΗ2 Ε3 𝑃𝑟
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𝜕𝜃
𝜕𝑓
𝜃′′ + 4 𝑓𝜃′ = 𝜉 [𝑓′ 𝜕𝜉 − 𝜃′ 𝜕𝜉 ],
(7) (8)
𝑓(𝜉 ,0) = 𝑓′(𝜉 ,0) = 𝜃(𝜉 ,0) − 1 = 0,
(9) 𝑓′(𝜉 , ∞) = 𝜃(𝜉 , ∞) = 0. Here Ε , Ε1 , Ε2 , Ε3 and Ε4 are the material parameters, 𝐺𝑟 is the Grashof number, Η is wavy contribution, 𝑃𝑟 is the Prandtl number, subscript ’ 𝜉 ’ represents derivative accordi---ng to 𝜉 and the ‘ ' ’ represents differentiation according to 𝜂. The wall shear stress 𝜏𝑤 = 𝜇𝑛𝑓 (∇𝑢̅ . 𝑛̂)𝑦=0 and
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ACCEPTED MANUSCRIPT the surface heat flux 𝑞𝑤 = −𝜅𝑛𝑓 (∇𝑇. 𝑛̂)𝑦=0 are physical quantities which are necessarily required for the determination of the local skin friction coefficient 𝐶𝑓𝑥 = 𝜏𝑤 /𝜌𝑓 𝑈 2 and local Nusselt
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number𝑁𝑢𝑥 = 𝑥̅ 𝑞𝑤 /𝜅𝑓 (𝑇𝑤 − 𝑇∞ ). In view of the non-dimensional transformations (6) their final
Η 𝑓′′ (𝜉 ,0), (1 − 𝜙)2.5 κ𝑛𝑓 κ𝑓
𝜃
′(𝜉 ,0)
(10)
,
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𝑁𝑢 = 𝑁𝑢𝑥 (𝐺𝑟𝑥 3 )−1/4 = Η
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1
𝐶𝑓 = 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4 =
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expressions read as:
2. Method of Solution
𝜎𝜉 1
, ).
Η Η
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where unit normal to the wavy surface is 𝑛̂ = (−
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Using implicit finite difference numerical scheme [33, 34, 35] the transport equations (7) and (8)
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together with the boundary conditions (9) have been solved. Any quantity 𝑓 in this scheme, at the point (𝜉𝑛 , 𝜂𝑗 ) is written as 𝑓𝑗𝑛 ; quantities at the midpoints of grid segments are assumed to second
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𝑛 𝑛 order as 𝑓𝑗𝑛−1/2 = 1/2(𝑓𝑗𝑛 + 𝑓𝑗𝑛−1 ), 𝑓𝑗−1/2 = 1/2(𝑓𝑗𝑛 + 𝑓𝑗−1 ), and the derivatives are assumed to 𝑛 second order as (𝜕𝑓/𝜕𝜉)𝑛−1/2 = 1/𝑘𝑛 (𝑓𝑗𝑛 + 𝑓𝑗𝑛−1 ), (𝜕𝑓/𝜕𝜂)𝑛𝑗−1/2 = 1/ℎ𝑗 (𝑓𝑗𝑛 + 𝑓𝑗−1 ) where 𝑓 is 𝑗
any dependent variable, 𝑛 and 𝑗 are the node locations along 𝜉 and 𝜂 directions correspondingly. By using the above central difference approximation in terms of the finite differences the differential equations are changed into the system of first order differential equations. According to Newton’s linearization process non-linear algebraic equations are linearized. Finally by tridiagonal block elimination method the system of linearized algebraic equations is solved. The solution was supposed to have converged and the iterative process was stopped if the comparative difference between the present and the previous iterations reached 10−5. In order to validate the precision of the current technique, we have matched our outcomes with those of Hearing et al. [3], Roy et al. [5], Singh et al. [8], Yih et al. [10] and Na et al. [16-17] for 8
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the skin friction coefficient 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4 and local Nusselt number 𝑁𝑢𝑥 (𝐺𝑟𝑥3 )−1/4 in Table 1. The comparisons are found to be superb match in all the above cases. After ensuring the accuracy of
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the current procedure we have employed it to the current problem.
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3. Results and Discussion
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In this exploring work we have shown the effects of amplitude of wavy surface, nanoparticle concentration, and cone half angle on the natural convection flow of nanofluid past a vertical
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wavy cone. The skin friction (𝐶𝑓 ) and the Nusselt number (𝑁𝑢) are the important physical
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quantities that are necessary to examine. The 𝐶𝑓 and 𝑁𝑢 plots are shown in Figs. 2-7 when 𝐶𝑢water based nanofluid is considered. Nanoparticle concentration 𝜙 varies from 0- 20 % and wavy
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amplitude 𝛼 varies from 0 to 0.2, the values of the cone half angle 𝛾 are taken 𝜋/12, 𝜋/6
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and 𝜋/4 for fixed Prandtl number (𝑃𝑟 = 7.0). The computation is performed for different values
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of the wavy amplitude 𝛼 , nanoparticle concentration parameter 𝜙 and the cone inclination half angle 𝛾. We used five nanoparticles viz. alumina (𝐴𝑙2 𝑂3), silver (𝐴𝑔), copper (𝐶𝑢), copper oxide (𝐶𝑢𝑂), and titania (𝑇𝑖𝑂2 ). Figures. 2 - 7 illustrate the change in 𝐶𝑓 and 𝑁𝑢 for increasing values of wavy amplitude 𝛼 , nanoparticle concentration parameter 𝜙 and the cone inclination half angle 𝛾 for Cu -water nanoparticle. These figures show that 𝐶𝑓 and 𝑁𝑢 enhance with the increase of nanoparticle concentration parameter 𝜙 and cone inclination angle 𝛾 whereas a decrease is depicted with an increase of wavy amplitude 𝛼. Figures 8 - 13 depict the variation of the 𝐶𝑓 and 𝑁𝑢 with the variation of wavy amplitude 𝛼 , nanoparticle volume fraction parameter 𝜙 and cone
inclination angle 𝛾 for the selected types of the nanoparticle. From these figures it is clear that the 𝑁𝑢 enhances with the increase of wavy amplitude 𝛼 , nanoparticle centration parameter 𝜙 and cone inclination angle 𝛾. The 𝐶𝑓 enhances with the increase of nanoparticle concentration parameter 𝜙 whereas decreases with the increase of wavy amplitude 𝛼 and cone inclination angle 9
ACCEPTED MANUSCRIPT 𝛾. Also it is noted that the 𝑁𝑢 is maximum in the case of 𝐶𝑢-nanoparticle whereas 𝐶𝑓 is maximum for 𝐴𝑔-nanoparticle; 𝐶𝑓 and 𝑁𝑢 are minimum for 𝑇𝑖𝑂2 -nanoparticle. Isotherms are
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plotted for some values of 𝛼 , 𝜙 (𝛼 = 𝜙 = 0.0, 0.1, 0.2) and 𝛾 (𝛾 = 𝜋/12, 𝜋/6, 𝜋/4) in
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Figures 14-16. The wavy pattern is prominent and the figures indicate that with the increase of
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these parameters the thermal boundary layer thickness increases.
Thermo-physical properties of the five nanoparticles 𝐶𝑢, 𝐶𝑢𝑂, 𝐴𝑙2 𝑂3, 𝐴𝑔 , 𝑇𝑖𝑂2 and water are
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shown in Table 2. The results of 𝐶𝑓 and 𝑁𝑢 are tabulated in Table 3. The values are reported for the selected values of 𝛼 , 𝜙 and 𝛾 when 𝑃𝑟 = 7.0 for copper (𝐶𝑢) nanoparticle along vertical
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wavy cone surface at 𝑥 = 0.5, 𝑥 = 1.0 and 𝑥 = 1.5. It is observed that the 𝑁𝑢 enhances with the
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increase of 𝛼 , 𝜙 and 𝛾 but 𝐶𝑓 decreases with the increase of 𝛼 and 𝛾 at crest and decreasing
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behavior at node and trough.
Fig. 2: Effect of 𝛼 on skin friction Fig. 3: Nusselt number plotted against 𝜉 for profile. variation of 𝛼 .
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Fig. 4: Influence of 𝜙 on skin friction.
Fig. 6: change of skin friction for different Fig. 7: Nusselt number graph for different 𝛾.
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𝛾.
Fig. 8: Effect of 𝜙 on Skin Friction for Fig. 9: Nusselt number profile against 𝜙 for different nanoparticles. different nanoparticles.
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Fig. 10: The effect of 𝛼 on Skin friction for Fig.11: Nusselt number against 𝛼 for different nanoparticles. different nanoparticles.
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Fig. 12: The effect of 𝛾 on 𝐶𝑓𝑥 (𝐺𝑟/𝑥)4for Fig.13: Change in Nusselt number against 𝛾 for different nanoparticles. different nanoparticles.
Fig. 14: Isotherms plotted at 𝛼 (𝛼 = Fig. 15: Effect of 𝜙 (𝜙 = 0.0, 0.1, 0.2) on Isotherms. 0.0, 0.1, 0.2).
Fig. 16: Isotherms at various values of 𝛾 (𝛾 = 𝜋/12, 𝜋/6, 𝜋/4). 12
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Table 1: Comparison with the existing data for 𝑓 ′′ (0, 0) and −𝜃 ′ (0, 0) when 𝛼 = 𝜉 = 0, 𝛾 =
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0, 𝜑 = 0.
Hering [4] 1.5166 1.3550 1.0960 0.7694 -
Roy [5] 0.8600 0.4899 0.2897 0.1661 0.0940
Yih et al. [10] 1.6006 1.5135 1.3551 1.0960 0.7699 0.4877 0.2896 0.1661 0.0940
Present
0.033845 0.038402 0.075460 0.211345 0.451095 0.510399 1.033989 1.922854 3.470171 6.200679
Na et.al [16, 17] 0.07493 0.45101 0.51039 1.03397 1.92197 3.470882 6.204813
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1.452616 1.440436 1.348483 1.095916 0.819591 0.769428 0.487697 0.289635 0.166145 0.094042
Singh et al. [8] 0.81959 -
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0.0001 0.001 0.01 0.1 0.7 1 10 100 1000 10000
Na et.al [16, 17] 0.81959 -
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present
−𝜃 ′ (0, 0)
Singh et al. [8] 0.45109 -
Yih et al. [9] 0.0079 0.0246 0.0749 0.2116 0.5109 1.0339 1.9226 3.4696 6.1984
Hering [4] 0.0247 0.0748 0.2113 0.5104 -
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Pr
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𝑓 ′′ (0, 0)
𝐶𝑝 (𝐽/ 𝐾𝑔𝐾) 383.1 535.6 765 235 686.2 4179
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Properties
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Table 2: Thermo-physical properties of nanoparticles and water.
𝐶𝑢 𝐶𝑢𝑂 𝐴𝑙2 𝑂3 𝐴𝑔 𝑇𝑖𝑂2 Fluid(water)
𝜌(𝐾𝑔/ 𝑚3 ) 8954 6500 3970 10500 4250 997.1
𝜅(𝑊 /𝑚𝐾) 386 20 40 429 8.9538 0.613
𝛽 × 10−5 (1/𝐾) 1.67 1.80 0.85 1.89 0.90 21.0
Table 3: Skin friction (𝐶𝑓 ) and Nusselt number (𝑁𝑢) data for different 𝛼, 𝜙 𝑎𝑛𝑑 𝛾 when 𝑃𝑟 = 7.0 for cupper (𝐶𝑢) nanoparticle. 𝜉
0.5 (crest)
𝜙 0.2
𝛼 0.0 0.1
𝛾 𝜋/4
0.2
0 𝜋/12
0 0.1
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𝐶𝑓 0.6401 0.6285 0.5100 0.5572 0.6452 0.6383
𝑁𝑢 1.1496 1.1542 0.9457 1.0529 1.1409 1.1442
Roy [5] 0.5275 1.0354 1.9229 3.4700 6.1998
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0.2 0.0 0.1
0 𝜋/12 𝜋/6 𝜋/4
0.2
0.0 0.1 0.0 0.1
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1.5 (trough)
𝜋/4
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0.2
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0 0.1
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1.0 (node)
𝜋/4
0 𝜋/12 𝜋/6 𝜋/4
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0.2
1.1492 1.1590 1.1496 1.2179 1.0007 1.1167 1.0717 1.1197 1.1697 1.2308 1.1496 1.1471 0.9156 1.0275 1.1298 1.1310 1.1332 1.1370
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0.2
0.6298 0.6170 0.6401 0.7183 0.5697 0.6233 0.4929 0.5490 0.6109 0.6914 0.6401 0.6621 0.5582 0.6160 0.6473 0.6584 0.6707 0.6869
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4. Concluding Remarks
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Heat transfer phenomena in natural convection flow has been investigated past a wavy cone for base fluid water and five nanoparticles, namely, alumina (𝐴𝑙2 𝑂3), copper (𝐶𝑢), copper oxide (𝐶𝑢𝑂), silver (𝐴𝑔), and titania (𝑇𝑖𝑂2). The non-linear transport equations have been solved numerically and the accuracy of the solution scheme has been proved by giving comparison with the already existing data. Results show that heat transfer rate is significantly increased by addition of nanoparticles with respect to the base liquid and heat transfer enhancement seems to be more prominent with the increase of the nanoparticle volume fraction. It is shown that Nusselt number is maximum in the case of 𝐶𝑢-nanoparticle and decreases successively for 𝐴𝑔nanoparticle, 𝐴𝑙2 𝑂3-nanoparticle, 𝐶𝑢𝑂-nanoparticle and 𝑇𝑖𝑂2 -nanoparticle. Furthermore, it is concluded that the nanofluid serves as very useful fluid in the expeditions of cooling and heating processes.
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ACCEPTED MANUSCRIPT 11. M.A. Hossain, S.C. Paul Free convection from a vertical permeable cone with nonuniform surface temperature. Acta Mech. 151 (2001), pp. 103-114
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ACCEPTED MANUSCRIPT 20. A. Rahman, M.M. Molla, M.M.A. Sarker, Natural convection flow along the vertical wavy cone in case of uniform surface heat flux where viscosity is an exponential function
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ACCEPTED MANUSCRIPT Highlights
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Tiwari and Das model of nanofluid has been utilized for this research article. Heat transfer rate is significantly increased by addition of nanoparticles with respect to the base liquid. Copper has higher heat transfer as compared to other nanoparticles considered. 𝑇𝑖𝑂2 -nanoparticle shown to have the maximum cooling performance.
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