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Heat transfer and flow analysis of Casson fluid enclosed in a partially heated trapezoidal cavity
International Communications in Heat and Mass Transfer 108 (2019) 104284
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Heat transfer and flow analysis of Casson fluid enclosed in a partially heated trapezoidal cavity
T
M. Hamida, M. Usmanb,c, Z.H. Khand,e, , R.U. Haqf, W. Wanga ⁎
a
School of Mathematical Sciences, Peking University, Beijing 100871, PR China BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China c State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China d State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, PR China e Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Tsinghua University, Beijing 100084, PR China f Department of Electrical Engineering, Bahria University, Islamabad 44000, Pakistan b
ARTICLE INFO
ABSTRACT
Keywords: Casson fluid Natural convection Trapezoidal enclosure Finite element method
In this article, we are reporting the mechanism of natural convection flow in a partially heated trapezoidal cavity containing non-Newtonian Casson fluid. A non-Newtonian model of Casson fluid is used to develop the governing flow equations. The thermal control inside the enclosure is managed using the partially heated lower wall. Most physical and significant conditions are used at the inclined walls of the cavity to attain the performance of heat management and streamlines. The Galerkin finite element technique (GFEM) is adopted to examine the solution of dimensionless PDEs system. The thermal and flow fields are envisioned through isotherms and streamlines while the simulations are performed for flow and thermal fields via various ranges of Rayleigh number (104 ≤ Ra ≤ 105.5), and different lengths of the heated part. Furthermore, outcomes are visualized for Nusselt number over the cavity wall under the influence of the Casson parameter. It is observed that as Casson fluid parameter decreases the velocities in x-direction and y-direction exhibit the dominant behavior. The influence of lower values of Casson fluid parameter on Nusselt number is significant at the middle of the cavity while the Nusselt number increases as increases the heated bottom wall in length.
1. Introduction The transport of heat is an essential topic of research due to its broad applications in industries, engineering, and applied sciences. For example, electronic equipments dispels heat which desires cooling, industrial process, cooling and heating systems in the buildings, petro chemical industries, phase change materials (PCMs), textile, vehicles and avionics, food and other plants etc. The stated applications gained a sound focus of research community to examine the transport of heat mechanism by means of experiments as well as theoretically which saves both time and other expenditures. The addition or removal of heat via fluids (Newtonian and non-Newtonian) through an efficient method has gain interest of many researches to examine this phenomena in various geometries. The readers are referred to references [1–6] to study more detailed literature related to heat transfer analysis via both Newtonian and non-Newtonian natured fluids. Recently, non-Newtonian fluids are acknowledged more appropriate for scientific and technological applications than the Newtonian
fluids. Several valuable applications in engineering, biological, physical and other sciences encouraged the scientific community towards the complex area of non-Newtonian fluids. The condensed milk, emulsions, melts, muds, glues, printing ink, sugar solution, soaps, shampoos, paints, tomato paste, are some non-Newtonian natured materials [6]. The potential applications of non-Newtonian materials in latent heat thermal energy storage and PCMs can be found in [7]. Since, the constitutive equations of non-Newtonian fluids are more complex and their properties cannot be described in the form of a single equation but a number of attempts have been made by scientists to define the rheological properties of fluids enclosing non-Newtonian behavior. In view of above discussion and usage of non-Newtonian materials in the industries various non-Newtonian fluid models are presented to define these complex nature phenomena. The Casson fluid model was presented by Casson (1995) and found that it doesn't obey the Newtonian law of viscosity. The said model is found to be more suitable to fit the rheological data than the other non-Newtonian models, such as blood and chocolate flows. The detailed analysis of Casson fluid through
⁎ Corresponding author at: State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, PR China. E-mail address: [email protected] (Z.H. Khan).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104284
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Nomenclature
X, Y
g H k L Nu p P Pr Ra T θ u, v U, V x, y
Greek symbols
Gravitational acceleration Height of the trapezium cavity Thermal conductivity of the fluid Length of the trapezium cavity Nusselt number Dimensional pressure Dimensionless pressure Prandtl number Rayleigh number Temperature Dimensionless temperature Velocity components Dimensionless velocity components Cartesian co-ordinates
ρ α μ ν β ϑ
Dimensionless Cartesian component
Density of fluid Thermal diffusivity Dynamic viscosity Kinematic viscosity Thermal expansion co-efficient Casson fluid parameter
Subscripts c h
different effects has been deliberated and refs are there in [8–12]. The mechanism of natural convection is encountered extensively due to its worthy scientific applications in electronic cooling, double pane windows, and heat exchanger. It is a topic of research in natural sciences because many scholars have examined convective heat transport in nanofluids. Khanfer et al. [13] did the pioneer efforts in this regard and examined the buoyancy driven heat transfer augmentation by taking two dimensional (2D) heated enclosure. It is reported that nanoparticles suspension enhances the rate of heat transfer for the specific values of Grashof number. The transfer of heat improvement in a differential heating square cavity occupied with Copper (Cu)-water nanofluid is examined by Santra et al. [14]. Bruggman and Maxwell Guarnett models are used in the reported investigation. The study of convective flow in trapezoidal enclosure is more complicated as compared to rectangular or square enclosure due to sloping walls. In this regards, a numerical study on trapezoidal enclosure with non-uniform and uniform heating bottom wall is presented by Basak et al. [15]. The finite element technique is adopted to analyze the problems numerically. Varol et al. [16–17] conducted an investigation in trapezoidal enclosure with cooled inclined wall. Apart of above mentioned investigations, many authors examined the flow of various Newtonian fluid for different geometries of cavity [18–28]. On the other hand, there are some other non-Newtonian natured fluid [29–31] which has not been reported for any closed domain and one can consider such fluid to examine the thermal analysis inside closed geometry.
Cold Hot/Heat
In reviewing the previous literature survey, it is noted that less attention is given to the flow of non-Newtonian fluids inside a closed enclosure. Herein, we are numerically reporting the natural convection transfer of heat in a trapezoidal cavity loaded with Casson fluid with various boundary conditions and configuration. The lower wall is partially heated while the upper wall is insulated. The inclined sides are retained cold whereas the flow and thermal fields are envisioned for various ranges of Rayleigh numbers, for various lengths of the heated portion, and volume fraction. The transport of heat performance is analyzed via local and average Nusselt numbers. 2. Physical and mathematical formulation The aim of the current physical formulation is to analyze the natural convection flow and transport of heat in a trapezium enclosure filled with Casson fluid. The lower wall of the cavity in computational trapezium domain is located at the origin O while the length of the wall is L. The lower wall is partially heated while the upper wall is insulated. At the lower wall, partially heated portion is defined into three portions. The mean position of the cavity is heated within the domain |ab| while the left and right parts are adiabatic. Keep this thing in notice that bottom heated domain |ab| can be enlarge or reduce in order to perform the variation of heat transfer. The inclined sides are retained cold whereas the flow and thermal fields are envisioned for various ranges of Rayleigh numbers. The buoyancy forces owed to the variance heating
Fig. 1. (a). Systematic diagram of the problem. (b). Flow chart of the methodology.
2
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 2. Variation of streamlines and isotherms for various partially heated domains with Ra = 105, γ = 0.3 and Pr = 20.
Fig. 3. Effect of Casson parameter on streamlines and isotherms for partially head domain (0.3 ≤ L ≤ 0.7), with Ra = 105 and Pr =20.
are performing on the fluid. The transport of heat because of natural convection mechanism is considered and simulations are executed for different dimensions of the heated portion, different ranges of Rayleigh number and Casson fluid parameter. The purpose is to notice the heat transfer rate at the heated wall. The thermal and flow field are
visualized in the forms of isotherms and streamlines while the physical characteristics of working fluid are assumed constant excluding density variance in the body force term under Boussinesq assumption. The effects of dissipation and radiation are neglected. In account of physical interpretation, the governing equations for incompressible, non3
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Ra = 10 4
Ra = 10 5
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
4
0.25
0.5
Ra = 10
q
0.75
5
0.75
1
Ra = 10 5.5
q
q 0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.062
0.25
0.5
0.75
0
1
0.25
0.5
0.75
1
(iii)
(ii)
(b) (i)
0.5
0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.063
0
1
0.25
(iii)
0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.063
0
3.591 3.079 2.567 2.054 1.542 1.030 0.518 -0.507 -1.019 -1.531 -2.044 -2.556 -3.068 -3.580
0
1
(ii) Ra = 10
Y
1.068 0.915 0.763 0.610 0.458 0.306 0.153 -0.152 -0.304 -0.456 -0.609 -0.761 -0.914 -1.066
0.072 0.062 0.051 0.041 0.031 0.020 0.010 -0.011 -0.021 -0.031 -0.042 -0.052 -0.062 -0.073
(a) (i)
Ra = 10 5.5
Y
Y
Fig. 4. Effect of Ra on streamlines and Isotherms for partially head domain (0.3 ≤ L ≤ 0.7), with γ = 0.1 and Pr =20.
viscous, laminar and steady flow of non-Newtonian Casson fluid are stated in (1)–(4).
u v + = 0, x y u
u u +v = x y
v v u +v = x y u
T T +v = x y
T. Now, consider the following non-dimensional variables.
(X , Y ) =
(1)
1 p + x 1 p + y 2T
x2
+
1+
1
1+ 2T
y2
1
2u + x2 2v
x2
+
2u
y2 2v
y2
= 0,
when x > bH .
Tc ) .
The above system reduced to the following nondimensional form:
,
(2)
+ g (T
Tc ),
(3)
.
when x < aH , , T = Th, at
Tc ) = (Th
(6)
(4)
LT = ab H ,
,
U
U U +V = X Y
1 P 1 + Pr 1 + X
2U
U
V V +V = X Y
1 P 1 + Pr 1 + Y
2V 2V + + Pr Ra , X2 Y2
U
The boundary conditions associated with the above set of Eqs. (1)–(4) are stated below: T At the top insulated wall y = 0,. For right and left inclined wall T = Tc,. For the bottom wall which is partially heated:
T = 0, y
pH 2 (x , y ) (uH , vH ) , (U , V ) = , P= , (T 2 H
X
+V
Y
=
2
X2
+
2
Y2
.
X2
+
2U
Y2
,
(7) (8) (9)
In the above expression (9), the Prandtl and Rayleigh numbers are g (Th Tc ) 3 H respectively, while the BCs are defined asPr = and Ra = reduced as: For right and left inclined wall: θ = 0,. At top wall: Y = 0,. At partially heated wall:
T y (5)
For all solid boundaries u = v = 0. In the above expression, the velocities along x and y − directions are u and v respectively. Where, L = |ab| is the partially heated length. The pressure and temperature terms are respectively indicated as p and
= 0, Y = 1, at
when X < a,
= 0,
when X > b.
Y 4
L = ab , (10)
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 5. Effect of Casson parameter on velocities for partially head domain (0.3 ≤ L ≤ 0.7), with Ra = 105 and Pr =20.
At all solid boundaries U = V = 0. The local and average Nusselt numbers along the domain of enclosure which is partially heated are described in Eq. (11).
Nu =
Y
, Nuavg = X =0
1 S
NudX. S
and we obtained the following final form,
(11)
U
U U +V = X Y
U
V V +V = X Y
X
V U + Y X
+ Pr 1 +
Y
V U + Y X
+ Pr 1 +
1
2U
X2 1
2V
X2
+
+
2U
Y2 2V
Y2
,
(12)
+ Pr Ra . (13)
3. Solution procedure In this section, we are presenting the solution procedure by means of Finite element method (FEM) to analyze the proposed complex nature model. The FEM along with the Galerkin weight residual scheme is used to examine the solution of the expressions (7)–(9) associated with BCs. The solution domain is discretized by means of finite number of elements to solve the given nonlinear system and then non-uniform elements are produced at partially heated domain. A grid sensitivity assessment is prepared for different elements and noticed better for accurate and precise results. One can read the additional details of the scheme in [32–33]. The FEM working steps in the form of a flow chart are presented in the Fig. 1(b). V U + X , The continuity equation and constraint equation P = Y has been used to eliminate the pressure term P, while the expression (1) is satisfied for higher values of δ = 107. Throughout the whole paper we considered δ = 107. The pressure term is replaced in the Eqs. (7)–(8)
4. Results and discussion This section is dedicated to a detailed evaluation of our outcomes and a worthy physical interpretation via set of graphs. Numerical results of thermal control and flow inside a partially heated trapezoidal cavity filled with Casson fluid is achieved in last section by means of penalty finite element method. Herein, a comprehensive study of the effect of Rayleigh number, Casson fluid parameter and bottom heated lengths on streams lines and isotherms is being made. Prandtl number is kept fixed at Pr = 20 throughout the calculations. Behavior of local and average Nusselt numbers due to the variation of various parameters is also considered. Fig. 2 demonstrated the significant behavior on stream lines and isotherms against different partially heated bottom boundary. It is observed that for each case of partially heated bottom we obtained the
5
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Ra = 10
4
Ra = 10
U
5
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
Ra = 10
0.25
(b) (i)
0.5
4
Ra = 10
V
0.75
1
0.25
0.5
0.75
1
(iii) 5
Ra = 10 5.5
V
V 34.393 30.326 26.259 22.192 18.124 14.057 9.990 5.923 1.855 -2.212 -6.279 -10.346 -14.414 -18.481 -22.548
9.510 8.351 7.192 6.032 4.873 3.714 2.554 1.395 0.236 -0.924 -2.083 -3.242 -4.402 -5.561 -6.720
0.590 0.513 0.437 0.361 0.284 0.208 0.132 0.055 -0.021 -0.098 -0.174 -0.250 -0.327 -0.403 -0.479
0
U 21.304 18.275 15.245 12.215 9.185 6.156 3.126 -2.933 -5.963 -8.993 -12.022 -15.052 -18.082 -21.111
0
(ii)
(a) (i)
5.5
6.605 5.663 4.721 3.780 2.838 1.896 0.955 -0.929 -1.870 -2.812 -3.754 -4.695 -5.637 -6.579
0.486 0.417 0.348 0.278 0.209 0.140 0.071 -0.068 -0.137 -0.207 -0.276 -0.345 -0.414 -0.484
0
Ra = 10
U
0
0.25
0.5
0.75
(ii)
1
0
0.25
0.5
0.75
1
(iii)
Fig. 6. Effect of Rayleigh number velocities for partially head domain (0.3 ≤ L ≤ 0.7), with γ = 0.1 and Pr =20.
symmetric behavior of streamlines about the x = 0.5 is line of symmetry. The Casson fluid particles dispenses in two symmetric bullous, one is left side of line x = 0.5 and other is on right side. As heated length increases it can be observed these bullous are decreasing gradually and occupying most part of the discussed cavity and that bullous don't break. On the other hand, isotherms behavior as varying the heated length is also symmetric about x = 0.5. It can be seen that the thermal boundary layers are growing from the left and right edges of the cavity. Again as heated length increases, it can be detected that isotherms are occupying most part of the cavity. Near the sides and upper walls most quantity of the Casson fluid becoming cooler. Also the uniform isotherms pattern specify strong conduction is leading in the bounded domain. The main characteristic of discussed non-Newtonian fluid have an additional viscosity of the liquid due to Casson parameter γ. Therefore Fig. 3 portrayed to show the effect of Casson fluid parameter on both stream lines and isotherms with bottom wall partially heated (0.3 ≤ L ≤ 0.7). As the Casson parameter increases, effective viscosity of the fluid is decreases gradually. So, the lower value of Casson fluid parameter characterizes a vital influence of viscosity. Isotherms pattern reflects more vital interaction between thermal boundary layers for high values of Casson fluid parameter with partial heated (0.3 ≤ L ≤ 0.7) bottom wall. It is essential to noted that as increasing Casson fluid parameter, velocity and thermal boundary layers
thicknesses is decreases gradually. Effect of Rayleigh number Ra with partial heated (0.3 ≤ L ≤ 0.7) bottom wall, on stream lines and isotherms are depicted in Fig. 4. Overall stream lines demonstrate two symmetric bullous along the side walls which is all because of the boundary conditions. Since, natural convection is associated with Rayleigh number. It is observed that for low Rayleigh number Ra = 104 magnitude of stream lines ψ is small which indicate the conductive heat transfer inside a cavity. For Rayleigh number Ra = 104 and Ra = 105 the stream line shows bullous which are symmetric about x = 0.5. For higher Rayleigh number it is noted that stream lines are occupying most part of the trapezoidal cavity. On the other hand, for small values of the Rayleigh number Ra = 104 isotherm pattern is very smooth symmetric about the line x = 0.5. It is also observed that the thermal boundary layers grow from the bottom heated portion and the then becoming thick consistently at the upper part of the trapezoidal cavity. Which leads the phenomena that conduction is effective in the whole domain. Almost similar behavior of isotherm for Ra = 105 is achieved. But for Ra = 105.5 the behavior of the isotherm is non uniform due to the convection. It can be noted that the thickness of thermal boundary layer decreases near the side walls for high values of Rayleigh number. This phenomena is because of the strong impact of buoyancy forces. Convection is effective overall in the trapezoidal cavity. Behavior of the velocities U and V under the influence of Casson
6
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 7. Effect of Casson parameter on Local Nusselt number for various partially heated domains.
fluid parameter and Rayleigh number are demonstrated in Figs. 5-6 with partial heated (0.3 ≤ L ≤ 0.7) bottom wall. It can be noted that as Casson fluid parameter decreases velocities U and V are occupying most part of the trapezoidal cavity. The flow pattern of these velocities are almost symmetric about x = 0.5. Similar behavior of velocities under the effects of Rayleigh number is observed. Flow is more disturbed by enhancing Rayleigh number. Fig. 7 exhibit the significant behavior of local Nusselt number under the influence of Casson fluid parameter with Ra = 104. From all these figures, it is observed that lower value Casson fluid parameter dominant effect at the middle of the cavity. As decreasing the heated wall the variation in Nusselt number is more visible. It is also noted that, near the side walls high values of Casson fluid parameter is more dominant. Fig. 8 is plotted to see the variation in average Nusselt number against the Rayleigh and partially heated wall. For small values of Rayleigh number average Nusselt number is not effected by varying the Casson fluid parameter but for higher value of Rayleigh number significant behavior of average Nusselt number is observed. Direct relation is achieved between average Nusselt number and Casson fluid parameter. Variation in average Nusselt number is very slow against lower values of Casson fluid parameter. Parabolic behavior of average Nusselt number is noted as varying heated bottom length. Also as enhancing the buoyancy forces average Nusselt number become dominant in the cavity.
5. Conclusion The purpose of this study is to analyze the numerical simulation of natural convection flow and thermal control in a partially heated trapezoidal cavity containing non-Newtonian Casson fluid. The natural convection in a trapezoidal cavity filled with Casson fluid for various lengths of heated portion by means of Finite Element method. The simulations are performed for flow and thermal fields via various ranges of Rayleigh numbers (104 ≤ Ra ≤ 105.5), and different lengths of the heated part, while the thermal and flow fields are envisioned through isotherms and streamlines. Hence findings from the analysis are listed below:
• Casson fluid parameter decreases velocities U and V are occupying • • •
7
most part of the trapezoidal cavity. The flow pattern of these velocities is almost symmetric. For higher Rayleigh number is it noted that stream lines are occupying most part of the trapezoidal cavity. Effect of lower value of Casson fluid parameter on local Nusselt number is dominant at the middle of the cavity. Also local Nusselt number increases as increases the heated bottom wall. Average Nusselt number exhibit increasing behavior under the impact of Rayleigh number and Casson fluid parameter.
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 8. Effect of Rayleigh number and Casson parameter on the average Nusselt number for various partially heated domains.
Acknowledgements [10]
The corresponding author is profoundly grateful to the financial support of the National Natural Science Foundation of China (Grant Nos. 51709191, 51706149 and 51606130), Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (Grant No. ARES-2018-10) and State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (Grant No. Skhl1803).
[11] [12] [13]
References
[14]
[1] G.K. Batchelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Q. Appl. Math. 12 (3) (1954) 209–233. [2] J.P. Hartnett, M. Kostic, Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts, Adv. Heat Tran. 19 (1989) 247–356. [3] B. Zalba, J.M. Marın, L.F. Cabeza, H. Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, Appl. Therm. Eng. 23 (3) (2003) 251–283. [4] I. Pop, M. Sheremet, Free convection in a square cavity filled with a Casson fluid under the effects of thermal radiation and viscous dissipation, Int. J. Numer. Methods Heat Fluid Flow 27 (10) (2017) 2318–2332. [5] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212–1221. [6] M. Hamid, M. Usman, T. Zubair, R.U. Haq, W. Wang, Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: a Galerkin approach, Int. J. Heat Mass Transf. 124 (2018) 706–714. [7] W. Aftab, X. Huang, W. Wu, Z. Liang, A. Mahmood, R. Zou, Nanoconfined phase change materials for thermal energy applications, Energy Environ. Sci. 11 (6) (2018) 1392–1424. [8] S.T. Mohyud-Din, M. Usman, W. Wang, M. Hamid, A study of heat transfer analysis for squeezing flow of a Casson fluid via differential transform method, Neural Comput. & Applic. (2017) 1–2. [9] S.T. Mohyud-Din, T. Zubair, M. Usman, M. Hamid, M. Rafiq, S. Mohsin, Investigation of
[15] [16] [17] [18] [19] [20] [21] [22]
8
heat and mass transfer under the influence of variable diffusion coefficient and thermal conductivity, Ind. J. Phys. (2018) 1–9. A. Hussanan, M.Z. Salleh, I. Khan, S. Shafie, Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation, Neural Comput. & Applic. 29 (12) (2018) 1507–1515. G.J. Reddy, B. Kethireddy, M. Kumar, M.M. Hoque, A molecular dynamics study on transient non-Newtonian MHD Casson fluid flow dispersion over a radiative vertical cylinder with entropy heat generation, J. Mol. Liq. 252 (2018) 245–262. M. Hamid, M. Usman, Z.H. Khan, R. Ahmad, W. Wang, Dual solutions and stability analysis of flow and heat transfer of Casson fluid over the stretching sheet, Phys. Lett. A 383 (20) (2019) 2400–2408. K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (19) (2003) 3639–3653. A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer characteristics of copper-water nanofluid in a differentially heated square cavity with different viscosity models, J. Enhanc. Heat Tran. 15 (4) (2008). T. Basak, S. Roy, I. Pop, Heat flow analysis for natural convection within trapezoidal enclosures based on heat line concept, Int. J. Heat Mass Transf. 52 (2009) 2471–2483. Y. Varol, H.F. Oztop, I. Pop, Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall, Int. Commun. Heat Mass Trans. 36 (1) (2009) 6–15. Y. Varol, H.F. Oztop, I. Pop, Entropy analysis due to conjugate-buoyant flow in a rightangle trapezoidal enclosure filled with a porous medium bounded by a solid vertical wall, Int. J. Therm. Sci. 48 (6) (2009) 1161–1175. R.U. Haq, S.N. Kazmi, T. Mekkaoui, Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM, Int. J. Heat Mass Transf. 112 (2017) 972–982. M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transf. 106 (2017) 745–755. R.U. Haq, Z. Hammouch, S.T. Hussain, T. Mekkaoui, MHD mixed convection flow along a vertically heated sheet, Int. J. Hydrog. Energy 42 (24) (2017) 15925–15932. M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids 30 (1) (2018) 012003. R.U. Haq, F.A. Soomro, Z. Hammouch, S.U. Rehman, Heat exchange within the partially
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
[23] [24] [25] [26] [27]
heated C-shape cavity filled with the water based SWCNTs, Int. J. Heat Mass Transf. 127 (2018) 506–514. R.U. Haq, F.A. Soomro, T. Mekkaoui, A.M. Al-Mdallal, MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium, Int. J. Heat Mass Transf. 121 (2018) 1168–1178. M. Sheikholeslami, Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424–438. M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transf. 118 (2018) 823–831. R.U. Haq, S. Aman, Water functionalized CuO nanoparticles filled in a partially heated trapezoidal cavity with inner heated obstacle: FEM approach, Int. J. Heat Mass Transf. 128 (2019) 401–417. Z. Boulahia, A. Wakif, A.J. Chamkha, R. Sehaqui, Numerical study of forced, mixed and natural convection of nanofluids inside a ventilated cavity containing different shapes of cold block, J. Nanofluids 8 (2) (2019) 439–447.
[28] M. Sheikholeslami, New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media, Comput. Methods Appl. Mech. Eng. 344 (2019) 319–333. [29] M. Hamid, T. Zubair, M. Usman, Z. H. Khan, W. Wang, Natural convection effects on heat and mass transfer of slip flow of time-dependent Prandtl fluid, J. Comput. Des. Eng. DOI: https://doi.org/10.1016/j.jcde.2019.03.004. [30] M. Hamid, M. Usman, Z.H. Khan, R.U. Haq, W. Wang, Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation, Eur. Phys. J. Plus 133 (12) (2018) 527. [31] M. Hamid, M. Usman, R.U. Haq, Z.H. Khan, W. Wang, Wavelets analysis of stagnation point flow of non-Newtonian nanofluid, Appl. Math. Mech. (2019), https://doi.org/10. 1007/s10483-019-2508-6 English edition. [32] C. Taylor, P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1) (1973) 73–100. [33] P. Dechaumphai, Finite Element Method in Engineering, second edition, Chulalongkorn University Press, Bangkok, 1999.
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Heat transfer and flow analysis of Casson fluid enclosed in a partially heated trapezoidal cavity
T
M. Hamida, M. Usmanb,c, Z.H. Khand,e, , R.U. Haqf, W. Wanga ⁎
a
School of Mathematical Sciences, Peking University, Beijing 100871, PR China BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China c State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China d State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, PR China e Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Tsinghua University, Beijing 100084, PR China f Department of Electrical Engineering, Bahria University, Islamabad 44000, Pakistan b
ARTICLE INFO
ABSTRACT
Keywords: Casson fluid Natural convection Trapezoidal enclosure Finite element method
In this article, we are reporting the mechanism of natural convection flow in a partially heated trapezoidal cavity containing non-Newtonian Casson fluid. A non-Newtonian model of Casson fluid is used to develop the governing flow equations. The thermal control inside the enclosure is managed using the partially heated lower wall. Most physical and significant conditions are used at the inclined walls of the cavity to attain the performance of heat management and streamlines. The Galerkin finite element technique (GFEM) is adopted to examine the solution of dimensionless PDEs system. The thermal and flow fields are envisioned through isotherms and streamlines while the simulations are performed for flow and thermal fields via various ranges of Rayleigh number (104 ≤ Ra ≤ 105.5), and different lengths of the heated part. Furthermore, outcomes are visualized for Nusselt number over the cavity wall under the influence of the Casson parameter. It is observed that as Casson fluid parameter decreases the velocities in x-direction and y-direction exhibit the dominant behavior. The influence of lower values of Casson fluid parameter on Nusselt number is significant at the middle of the cavity while the Nusselt number increases as increases the heated bottom wall in length.
1. Introduction The transport of heat is an essential topic of research due to its broad applications in industries, engineering, and applied sciences. For example, electronic equipments dispels heat which desires cooling, industrial process, cooling and heating systems in the buildings, petro chemical industries, phase change materials (PCMs), textile, vehicles and avionics, food and other plants etc. The stated applications gained a sound focus of research community to examine the transport of heat mechanism by means of experiments as well as theoretically which saves both time and other expenditures. The addition or removal of heat via fluids (Newtonian and non-Newtonian) through an efficient method has gain interest of many researches to examine this phenomena in various geometries. The readers are referred to references [1–6] to study more detailed literature related to heat transfer analysis via both Newtonian and non-Newtonian natured fluids. Recently, non-Newtonian fluids are acknowledged more appropriate for scientific and technological applications than the Newtonian
fluids. Several valuable applications in engineering, biological, physical and other sciences encouraged the scientific community towards the complex area of non-Newtonian fluids. The condensed milk, emulsions, melts, muds, glues, printing ink, sugar solution, soaps, shampoos, paints, tomato paste, are some non-Newtonian natured materials [6]. The potential applications of non-Newtonian materials in latent heat thermal energy storage and PCMs can be found in [7]. Since, the constitutive equations of non-Newtonian fluids are more complex and their properties cannot be described in the form of a single equation but a number of attempts have been made by scientists to define the rheological properties of fluids enclosing non-Newtonian behavior. In view of above discussion and usage of non-Newtonian materials in the industries various non-Newtonian fluid models are presented to define these complex nature phenomena. The Casson fluid model was presented by Casson (1995) and found that it doesn't obey the Newtonian law of viscosity. The said model is found to be more suitable to fit the rheological data than the other non-Newtonian models, such as blood and chocolate flows. The detailed analysis of Casson fluid through
⁎ Corresponding author at: State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, PR China. E-mail address: [email protected] (Z.H. Khan).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104284
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Nomenclature
X, Y
g H k L Nu p P Pr Ra T θ u, v U, V x, y
Greek symbols
Gravitational acceleration Height of the trapezium cavity Thermal conductivity of the fluid Length of the trapezium cavity Nusselt number Dimensional pressure Dimensionless pressure Prandtl number Rayleigh number Temperature Dimensionless temperature Velocity components Dimensionless velocity components Cartesian co-ordinates
ρ α μ ν β ϑ
Dimensionless Cartesian component
Density of fluid Thermal diffusivity Dynamic viscosity Kinematic viscosity Thermal expansion co-efficient Casson fluid parameter
Subscripts c h
different effects has been deliberated and refs are there in [8–12]. The mechanism of natural convection is encountered extensively due to its worthy scientific applications in electronic cooling, double pane windows, and heat exchanger. It is a topic of research in natural sciences because many scholars have examined convective heat transport in nanofluids. Khanfer et al. [13] did the pioneer efforts in this regard and examined the buoyancy driven heat transfer augmentation by taking two dimensional (2D) heated enclosure. It is reported that nanoparticles suspension enhances the rate of heat transfer for the specific values of Grashof number. The transfer of heat improvement in a differential heating square cavity occupied with Copper (Cu)-water nanofluid is examined by Santra et al. [14]. Bruggman and Maxwell Guarnett models are used in the reported investigation. The study of convective flow in trapezoidal enclosure is more complicated as compared to rectangular or square enclosure due to sloping walls. In this regards, a numerical study on trapezoidal enclosure with non-uniform and uniform heating bottom wall is presented by Basak et al. [15]. The finite element technique is adopted to analyze the problems numerically. Varol et al. [16–17] conducted an investigation in trapezoidal enclosure with cooled inclined wall. Apart of above mentioned investigations, many authors examined the flow of various Newtonian fluid for different geometries of cavity [18–28]. On the other hand, there are some other non-Newtonian natured fluid [29–31] which has not been reported for any closed domain and one can consider such fluid to examine the thermal analysis inside closed geometry.
Cold Hot/Heat
In reviewing the previous literature survey, it is noted that less attention is given to the flow of non-Newtonian fluids inside a closed enclosure. Herein, we are numerically reporting the natural convection transfer of heat in a trapezoidal cavity loaded with Casson fluid with various boundary conditions and configuration. The lower wall is partially heated while the upper wall is insulated. The inclined sides are retained cold whereas the flow and thermal fields are envisioned for various ranges of Rayleigh numbers, for various lengths of the heated portion, and volume fraction. The transport of heat performance is analyzed via local and average Nusselt numbers. 2. Physical and mathematical formulation The aim of the current physical formulation is to analyze the natural convection flow and transport of heat in a trapezium enclosure filled with Casson fluid. The lower wall of the cavity in computational trapezium domain is located at the origin O while the length of the wall is L. The lower wall is partially heated while the upper wall is insulated. At the lower wall, partially heated portion is defined into three portions. The mean position of the cavity is heated within the domain |ab| while the left and right parts are adiabatic. Keep this thing in notice that bottom heated domain |ab| can be enlarge or reduce in order to perform the variation of heat transfer. The inclined sides are retained cold whereas the flow and thermal fields are envisioned for various ranges of Rayleigh numbers. The buoyancy forces owed to the variance heating
Fig. 1. (a). Systematic diagram of the problem. (b). Flow chart of the methodology.
2
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 2. Variation of streamlines and isotherms for various partially heated domains with Ra = 105, γ = 0.3 and Pr = 20.
Fig. 3. Effect of Casson parameter on streamlines and isotherms for partially head domain (0.3 ≤ L ≤ 0.7), with Ra = 105 and Pr =20.
are performing on the fluid. The transport of heat because of natural convection mechanism is considered and simulations are executed for different dimensions of the heated portion, different ranges of Rayleigh number and Casson fluid parameter. The purpose is to notice the heat transfer rate at the heated wall. The thermal and flow field are
visualized in the forms of isotherms and streamlines while the physical characteristics of working fluid are assumed constant excluding density variance in the body force term under Boussinesq assumption. The effects of dissipation and radiation are neglected. In account of physical interpretation, the governing equations for incompressible, non3
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Ra = 10 4
Ra = 10 5
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
4
0.25
0.5
Ra = 10
q
0.75
5
0.75
1
Ra = 10 5.5
q
q 0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.062
0.25
0.5
0.75
0
1
0.25
0.5
0.75
1
(iii)
(ii)
(b) (i)
0.5
0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.063
0
1
0.25
(iii)
0.938 0.875 0.813 0.750 0.688 0.625 0.563 0.500 0.438 0.375 0.313 0.250 0.188 0.125 0.063
0
3.591 3.079 2.567 2.054 1.542 1.030 0.518 -0.507 -1.019 -1.531 -2.044 -2.556 -3.068 -3.580
0
1
(ii) Ra = 10
Y
1.068 0.915 0.763 0.610 0.458 0.306 0.153 -0.152 -0.304 -0.456 -0.609 -0.761 -0.914 -1.066
0.072 0.062 0.051 0.041 0.031 0.020 0.010 -0.011 -0.021 -0.031 -0.042 -0.052 -0.062 -0.073
(a) (i)
Ra = 10 5.5
Y
Y
Fig. 4. Effect of Ra on streamlines and Isotherms for partially head domain (0.3 ≤ L ≤ 0.7), with γ = 0.1 and Pr =20.
viscous, laminar and steady flow of non-Newtonian Casson fluid are stated in (1)–(4).
u v + = 0, x y u
u u +v = x y
v v u +v = x y u
T T +v = x y
T. Now, consider the following non-dimensional variables.
(X , Y ) =
(1)
1 p + x 1 p + y 2T
x2
+
1+
1
1+ 2T
y2
1
2u + x2 2v
x2
+
2u
y2 2v
y2
= 0,
when x > bH .
Tc ) .
The above system reduced to the following nondimensional form:
,
(2)
+ g (T
Tc ),
(3)
.
when x < aH , , T = Th, at
Tc ) = (Th
(6)
(4)
LT = ab H ,
,
U
U U +V = X Y
1 P 1 + Pr 1 + X
2U
U
V V +V = X Y
1 P 1 + Pr 1 + Y
2V 2V + + Pr Ra , X2 Y2
U
The boundary conditions associated with the above set of Eqs. (1)–(4) are stated below: T At the top insulated wall y = 0,. For right and left inclined wall T = Tc,. For the bottom wall which is partially heated:
T = 0, y
pH 2 (x , y ) (uH , vH ) , (U , V ) = , P= , (T 2 H
X
+V
Y
=
2
X2
+
2
Y2
.
X2
+
2U
Y2
,
(7) (8) (9)
In the above expression (9), the Prandtl and Rayleigh numbers are g (Th Tc ) 3 H respectively, while the BCs are defined asPr = and Ra = reduced as: For right and left inclined wall: θ = 0,. At top wall: Y = 0,. At partially heated wall:
T y (5)
For all solid boundaries u = v = 0. In the above expression, the velocities along x and y − directions are u and v respectively. Where, L = |ab| is the partially heated length. The pressure and temperature terms are respectively indicated as p and
= 0, Y = 1, at
when X < a,
= 0,
when X > b.
Y 4
L = ab , (10)
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 5. Effect of Casson parameter on velocities for partially head domain (0.3 ≤ L ≤ 0.7), with Ra = 105 and Pr =20.
At all solid boundaries U = V = 0. The local and average Nusselt numbers along the domain of enclosure which is partially heated are described in Eq. (11).
Nu =
Y
, Nuavg = X =0
1 S
NudX. S
and we obtained the following final form,
(11)
U
U U +V = X Y
U
V V +V = X Y
X
V U + Y X
+ Pr 1 +
Y
V U + Y X
+ Pr 1 +
1
2U
X2 1
2V
X2
+
+
2U
Y2 2V
Y2
,
(12)
+ Pr Ra . (13)
3. Solution procedure In this section, we are presenting the solution procedure by means of Finite element method (FEM) to analyze the proposed complex nature model. The FEM along with the Galerkin weight residual scheme is used to examine the solution of the expressions (7)–(9) associated with BCs. The solution domain is discretized by means of finite number of elements to solve the given nonlinear system and then non-uniform elements are produced at partially heated domain. A grid sensitivity assessment is prepared for different elements and noticed better for accurate and precise results. One can read the additional details of the scheme in [32–33]. The FEM working steps in the form of a flow chart are presented in the Fig. 1(b). V U + X , The continuity equation and constraint equation P = Y has been used to eliminate the pressure term P, while the expression (1) is satisfied for higher values of δ = 107. Throughout the whole paper we considered δ = 107. The pressure term is replaced in the Eqs. (7)–(8)
4. Results and discussion This section is dedicated to a detailed evaluation of our outcomes and a worthy physical interpretation via set of graphs. Numerical results of thermal control and flow inside a partially heated trapezoidal cavity filled with Casson fluid is achieved in last section by means of penalty finite element method. Herein, a comprehensive study of the effect of Rayleigh number, Casson fluid parameter and bottom heated lengths on streams lines and isotherms is being made. Prandtl number is kept fixed at Pr = 20 throughout the calculations. Behavior of local and average Nusselt numbers due to the variation of various parameters is also considered. Fig. 2 demonstrated the significant behavior on stream lines and isotherms against different partially heated bottom boundary. It is observed that for each case of partially heated bottom we obtained the
5
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Ra = 10
4
Ra = 10
U
5
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
Ra = 10
0.25
(b) (i)
0.5
4
Ra = 10
V
0.75
1
0.25
0.5
0.75
1
(iii) 5
Ra = 10 5.5
V
V 34.393 30.326 26.259 22.192 18.124 14.057 9.990 5.923 1.855 -2.212 -6.279 -10.346 -14.414 -18.481 -22.548
9.510 8.351 7.192 6.032 4.873 3.714 2.554 1.395 0.236 -0.924 -2.083 -3.242 -4.402 -5.561 -6.720
0.590 0.513 0.437 0.361 0.284 0.208 0.132 0.055 -0.021 -0.098 -0.174 -0.250 -0.327 -0.403 -0.479
0
U 21.304 18.275 15.245 12.215 9.185 6.156 3.126 -2.933 -5.963 -8.993 -12.022 -15.052 -18.082 -21.111
0
(ii)
(a) (i)
5.5
6.605 5.663 4.721 3.780 2.838 1.896 0.955 -0.929 -1.870 -2.812 -3.754 -4.695 -5.637 -6.579
0.486 0.417 0.348 0.278 0.209 0.140 0.071 -0.068 -0.137 -0.207 -0.276 -0.345 -0.414 -0.484
0
Ra = 10
U
0
0.25
0.5
0.75
(ii)
1
0
0.25
0.5
0.75
1
(iii)
Fig. 6. Effect of Rayleigh number velocities for partially head domain (0.3 ≤ L ≤ 0.7), with γ = 0.1 and Pr =20.
symmetric behavior of streamlines about the x = 0.5 is line of symmetry. The Casson fluid particles dispenses in two symmetric bullous, one is left side of line x = 0.5 and other is on right side. As heated length increases it can be observed these bullous are decreasing gradually and occupying most part of the discussed cavity and that bullous don't break. On the other hand, isotherms behavior as varying the heated length is also symmetric about x = 0.5. It can be seen that the thermal boundary layers are growing from the left and right edges of the cavity. Again as heated length increases, it can be detected that isotherms are occupying most part of the cavity. Near the sides and upper walls most quantity of the Casson fluid becoming cooler. Also the uniform isotherms pattern specify strong conduction is leading in the bounded domain. The main characteristic of discussed non-Newtonian fluid have an additional viscosity of the liquid due to Casson parameter γ. Therefore Fig. 3 portrayed to show the effect of Casson fluid parameter on both stream lines and isotherms with bottom wall partially heated (0.3 ≤ L ≤ 0.7). As the Casson parameter increases, effective viscosity of the fluid is decreases gradually. So, the lower value of Casson fluid parameter characterizes a vital influence of viscosity. Isotherms pattern reflects more vital interaction between thermal boundary layers for high values of Casson fluid parameter with partial heated (0.3 ≤ L ≤ 0.7) bottom wall. It is essential to noted that as increasing Casson fluid parameter, velocity and thermal boundary layers
thicknesses is decreases gradually. Effect of Rayleigh number Ra with partial heated (0.3 ≤ L ≤ 0.7) bottom wall, on stream lines and isotherms are depicted in Fig. 4. Overall stream lines demonstrate two symmetric bullous along the side walls which is all because of the boundary conditions. Since, natural convection is associated with Rayleigh number. It is observed that for low Rayleigh number Ra = 104 magnitude of stream lines ψ is small which indicate the conductive heat transfer inside a cavity. For Rayleigh number Ra = 104 and Ra = 105 the stream line shows bullous which are symmetric about x = 0.5. For higher Rayleigh number it is noted that stream lines are occupying most part of the trapezoidal cavity. On the other hand, for small values of the Rayleigh number Ra = 104 isotherm pattern is very smooth symmetric about the line x = 0.5. It is also observed that the thermal boundary layers grow from the bottom heated portion and the then becoming thick consistently at the upper part of the trapezoidal cavity. Which leads the phenomena that conduction is effective in the whole domain. Almost similar behavior of isotherm for Ra = 105 is achieved. But for Ra = 105.5 the behavior of the isotherm is non uniform due to the convection. It can be noted that the thickness of thermal boundary layer decreases near the side walls for high values of Rayleigh number. This phenomena is because of the strong impact of buoyancy forces. Convection is effective overall in the trapezoidal cavity. Behavior of the velocities U and V under the influence of Casson
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International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 7. Effect of Casson parameter on Local Nusselt number for various partially heated domains.
fluid parameter and Rayleigh number are demonstrated in Figs. 5-6 with partial heated (0.3 ≤ L ≤ 0.7) bottom wall. It can be noted that as Casson fluid parameter decreases velocities U and V are occupying most part of the trapezoidal cavity. The flow pattern of these velocities are almost symmetric about x = 0.5. Similar behavior of velocities under the effects of Rayleigh number is observed. Flow is more disturbed by enhancing Rayleigh number. Fig. 7 exhibit the significant behavior of local Nusselt number under the influence of Casson fluid parameter with Ra = 104. From all these figures, it is observed that lower value Casson fluid parameter dominant effect at the middle of the cavity. As decreasing the heated wall the variation in Nusselt number is more visible. It is also noted that, near the side walls high values of Casson fluid parameter is more dominant. Fig. 8 is plotted to see the variation in average Nusselt number against the Rayleigh and partially heated wall. For small values of Rayleigh number average Nusselt number is not effected by varying the Casson fluid parameter but for higher value of Rayleigh number significant behavior of average Nusselt number is observed. Direct relation is achieved between average Nusselt number and Casson fluid parameter. Variation in average Nusselt number is very slow against lower values of Casson fluid parameter. Parabolic behavior of average Nusselt number is noted as varying heated bottom length. Also as enhancing the buoyancy forces average Nusselt number become dominant in the cavity.
5. Conclusion The purpose of this study is to analyze the numerical simulation of natural convection flow and thermal control in a partially heated trapezoidal cavity containing non-Newtonian Casson fluid. The natural convection in a trapezoidal cavity filled with Casson fluid for various lengths of heated portion by means of Finite Element method. The simulations are performed for flow and thermal fields via various ranges of Rayleigh numbers (104 ≤ Ra ≤ 105.5), and different lengths of the heated part, while the thermal and flow fields are envisioned through isotherms and streamlines. Hence findings from the analysis are listed below:
• Casson fluid parameter decreases velocities U and V are occupying • • •
7
most part of the trapezoidal cavity. The flow pattern of these velocities is almost symmetric. For higher Rayleigh number is it noted that stream lines are occupying most part of the trapezoidal cavity. Effect of lower value of Casson fluid parameter on local Nusselt number is dominant at the middle of the cavity. Also local Nusselt number increases as increases the heated bottom wall. Average Nusselt number exhibit increasing behavior under the impact of Rayleigh number and Casson fluid parameter.
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
Fig. 8. Effect of Rayleigh number and Casson parameter on the average Nusselt number for various partially heated domains.
Acknowledgements [10]
The corresponding author is profoundly grateful to the financial support of the National Natural Science Foundation of China (Grant Nos. 51709191, 51706149 and 51606130), Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (Grant No. ARES-2018-10) and State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (Grant No. Skhl1803).
[11] [12] [13]
References
[14]
[1] G.K. Batchelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Q. Appl. Math. 12 (3) (1954) 209–233. [2] J.P. Hartnett, M. Kostic, Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts, Adv. Heat Tran. 19 (1989) 247–356. [3] B. Zalba, J.M. Marın, L.F. Cabeza, H. Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, Appl. Therm. Eng. 23 (3) (2003) 251–283. [4] I. Pop, M. Sheremet, Free convection in a square cavity filled with a Casson fluid under the effects of thermal radiation and viscous dissipation, Int. J. Numer. Methods Heat Fluid Flow 27 (10) (2017) 2318–2332. [5] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212–1221. [6] M. Hamid, M. Usman, T. Zubair, R.U. Haq, W. Wang, Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: a Galerkin approach, Int. J. Heat Mass Transf. 124 (2018) 706–714. [7] W. Aftab, X. Huang, W. Wu, Z. Liang, A. Mahmood, R. Zou, Nanoconfined phase change materials for thermal energy applications, Energy Environ. Sci. 11 (6) (2018) 1392–1424. [8] S.T. Mohyud-Din, M. Usman, W. Wang, M. Hamid, A study of heat transfer analysis for squeezing flow of a Casson fluid via differential transform method, Neural Comput. & Applic. (2017) 1–2. [9] S.T. Mohyud-Din, T. Zubair, M. Usman, M. Hamid, M. Rafiq, S. Mohsin, Investigation of
[15] [16] [17] [18] [19] [20] [21] [22]
8
heat and mass transfer under the influence of variable diffusion coefficient and thermal conductivity, Ind. J. Phys. (2018) 1–9. A. Hussanan, M.Z. Salleh, I. Khan, S. Shafie, Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation, Neural Comput. & Applic. 29 (12) (2018) 1507–1515. G.J. Reddy, B. Kethireddy, M. Kumar, M.M. Hoque, A molecular dynamics study on transient non-Newtonian MHD Casson fluid flow dispersion over a radiative vertical cylinder with entropy heat generation, J. Mol. Liq. 252 (2018) 245–262. M. Hamid, M. Usman, Z.H. Khan, R. Ahmad, W. Wang, Dual solutions and stability analysis of flow and heat transfer of Casson fluid over the stretching sheet, Phys. Lett. A 383 (20) (2019) 2400–2408. K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (19) (2003) 3639–3653. A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer characteristics of copper-water nanofluid in a differentially heated square cavity with different viscosity models, J. Enhanc. Heat Tran. 15 (4) (2008). T. Basak, S. Roy, I. Pop, Heat flow analysis for natural convection within trapezoidal enclosures based on heat line concept, Int. J. Heat Mass Transf. 52 (2009) 2471–2483. Y. Varol, H.F. Oztop, I. Pop, Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall, Int. Commun. Heat Mass Trans. 36 (1) (2009) 6–15. Y. Varol, H.F. Oztop, I. Pop, Entropy analysis due to conjugate-buoyant flow in a rightangle trapezoidal enclosure filled with a porous medium bounded by a solid vertical wall, Int. J. Therm. Sci. 48 (6) (2009) 1161–1175. R.U. Haq, S.N. Kazmi, T. Mekkaoui, Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM, Int. J. Heat Mass Transf. 112 (2017) 972–982. M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transf. 106 (2017) 745–755. R.U. Haq, Z. Hammouch, S.T. Hussain, T. Mekkaoui, MHD mixed convection flow along a vertically heated sheet, Int. J. Hydrog. Energy 42 (24) (2017) 15925–15932. M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids 30 (1) (2018) 012003. R.U. Haq, F.A. Soomro, Z. Hammouch, S.U. Rehman, Heat exchange within the partially
International Communications in Heat and Mass Transfer 108 (2019) 104284
M. Hamid, et al.
[23] [24] [25] [26] [27]
heated C-shape cavity filled with the water based SWCNTs, Int. J. Heat Mass Transf. 127 (2018) 506–514. R.U. Haq, F.A. Soomro, T. Mekkaoui, A.M. Al-Mdallal, MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium, Int. J. Heat Mass Transf. 121 (2018) 1168–1178. M. Sheikholeslami, Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424–438. M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transf. 118 (2018) 823–831. R.U. Haq, S. Aman, Water functionalized CuO nanoparticles filled in a partially heated trapezoidal cavity with inner heated obstacle: FEM approach, Int. J. Heat Mass Transf. 128 (2019) 401–417. Z. Boulahia, A. Wakif, A.J. Chamkha, R. Sehaqui, Numerical study of forced, mixed and natural convection of nanofluids inside a ventilated cavity containing different shapes of cold block, J. Nanofluids 8 (2) (2019) 439–447.
[28] M. Sheikholeslami, New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media, Comput. Methods Appl. Mech. Eng. 344 (2019) 319–333. [29] M. Hamid, T. Zubair, M. Usman, Z. H. Khan, W. Wang, Natural convection effects on heat and mass transfer of slip flow of time-dependent Prandtl fluid, J. Comput. Des. Eng. DOI: https://doi.org/10.1016/j.jcde.2019.03.004. [30] M. Hamid, M. Usman, Z.H. Khan, R.U. Haq, W. Wang, Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation, Eur. Phys. J. Plus 133 (12) (2018) 527. [31] M. Hamid, M. Usman, R.U. Haq, Z.H. Khan, W. Wang, Wavelets analysis of stagnation point flow of non-Newtonian nanofluid, Appl. Math. Mech. (2019), https://doi.org/10. 1007/s10483-019-2508-6 English edition. [32] C. Taylor, P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1) (1973) 73–100. [33] P. Dechaumphai, Finite Element Method in Engineering, second edition, Chulalongkorn University Press, Bangkok, 1999.
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