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Free Convective Analysis in a Solar Collector Using
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Nanofluid
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Rehena Nasrin* and M. A. Alim (insert full mane)
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Department of Mathematics, Bangladesh University of Engineering & Technology,
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Dhaka-1000, Bangladesh. E-mail: rehena@math.buet.ac.bd
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*
Corresponding author
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Abstract
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A numerical study has been conducted to investigate the natural convection
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inside a solar collector having the flat-plate cover and the sine-wave absorber.
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The water alumina nanofluid is used as the working fluid inside the solar
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collector. The governing differential equations with boundary conditions are
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solved by Penalty Finite Element Method using Galerkin’s weighted residual
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scheme. The effects of major system parameters on the natural convection heat
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transfer are simulated. These parameters include the Prandtl number Pr and the
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Rayleigh number Ra. Comprehensive average Nusselt number, average
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temperature and mean velocity field for both nanofluid and base fluid within the
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collector are presented as functions of the governing parameters mentioned
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above. The numerical results show that the highest heat transfer rate is observed
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for both the largest Pr and Ra.
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Keywords: Natural convection, solar collector, finite element method, water-
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Al2O3 nanofluid
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Nomenclature
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A
Area of glass cover plate (m2)
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Am
Dimensionless amplitude of wave
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Cp
Specific heat at constant pressure (kJ kg-1 K-1)
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g
Gravitational acceleration (m s-2)
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h
Local heat transfer coefficient (W m-2 K-1)
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k
Thermal conductivity (W m-1 K-1)
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L
Lengh of the solar collector (m)
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Nu
Nusselt number, Nu = hL/kf
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Pr
Prandtl number, Pr   f /  f
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Ra
Rayleigh number, Ra 
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T
Dimensional temperature (oK)
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Ti
Initial temperature of nanofluid (oK)
g  f L3 Tw  Tc 
 f f
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u, v
Dimensional x and y components of velocity (m s-1)
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U, V
Dimensionless velocities, U 
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X, Y
Dimensionless coordinates, X  x / L, Y  y / L
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x, y
Dimensional coordinates (m)
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Greek Symbols
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α
Fluid thermal diffusivity (m2 s-1)
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β
Thermal expansion coefficient (K-1)
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
Emissivity
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
Nanoparticles volume fraction
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ν
Kinematic viscosity (m2 s-1)
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θ
Dimensionless temperature,   T  Tc  / Th  Tc 
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ρ
Density (kg m-3)
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
Number of wave
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μ
Dynamic viscosity (N s m-2)
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
Stefan Boltzmann constant
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V
Dimensionless velocity field
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Subscripts
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av
average
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c
cold
uL
f
, V
vL
f
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f
fluid
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nf
nanofluid
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s
solid particle
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w
top wall
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Introduction
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Because of the desirable environmental and safety aspects it is widely believed that
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solar energy should be utilized instead of other alternative energy forms, even when
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the costs involved are slightly higher. The flat-plate solar collector is commonly used
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today for the collection of low temperature solar thermal energy. Solar collectors are
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key elements in many applications, such as building heating systems, solar drying
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devices, etc. Solar energy has the greatest potential of all the sources of renewable
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energy especially when other sources in the country have depleted. The fluids with
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solid-sized nanoparticles suspended in them are called “nanofluids.” The natural
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convection in enclosures continues to be a very active area of research during the past
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few decades. Applications of nanoparticles in thermal field are to enhance heat
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transfer from solar collectors to storage tanks, to improve efficiency of coolants in
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transformers.
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The sun is a sphere of intensely hot gaseous matter with a diameter of
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1.39x109 m. The solar energy strikes our planet a mere 8 min and 20 s after leaving
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the giant furnace. The sun has an effective blackbody temperature of 5762 K (Kreith
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and Kreider, 1978). Conventional analysis and design of solar collector is based on a
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one-dimensional conduction equation formulation [2]. The analysis has been
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substantially assisted by the derivation of plate-fin efficiency factors. The factors
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relate the design and operating conditions of the collector in a systematic manner that
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facilitates prediction of heat collection rates at the design stage. The one-dimensional
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analysis offers a desired accuracy required in a routine analysis even though a two-
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dimensional temperature distribution exists over the absorber plate of the collector.
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Therefore, for more accurate analysis at low mass flow rates, a two-dimensional
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temperature distribution must be considered. Various investigators have used two
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dimensional conduction equations in their analysis with different boundary
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conditions. Rao [3] proposed a two-dimensional model for the absorber plate
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conduction to compute the fluid temperature rise. Lund [4] used a two-dimensional
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model for the transfer of heat in flat plate solar collector absorbers with adiabatic
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boundary conditions at the upper and lower edges of the collector. He solved
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approximately the governing partial differential equations analytically in terms of
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perturbation series.
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Nag et al. [5] used the two-dimensional model proposed by Lund, but with
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convection boundary condition at the upper and lower edge of the absorber plate.
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They solved the governing equations using finite element method. They concluded
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that the isotherms deviated from a one-dimensional pattern for a high flow rate to a
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predominantly two-dimensional distribution for a low mass flow rate. Stasiek [6]
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made experimental studies of heat transfer and fluid flow across corrugated and
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undulated heat exchanger surfaces. Piao et al. [7, 8] investigated experimentally
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natural, forced and mixed convective heat transfer in a cross-corrugated channel solar
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air heater. Noorshahi et al. [9] studied numerically the natural convection effect in a
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corrugated enclosure with mixed boundary conditions. Detailed experimental and
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numerical studies on the performance of the solar air heater were made by Gao [10].
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There are so many methods introduced to increase the efficiency of the solar
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water heater [11–14]. But the novel approach is to introduce the nanofluids in solar
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water heater instead of conventional heat transfer fluids (like water). The poor heat
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transfer properties of these conventional fluids compared to most solids are the
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primary obstacle to the high compactness and effectiveness of the system. The
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essential initiative is to seek the solid particles having thermal conductivity of several
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hundred times higher than those of conventional fluids. An innovative idea is to
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suspend ultra fine solid particles in the fluid for improving the thermal conductivity
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of the fluid [15]. These early studies, however, used suspensions of millimeter- or
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micrometer-sized particles, which, although showed some enhancement, experienced
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problems such as poor suspension stability and hence channel clogging, which are
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particularly serious for systems using mini sized and micro sized particles. The
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suspended metallic or nonmetallic nanoparticles change the transport properties and
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heat transfer characteristics of the base fluid. Stability and thermal conductivity
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characteristics of nanofluids was performed by Hwang et al. [16]. In this study, they
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concluded that the thermal conductivity of ethylene glycol was increased by 30%.
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The absorptance of the collector surface for shortwave solar radiation depends
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on the nature and colour of the coating and on the incident angle. Usually black
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colour is used. Various colour coatings had been proposed in [17–19] mainly for
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aesthetic reasons. A low-cost mechanically manufactured selective solar absorber
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surface method had been proposed by Konttinen et al. [20]. Another category of
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collectors is the uncovered or unglazed solar collector [21]. These are usually low-
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cost units which can offer cost effective solar thermal energy in applications such as
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water preheating for domestic or industrial use, heating of swimming pools [22],
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space heating and air heating for industrial or agricultural applications. The principal
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requirement of the solar collector is a large contact area between the absorbing
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surface and the air. Various applications of solar air collectors were reported by Kolb
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et al. [23].
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In this paper, we investigate numerically the natural convection inside the
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solar collector having the flat-plate cover and wavelike absorber. The objective of
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this paper is to present flow and heat transfer used to harness solar energy.
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Problem Formulation
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Figure 1 shows a schematic diagram of a solar collector. The fluid in the collector is
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water-based nanofluid containing Al2O3 nanoparticles. The nanofluid is assumed
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incompressible and the flow is considered to be laminar. It is taken that water and
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nanoparticles are in thermal equilibrium and no slip occurs between them. The cold
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water enters on the left side of the absorber and become hot. Then this water leaves
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from the right side. The heated water is used for various purposes. The solar collector
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is a metal box with a glass cover plate on the top surface and a dark colored
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sinusoidal wavy absorber plate on the bottom. The length and height of the collector
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are L and H respectively. The transparent cover plate is continuously absorbing solar
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energy, while bottom wavy wall is uniformly maintained temperature Tc. The density
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of the nanofluid is approximated by the Boussinesq model. Amplitude of wave Am =
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0.075 and number of wave λ = 3.5 are assumed. Only steady state case is considered.
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The thermophysical properties of the nanofluid are taken from Lin and Violi
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[24] and given in Table 1. The density of the nanofluid is approximated by the
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Boussinesq model. Only steady state case is considered.
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The governing equations for laminar natural convection in a solar collector
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filled with water-alumina nanofluid in terms of the Navier-Stokes and energy
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equation (dimensional form) are given as:
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Continuity equation:
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u v

0
x y
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(
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1)
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x-momentum equation:
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 u
u 
p
nf  u  v     nf
y 
x
 x
  2u  2u 
 2 2
 x y 
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(
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2)
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y-momentum equation:
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nf  u
 v
v 
p
 v     nf
y 
y
 x
  2v  2v 
 2  2   g   nf (T  Tc )
 x y 
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(
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3)
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Energy equation:
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u
T
T
v
  nf
x
y
  2T  2T 
 2  2
y 
 x
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(
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4)
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where, nf  1     f  s is the density,
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 C 
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  nf  1      f
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 nf  knf
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the dynamic viscosity of Brinkman model [25] is nf   f 1   
p nf
 1      C p      C p  is the heat capacitance,
f
 C 
p nf
s
    s is the thermal expansion coefficient,
is the thermal diffusivity,
2.5
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and the thermal conductivity of Maxwell Garnett (MG) model [26] is
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knf  k f
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Radiation heat transfer by the top glass cover surface must account for thermal
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radiation which can be absorbed, reflected, or transmitted. This decomposition can be
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expressed by,
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qnet  qabsorbed  qtransmitted  qreflected
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Outside the boundary layer, the amount of energy qreflected is neglected.
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Thus total energy of the glass cover plate becomes qnet  qabsorbed  qtransmitted
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Now the amount of transmitted energy is radiated from the cover plate to the bottom
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wavy absorber without any medium as:
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qtransmitted  qr   A Tw  Tc4
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Here  is emissivity of the glass cover plate,  is Stefan Boltzmann constant
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5.670400 x 10-8 Js-1m-2K-4 and Tw is the variable temperature of the top wall. Again,
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the amount of absorbed energy is transferred from cover plate to bottom absorber by
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natural convection where medium is nanofluid as:
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qabsorbed  qc  hA Tw  Ti 
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So
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qnet  hA Tw  Ti    A Tw  Ti 4
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The boundary conditions are:
ks  2k f  2  k f  ks 
k s  2k f    k f  k s 

total
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energy
.

gained

4
or

loosed
by
the
cover
plate
is
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at all solid boundaries u = v = 0
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at the top cover plate q  hA Tw  Ti    A Tw  Ti 4
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at the vertical walls
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at the bottom wavy absorber T = Tc

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204
4

T
0
x
The above equations are non-dimensionalized by using the following
dimensionless dependent and independent variables:
X
T205
 Tc
x
y
uL
vL
pL2
, Y , U
, V
, P
,


2
L
L
f
f
Tw  Tc
 f f
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Then the non-dimensional governing equations are
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U V

0
X Y
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U
 f P  nf   2U  2U 
U
U
V





X
Y
nf X  f  X 2 Y 2 
(6)
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U
 f P  nf   2V  2V  Ra 1      f     s
V
V
V






X
Y
 nf Y  f  X 2 Y 2  Pr
 nf  f
(7)
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U


1   2  2 
V




X
Y Pr  X 2 Y 2 
(8)
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where Pr 
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νf
αf
(5)
is the Prandtl number, Ra 
g  f L3 Tw  Tc 
 f f
is the Rayleigh number.
The corresponding boundary conditions take the following form:
at all solid boundaries U = V = 0
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
0
X
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at the vertical walls
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at the bottom wavy absorber  = 0
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Average Nusselt number
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The average Nusselt number (Nu) is expected to depend on a number of
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factors such as thermal conductivity, heat capacitance, viscosity, flow structure of
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nanofluids, volume fraction, dimensions and fractal distributions of nanoparticles.
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The non-dimensional form of local convective heat transfer is Nu c  
knf 
.
k f Y
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By integrating the local Nusselt number over the top heated surface, the
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average convective heat transfer along the heated wall of the collector is used by
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Saleh et al. [27] as Nuc   Nuc dX .
1
0
1
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The radiated heat transfer rate is expressed as Nur   qr dX .
0
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The average Nusselt number is Nu  Nuc  Nur
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The mean bulk temperature and average sub domain velocity of the fluid
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inside the collector may be written as  av    dV / V and Vav   V dV / V , where
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V is the volume of the collector.
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Numerical Implementation
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The Galerkin finite element method [28, 29] is used to solve the non-dimensional
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governing equations along with boundary conditions for the considered problem. The
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equation of continuity has been used as a constraint due to mass conservation and this
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restriction may be used to find the pressure distribution. The penalty finite element
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method [30] is used to solve the Equations (6)-(8), where the pressure P is eliminated
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by a penalty constraint. The continuity equation is automatically fulfilled for large
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values of this penalty constraint. Then the velocity components (U, V) and
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temperature (θ) are expanded using a basis set. The Galerkin finite element technique
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yields the subsequent nonlinear residual equations. Three points Gaussian quadrature
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is used to evaluate the integrals in these equations. The non-linear residual equations
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are solved using Newton–Raphson method to determine the coefficients of the
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expansions. The convergence of solutions is assumed when the relative error for each
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variable between consecutive iterations is recorded below the convergence criterion
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such that  n 1  n  104 , where n is the number of iteration and Ψ is a function of
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U, V and θ.
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Mesh Generation
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In the finite element method, the mesh generation is the technique to
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subdivide a domain into a set of sub-domains, called finite elements, control volume,
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etc. The discrete locations are defined by the numerical grid, at which the variables
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are to be calculated. It is basically a discrete representation of the geometric domain
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on which the problem is to be solved. The computational domains with irregular
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geometries by a collection of finite elements make the method a valuable practical
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tool for the solution of boundary value problems arising in various fields of
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engineering. Fig. 2 displays the finite element mesh of the present physical domain.
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Grid Independent Test
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An extensive mesh testing procedure is conducted to guarantee a grid-
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independent solution for Ra = 105 and Pr = 6.2 in a solar collector. In the present
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work, we examine five different non-uniform grid systems with the following number
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of elements within the resolution field: 2969, 5130, 6916, 9057, and 11426. The
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numerical scheme is carried out for highly precise key in the average convective and
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radiated Nusselt numbers namely Nuc and Nur for the aforesaid elements to develop
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an understanding of the grid fineness as shown in Table 2 and Fig. 3. The scale of the
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average Nusselt and Sherwood numbers for 9057 elements shows a little difference
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with the results obtained for the other elements. Hence, considering the non-uniform
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grid system of 9057 elements is preferred for the computation.
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Code Validation
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The present numerical solution is validated by comparing the current code
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results for heat transfer - temperature difference profile at Pr = 0.73, Gr = 104 with
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the graphical representation of Gao et al. [31] which was reported for heat transfer
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augmentation inside a channel between the flat-plate cover and sine-wave absorber of
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a cross-corrugated solar air heater. Fig. 4 demonstrates the above stated comparison.
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As shown in Fig. 4, the numerical solutions (present work and Gao et al. [31]) are in
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good agreement.
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In addition, the current code results for streamlines and isotherms at Pr =
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0.71, A = 2, L = 2,  = 400, Ra = 106 with that of Varol and Oztop [32] are compared
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and displayed by the Fig. 5. They studied Buoyancy induced heat transfer and fluid
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flow inside a tilted wavy solar collector.
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Results and Discussion
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In this section, numerical results of streamlines and isotherms for various values of
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Rayleigh number (Ra) and Prandtl number (Pr) with Al2O3 /water nanofluid in a solar
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collector are displayed. The considered values of Ra and Pr are Ra = (104, 105, 106,
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and 5*106), Pr (= 4.2, 5.8, 6.2, and 8.8) while the solid volume fraction  = 5%, the
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emissivity  = 0.9, the wave amplitude of bottom surface Am = 0.075 and number of
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wave  = 3.5. In addition, the values of the average Nusselt number both for
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convection and radiation as well as mean bulk temperature and average sub domain
293
velocity profile are shown graphically.
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The effect of Ra on the thermal and flow fields are presented in Fig. 6 (a) - (b)
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while Pr = 6.2. The strength of the flow circulation and the thermal current activities
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is much more activated with escalating Ra. Isotherms are almost similar to the active
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parts for both nanofluid and base fluid. Increasing Ra, the temperature lines at the
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middle part of the collector become horizontal whereas initially they are almost wavy
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pattern due to convection is dominated across the solar collector. Due to rising values
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of Ra, the temperature distributions become distorted resulting in an increase in the
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overall heat transfer. This result can be attributed to the dominance of the buoyancy
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force. It is worth noting that as the Rayleigh number increases, the thickness of the
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thermal boundary layer near the top cover plate enhances which indicates a steep
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temperature gradients and hence, an increase in the overall heat transfer from the
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transparent cover plate to the wavelike absorber. On the other hand, for both type of
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fluids, six primary recirculation cells occupying the whole cavity is found at the
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lowest value of the Rayleigh number (Ra = 104). As well as two tiny eddies appear
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near the left and right vertical walls in this case. In each wave, the right and left
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vortices rotate in counter clockwise and clockwise direction respectively. The size of
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these cells becomes larger for clear water than the water alumina nanofluid. This
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happens due to greater velocity of base fluid than nanofluid. For increasing Ra,
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velocity of both type of fluids increases and thus size of the created eddies becomes
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larger. In addition more perturbation is observed in the streamlines at Ra = 5*106
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because of rising buoyancy force.
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Fig. 7 (a)–(b) exposes the heat transfer and fluid flow for various Prandtl
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number Pr (= 4.2-8.8). In this figure we observe that as the Prandtl number enhances
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from 4.2 to 8.8, the isothermal contours tend to get affected considerably. In addition,
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318
these lines corresponding to Pr = 8.8 become less bended whereas initially (Pr = 4.2)
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the lines take sinusoidal wavelike form. The isotherms cover the whole region of the
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solar collector due to comparatively higher temperature of the working water-Al2O3
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nanofluid at the lowest Pr where the contour lines mimic the wall’s (absorber)
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profile. Rising Prandtl number leads to deformation of the thermal boundary layer at
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the cold wavy absorber. However, the increase in the thermal gradients at the upper
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horizontal wall is much higher for the considered nanofluid than for the clear water.
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This means that higher heat transfer rate is predicted by the nanofluid than the base
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fluid (water). The fluid flow covers the entire collector at the lowest Pr forming few
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eddies. The streamlines have no significant change due to rising Pr except the core
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of the vortices becomes slightly smaller. This is expected because highly viscous
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fluid having larger Prandtl number does not move freely.
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The average Nusselt (convective and radiated) numbers, average temperature
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(θav) and mean sub domain velocity (Vav) profile along with the Rayleigh number
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(Ra) for both type of fluids are depicted in Fig. 8(i)-(iii). It is seen from Fig. 8(i) that
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Nuc and Nur enhance sharply due to effective buoyancy force. The rate of heat
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transfer for water-alumina nanofluid is found to be more effective than the clear water
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due to higher thermal conductivity of solid nanoparticles. Rate of convective heat
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transfer enhances by 16% and 10% for nanofluid and base fluid respectively whereas
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this rate for radiation is 5% with the variation of Ra from 104 to 5*106. It is well
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known that heat transfer rate is always higher for convection than radiation and is
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justified by the current investigation. Consequently Fig. 8(ii) shows that (θav) falls
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340
sequentially for all Ra. Fig. 8(iii) describes the enhancing phenomena in the Vav -Ra
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profile. Here base fluid has higher mean velocity than the nanofluid.
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Fig. 9(i)-(iii) displays the Nuc, Nur, θav and Vav for the effect of Prandtl
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number Pr. Mounting Pr enhances average Nusselt number for both convection and
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radiation. From Fig. 9 (i), it is found that, rate of convective heat transfer enhances by
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25% and 17% for nanofluid and base fluid respectively whereas this rate for radiation
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is 7% with the increasing values of Pr from 4.2 to 8.8. On the other hand, θav
347
devalues due to the variation of Pr as temperature of fluid grows down with growing
348
Pr. Vav has notable changes with different values of Prandtl number.
349
350
Conclusions
351
The influences of Rayleigh number and Prandtl number on natural convection
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boundary layer flow inside a solar collector with water-Al2O3 nanofluid are
353
accounted. Various Rayleigh number and Prandtl number have been considered for
354
the flow and temperature fields as well as the convective and radiated heat transfer
355
rates, mean bulk temperature of the fluids and average velocity field in the collector
356
while  and  are fixed at 5% and 0.9 respectively. The results of the numerical
357
analysis lead to the following conclusions:
358
359

The structure of the fluid streamlines and isotherms within the solar collector
is found to significantly depend upon the Ra and Pr.
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
360
361
The Al2O3 nanoparticles with the highest Ra and Pr are established to be
most effective in enhancing performance of heat transfer rate than base fluid.
362

Average heat transfer is obtained higher for convection than radiation.
363

Mean temperature diminishes for both fluids with rising both parameters.
364

Average velocity field increases due to growing Ra and falling Pr.
365
Overall the analysis also defines the operating range where water-Al2O3 nanofluid
366
can be considered effectively in determining the level of heat transfer augmentation.
367
368
References
369
Kreith, F. and Kreider, J.F. (1978). Principles of solar engineering. Insert book
370
371
edition. McGraw-Hill, NY, insert total number of pages.
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[2]
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[3]
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Sukhatme, S.P. (1991). Solar energy, principles of thermal collection and
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[4]
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study of parallel flow flat plate solar collector using finite element method. In:
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Numerical Methods in Thermal Problems, Proceedings of the 6th International
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Conference, Swansea, UK.
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[5]
Stasiek, J.A. (1998). Experimental studies of heat transfer and fluid flow
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Piao, Y. (1992). Natural, forced and mixed convection in a vertical cross
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corrugated channel. MS thesis, The University of British Columbia,
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Corrugated Enclosure with Mixed Boundary Conditions. ASME Journal of
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[9]
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corrugated absorber and back-plate. MS thesis, Yunnan Normal University,
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Kunming.
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[10]
Ho, C.D., Chen, T.C. (2006). The recycle effect on the collector efficiency
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improvement of double-pass sheet-and-tube solar water heaters with external
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recycle. Renew Energy, 31 (7), 953–97.
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Hussain, A. (2006) The performance of a cylindrical solar water heater.
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new-style double-tube heat exchanger for heating crude oil using solar hot
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B.C., Jang, S.P. (2007). Stability and thermal conductivity characteristics of
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nanofluids. Thermochimica Acta, 455 (1–2), 70–74.
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with colored absorbers. Solar Energy, 68, 343–356.
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nickel-pigmented aluminium oxide selective absorber. Renewable Energy, 27,
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277–292.
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absorbers in different non-black colours. Proceedings of WREC VII, Cologne
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on CD-ROM.
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Konttinen, P., Lund, P.D., Kilpi, R.J. (2003). Mechanically manufactured
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selective solar absorber surfaces. Solar Energy Mater Solar Cells, 79 (3), 273–
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283.
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Soltau, H. (1992). Testing the thermal performance of uncovered solar
collectors. Solar Energy, 49 (4), 263–272.
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Molineaux, B., Lachal, B., Gusian, O. (1994). Thermal analysis of five
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outdoor swimming pools heated by unglazed solar collectors. Solar Energy,
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[22]
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Kolb, A., Winter, E.R.F., Viskanta, R. (1999). Experimental studies on a solar
air collector with metal matrix absorber. Solar Energy, 65 (2), 91–98.
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vertical cavity: Effects of non-uniform particle diameter and temperature on
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thermal conductivity. Int. J. of Heat and Fluid Flow, 31, 236–245.
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solution. J. Chem. Phys. 20, 571–581.
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Brinkman, H.C. (1952). The viscosity of concentrated suspensions and
Maxwell-Garnett, J.C. (1904). Colours in metal glasses and in metallic films.
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Saleh, H., Roslan, R., Hashim, I. (2011). Natural convection heat transfer in a
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nanofluid-filled trapezoidal enclosure. Int. J. of Heat and Mass Trans., 54,
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194–201.
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[27]
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equations using finite element technique. Computer and Fluids, 1, 73–89.
[28]
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443
Taylor, C., Hood, P. (1973). A numerical solution of the Navier-Stokes
Dechaumphai, P. (1999). Finite Element Method in Engineering. 2nd ed.,
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[29]
Basak, T., Roy, S., Pop, I. (2009). Heat flow analysis for natural convection
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within trapezoidal enclosures based on heatline concept, Int. J. Heat Mass
445
Trans., 52, 2471–2483.
446
[30]
Gao, W., Lin, W., Lu, E. (2000). Numerical study on natural convection
447
inside the channel between the flat-plate cover and sine-wave absorber of a
448
cross-corrugated solar air heater, Energy Conversion & Management, 41, 145-
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151.
450
[31]
Varol, Y. and Oztop, H.F. (2007). Buoyancy induced heat transfer and fluid
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flow inside a tilled wavy solar collector. Building Environment, 42, 2062-
452
2071.
453
454
455
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456
Table 1. Thermo physical properties of fluid and nanoparticles
Physical Properties
Fluid phase
Al2O3
(Water)
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458
Cp(J/kgK)
4179
850
 (kg/m3)
997.1
3900
k (W/mK)
0.6
46
105 (1/K)
21
1.67
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459
Table 2: Grid Sensitivity Check at Pr = 6.2,  = 5% and Ra = 105
460
461
Nodes
6224
10982
13538
20295
27524
462
(elements)
(2969)
(5130)
(6916)
(9057)
(11426)
463
Nuc
6.82945
7.98176
8.77701 9.088308 9.089308
Nur
4.32945
5.38176
5.98701 6.454909 6.455909
Time (s)
226.265
292.594
388.157
464
465
466
467
468
421.328
627.375
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469
Solar Energy
470
Th
Water-alumina nanofluid
Wave like Absorber
Tc
474
y = Am[1−cos(2λπx)]
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Cold water input
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477
Insulation
y
g
x
478
479
480
481
482
483
484
485
Insulation
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Cover plate
L
H
472
Insulation
471
Figure 1. Schematic diagram of the Solar Collector
Hot water output
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486
487
488
489
490
491
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Figuer 2. Mesh generation of the collector
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494
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497
498
499
500
501
502
503
504
505
506
507
Figure 3. Grid test for the geometry of Pr = 6.2 and Ra = 105
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508
509
510
511
512
513
514
515
∆T
516
517
518
Figure 4. Comparison between present code and Gao et al. [31] at Pr = 0.73 and
519
Ra = 104
520
521
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523
524
525
Streamlines
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526
528
529
Isotherms
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530
531
Varol and Oztop [32]
Present Work
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533
534
535
536
Figure 5. Comparison of present code with Varol and Oztop [32]
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538
Ra =
5*106
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540
541
Ra = 106
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Ra = 105
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546
547
Ra = 104
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548
549
(a)
(b)
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Figure 6. Effect of Ra on (a) Isotherms and (b) Streamlines at Pr = 6.2 (solid lines
552
for nanofluid and dashed lines for base fluid)
553
554
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556
557
Pr = 8.8
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559
560
Pr = 6.2
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561
563
Pr = 5.8
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564
566
567
568
Pr = 4.2
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(a)
(b)
569
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Figure 7. Effect of Pr on (a) Isotherms and (b) Streamlines at Ra = 105 (solid lines
572
for nanofluid and dashed lines for base fluid)
573
574
575
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576
577
578
579
580
(
a
)
)
581
582
583
584
(b)
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586
587
588
589
590
(c)
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592
593
594
595
596
597
Figure 8. Effect of Ra on (a) Nu, (b) θav and (c) Vav at Pr = 6.2
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598
599
600
601
(
a
)
602
603
604
605
606
(b)
607
608
609
610
611
612
613
(c)
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615
616
617
Figure 9. Effect of Pr on (a) Nu (b) θav and (c) Vav at Ra = 105