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Numerical study of pulsatile MHD counter-current nanofluid flows through two elastic coaxial pipes containing porous blocks
International Journal of Heat and Mass Transfer 113 (2017) 1265–1280
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Numerical study of pulsatile MHD counter-current nanofluid flows through two elastic coaxial pipes containing porous blocks Victor M. Job, Sreedhara Rao Gunakala ⇑ Department of Mathematics and Statistics, The University of the West Indies, St. Augustine, Trinidad and Tobago
a r t i c l e
i n f o
Article history: Received 22 March 2017 Received in revised form 8 June 2017 Accepted 14 June 2017
Keywords: Two-phase nanofluid MHD Mixed convection Elastic pipes Pulsatile flow Porous blocks
a b s t r a c t We investigate the MHD convective pulsatile flow through a counter-current two-nanofluid (Cu-water and CuO-water) system. This system consists of two elastic coaxial pipes, with porous blocks mounted on the walls of each channel. Hot CuO-water nanofluid flows upwards in the inner channel, while cold Cu-water nanofluid flows downwards in the outer channel. A two-phase model that considers the effects of thermophoresis and brownian motion on the concentration of nanoparticles within the fluid is used to describe heat and mass transfer within the system. The governing equations with the associated initial and boundary conditions are solved using the mixed finite element method with P2 P 1 Taylor-Hood elements. The influence of time, pulsation frequencies and amplitudes of the inner and outer fluids, elastic modulus, Reynolds number, solid volume fractions of the inner and outer nanofluids, and the diameters of the Cu and CuO nanoparticles on fluid flow, heat transfer, nanoparticle concentration and pressure drop is studied. We found that heat transfer may be enhanced by increasing parameters such as the elastic modulus of the pipes, Reynolds number, solid volume fraction of nanoparticles in the inner fluid and nanoparticle size in the outer fluid. Similarly, heat transfer is enhanced by reducing the solid volume fraction of nanoparticles in the outer fluid and nanoparticle size in the inner fluid. In most cases, the heat transfer enhancement observed in the study is accompanied by increases in pressure drop. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Convective fluid flows through channels containing porous blocks have been studied by many authors. This is due to their importance in heat transfer systems such as heat exchangers, electronic cooling devices and thermal regenerators [1]. Ko and Anand [1] conducted an experimental study on heat transfer enhancement with or without the use of wall-mounted porous baffles in a uniformly heated rectangular channel. In their study, the authors found that when porous baffles are used the heat transfer is enhanced by 300 percent, as compared to the case of no baffles. Furthermore, an increase in the thickness of the porous baffles results in enhanced heat transfer. However, a significant increase in pumping power was observed with the use of porous baffles. Forced convective heat transfer in a horizontal channel containing staggered wall-mounted porous baffles was investigated in 2004 by Miranda and Anand [2]. In this connection, the heat transfer enhancement ratio for solid baffles was found to be greater than that of porous baffles. However, porous baffles show greater over-
⇑ Corresponding author. E-mail address: [email protected] (S.R. Gunakala). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.062 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
all heat transfer performance than solid baffles since the Darcy friction factor ratio is smaller in the case of porous baffles. The flow of fluids through coaxial pipes has wide applications to heat exchange in refrigerators, automobile engine cooling, solar heating and air conditioning systems [3,4]. Furthermore, due to the growing need for the development of fluids with better heat transfer properties than conventional coolants, the study of nanofluids in heat exchangers is increasingly important. The heat transfer performance of TiO2 -water nanofluid in a double tube counter flow heat exchanger was investigated experimentally by Duangthongsuk and Wongwises [5]. The study showed that the use of this nanofluid gives significant heat transfer enhancement compared to water. Also, no additional pumping power is required, since the pressure drop and friction factor as the same as those of water. Moreover, an increase in the rate of heat transfer was observed as the Reynolds number is increased. The effects of an applied radial magnetic field on water based Mn-Zn ferrite nanofluid in a counter-flow double-pipe heat exchanger was examined by Bahiraei and Hangi [6]. The authors of this study observed that the pressure drop and heat transfer increase with an increase in the magnetic field strength and nanoparticle concentration. Targui and Kahalerras [7] analyzed the heat transfer performance of forced convective nanofluid flows through a double pipe parallel flow
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Nomenclature A1 A2 cp cF Da DB DT ds1 ds2 E Er g Gr h1 h2 Ha K kB Nu NBT L p Pr R R0 Re Sc St t T
inner pipe pulsation amplitude outer pipe pulsation amplitude 1 specific heat capacity, J kg K1 form drag coefficient Darcy number Brownian diffusivity thermophoretic diffusivity nanoparticle diameter in inner nanofluid nanoparticle diameter in outer nanofluid elastic modulus, kg m1 s2 dimensionless elastic modulus gravitational acceleration Grashof number porous block width, m pipe wall thickness, m Hartmann number permeability, m2 Boltzmann constant Nusselt number Browniannthermophoretic diffusivity ratio initial length of elastic pipe pressure, kg m1 s2 Prandtl number initial radius of the inner pipe initial radius of the outer pipe Reynolds number Schmidt number Strouhal number time, s temperature, K
heat exchanger with porous baffles. In their study, the authors examined water-based nanofluids that contain Cu, Ag, Al2 O3 , CuO or TiO2 nanoparticles. It was found that heat transfer is enhanced with increased solid volume fraction of these nanofluids. The highest rate of heat transfer was obtained with the Ag-water nanofluid, whereas the TiO2 -water nanofluid gave the lowest heat transfer rates. In each case, the addition of nanoparticles to the base fluid increases the pressure drop within the heat exchanger, and the highest values are obtained for the Ag-water nanofluid. Finally, it was concluded that the best heat transfer performance is achieved when nanoparticles are added only to the cold fluid in the annular gap. The use of pulsating flow is a useful method of heat transfer enhancement in many engineering systems such as heat exchangers in stirling or pulse tube cryocoolers. Pulsating flow phenomena may also be found in human respiratory and circulatory systems [4,8,9]. The effects of pulsation on forced convection within a channel with two wall-mounted porous blocks was considered by Huang and Yang [9]. An increase in the rate of heat transfer was observed with an increase in pulsating amplitude. It was also shown that critical values of the permeability of the porous blocks and the pulsation frequency exist at which the heat transfer is minimum and maximum respectively. Targui and Kahalerras [4] examined the use of porous baffles and pulsating flow for heat transfer enhancement in double pipe parallel flow heat exchangers. The results of this study showed that the greatest heat transfer enhancement occurs when pulsation is applied only to the hot fluid. It was observed that an increase in the amplitude and frequency of pulsation cause an enhancement in heat transfer performance. Furthermore, the heat transfer performance is enhanced by a decrease in the permeability of the porous baffles. A numerical
u,v U1 U2 x,r b
e g j l m n
q r /
x
velocity components, m s1 cycle-averaged inlet velocity of inner channel, m s1 cycle-averaged inlet velocity of outer channel, m s1 space coordinates, m coefficient of thermal expansion, K1 porosity radial deformation, m thermal conductivity, W m1 K1 viscosity, kg m1 s1 Poisson’s ratio axial deformation, m density, kg m3 electrical conductivity, X1 m1 nanoparticle solid volume fraction pulsation frequency, s1
Subscripts 1 inner channel 2 outer channel eff effective f fluid m nanofluid in porous layer nf nanofluid in free-fluid region ps porous solid av average w pipe wall wi inner pipe wall wo outer pipe wall
Fig. 1. Schematic diagram of the problem.
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Fig. 2. Schematic diagram of the problem (axisymmetric view).
study was conducted by Nandi and Chattopadhyay [10] on unsteady forced convective flow through a wavy channel with pulsation at the inlet. This investigation determined that the inlet pulsation enhances heat transfer and causes a reduction in the pressure drop within the channel. Akdag et al. [11] numerically investigated the heat transfer performance of pulsating Al2 O3 water nanofluid flow in a wavy channel. It was observed in this study that the average Nusselt number is enhanced by increases in nanoparticle concentration, pulsation frequency and pulsation amplitude. Moreover, the pressure drop in the channel increases with increased nanoparticle concentration and pulsation amplitude.
Table 1 Thermophysical properties of pure water, Cu and CuO at 25 C. Quantity
Pure water
Cu
CuO
q (kg m )
997.1 4179
8933 385
6510 540
3
1
cp (J kg
j (W m
1
K1 ) K1 )
b (K1 )
r (X1 m1 )
0.613
401
18
21 105 0.05
1:67 105
8:5 106
5:96 107
1:1 103
Extensive research has been conducted on fluid flows through rigid tubes; however, only a few studies exist that consider the flow of fluids through flexible or elastic tubes. In particular, the study of convective heat transfer through flexible or elastic tubes has potential applications in refrigeration systems and in bioheat transfer [12,13]. The pressure-driven pulsatile flow through microtubes having thin elastic walls was investigated analytically by San and Staples [14]. It was observed that the shear stress at the walls of elastic tubes is greater than it is at the walls of rigid tubes for small values of the Womersley number Wo. Furthermore, the shear stress decreases with increased Wo. De Andrade et al. [15] studied three-dimensional pulsatile flows in rigid and elastic curved pipes using the CFD commercial software STAR-CCM+. In the case of the elastic curved pipe, the wall deformation increases with a decrease in Young’s modulus and an increase in Womersley number. Coccarelli and Nithiarasu [13] used the finite element method to analyze pulsatile fluid flow and heat transfer in rigid and elastic tubes and tube networks with convective thermal boundary conditions imposed at the elastic walls. The authors found that a decrease in the material parameter of the wall increases the average flow speed via a reduction in flow resistance. The average temperature of the fluid also increases as the material parameter of the wall is increased. In this paper, we consider the mixed convective counter-current flow of CuO-water (inner pipe) and Cu-water (outer pipe) nanofluids in elastic coaxial pipes containing wall-mounted porous blocks in the presence of a uniform radial magnetic field. The nanofluid heat and mass transfer is described using a two-phase model that incorporates the effects of thermophoresis and brownian motion on the concentration of nanoparticles within the fluid. Using the mixed finite element method, the governing system of equations is solved and the effects of pertinent parameters on fluid flow, heat transfer, nanoparticle concentration distribution and pressure drop within the system are investigated. 2. Problem formulation We consider a system that is composed of two vertical nanofluid-filled circular coaxial elastic pipes with initial length L and initial radii R and R0 (0 < R < R0 ) as shown in Figs. 1 and 2. CuO-water nanofluid enters the inner channel at x ¼ 0 with pulsating velocity U 1 ½1 þ A1 sinð2px1 tÞ in the positive x-direction and concentration /1 . Cu-water nanofluid enters the outer channel at x ¼ L with velocity U 2 ½1 þ A2 sinð2px2 tÞ in the opposite direction of the flow in the inner channel and concentration /2 . In this study, A1 ; x1 and A2 ; x2 denote the amplitude and frequency of pulsation in the inner and outer pipe respectively, and U i (i ¼ 1; 2) denotes the cycle-averaged velocity at the inlet of the inner and outer channels respectively. Furthermore, the pulsation frequency is taken to be equal for both nanofluids (x1 ¼ x2 ¼ x) [4]. We consider the nanoparticles in the Cu-water and CuO-water nanofluids to be spherical in shape with diameters ds1 and ds2 respectively. The inner pipe undergoes time-dependent axial and radial deformations nwi
Table 2 1 Grid-independence test values at Pr ¼ 6:2; jps ¼ 8 W m1 K1 ; qps ¼ 200 kg m3 ; ðcp Þps ¼ 900 J kg K1 ; cF ¼ 0:06; t ¼ 10; Re ¼ 10; U r ¼ 1; Ha ¼ 10; Gr ¼ 10; Da ¼ 104 ; h1 ¼ 0:03; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; N BT ¼ 1; ds1 ¼ ds2 ¼ 50 nm; Sc ¼ 50; Er ¼ 2 104 ; qr ¼ 5; hr ¼ 102 ; m ¼ 0:4 and L ¼ 3.
e ¼ 0:95;
Number of elements
Nuav
P drop
Time (s)
5363 6823 7556 7843 8813 9273
0.4282 0.4647 0.4814 0.4831 0.4846 0.4855
32.9062 37.7851 37.8102 40.2594 40.3884 40.4781
6552 8297 10127 12011 12309 12958
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Fig. 3. Comparison of heat transfer enhancement ratio between present code and Akdag et al. [11].
nel. The blocks in the inner and outer channels have lengths R =2 and ðR0 R Þ=2 respectively. The nanofluid flows are influenced by the presence of a uniform radial magnetic field with strength B0 , and free convection due to the presence of buoyancy forces. In order to analyze the present problem, we make the following assumptions: 1. The fluid flow is incompressible and axially symmetric. 2. The volume charge density is very small compared to the Lorentz force and the magnetic Reynolds number is very small. 3. The nanofluids are modelled as two-phase Newtonian fluids which have low nanoparticle concentrations. 4. The Cu and CuO nanoparticles have uniform size and are spherical in shape. 5. Brownian diffusivity and thermophoresis are the only important nanoparticle-fluid slip mechanisms in nanofluids. 6. Heat transfer associated with nanoparticle dispersion, thermal radiation and viscous and Joule dissipation are negligible. 7. The Boussinesq approximation is applicable. 8. The fluid and porous medium are in local thermal equilibrium. 9. The thickness of each elastic pipe is very small compared to its radius. 10. Axial and radial deformations are sufficiently small so that the inner and outer pipes exhibit linear elastic behaviour. Using these assumptions, we obtain the following governing equations for the problem in dimensionless form. Continuity Equation for Nanofluids Fig. 4. Comparison between isotherm plots Akdag et al. [11] present code.
@u 1 @ þ ðrv Þ ¼ 0 @x r @r
ð1Þ
Continuity Equation for Nanoparticles
and gwi , while the outer pipe undergoes axial and radial deformations nwo and gwo . The inlet temperature of the inner and outer channels are T 1 and T 2 respectively (T 1 > T 2 ) and the outer channel wall is thermally insulated. Porous blocks with porosity e, permeability K and width h1 are inserted in a staggered configuration within the annular gap and on the inside surface of the inner chan-
" # @/ 1 @/ @/ 1 @2/ 1 @ @/ ¼ þ þv r u þ @t e @x @r ReSc @x2 r @r @r " # 2 1 @ T 1 @ @T þ r þ ReScNBT @x2 r @r @r
ð2Þ
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Fig. 5. Comparison of efficiency ratio between present code and Targui and Kahalerras [7].
Fig. 6. Plot of axial velocity u over time at monitoring points ðx; rÞ ¼ ð0:5; 0:25Þ and ðx; rÞ ¼ ð2:5; 0:75Þ.
X-Momentum Equation
" 1 @u 1 @u @u @p 1 @ u þv þ ¼ þ @r @x eRe @x e @t e2 @x
lnf lf rnf rf
R-Momentum Equation
! !# lnf @u 1 @ lnf @u þ r r @r lf @x lf @r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p 1 cF u pffiffiffiffiffiffi u2 þ v 2 u ReDa Da ðqbÞnf Gr Ha2 T ð3Þ uþ Re ðqbÞf Re2
" 1 @v 1 @v @v @p 1 @ ¼ þ u þv þ @r eRe @x e @t e2 @x @r
!
lnf @ v 1 @ lnf @ v þ r r @r lf @x lf @r
1 lnf v lnf 1 v eRe lf r2 lf ReDa cF pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi u2 þ v 2 v Da
!#
ð4Þ
Energy Equation
ðqcp Þeff @T ðqcp Þnf @T @T þ þv u @x @r ðqcp Þf @t ðqcp Þf jeff @T 1 @ jeff @T 1 @ þ r ¼ RePr @x jf @x r @r jf @r
ð5Þ
The effective heat capacity ðqcp Þeff and effective thermal conductivity
jeff are given by [16]:
ðqcp Þeff ¼ ð1 eÞðqcp Þps þ eðqcp Þnf
jeff ¼ ð1 eÞjps þ ejnf The governing equations in the free-fluid region are obtained from Eqs. (1)–(5) when e ¼ 1 and K ! 1. The axial and radial deformations nwi ,nwo , gwi and gwo of the inner and outer pipe are described by the following dimensionless equations [17].
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Fig. 7. Streamline plots with A1 ¼ 0:2 ¼ A2 , t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
Axial Deformation of the Inner Pipe
@ nwi @ 2 nwi @g ¼ E þ m wi r @x2 @x @t2 2
!
1 þ qr hr Re
lnf @u @ v þ lf @r @x wi
Axial Deformation of the Outer Pipe
ð6Þ
Radial Deformation of the Inner Pipe
@ 2 gwi pwi @nwi ¼ E g þ m r wi qr hr @x @t 2
Fig. 8. Streamline plots with A1 ¼ 0:2, A2 ¼ 0:8, t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
@ 2 nwo @ 2 nwo @g ¼ Er þ m wo 2 @x2 @x @t
!
þ
1 qr hr Re
lnf @u @ v þ lf @r @x wo
ð8Þ
Radial Deformation of the Outer Pipe
ð7Þ
@ 2 gwo pwo @nwo ¼ E g þ m r wo qr hr @x @t2
ð9Þ
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Fig. 9. Streamline plots with A1 ¼ 0:8, A2 ¼ 0:2, t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
where Gr ¼
Re ¼
q2f bf gðT 1 T 2 Þh3 , l2f
qf U 1 R0 lf
; Da ¼ RK2 , 0
Er ¼ q
E
2 w U 1 ð1mÞ
Ha ¼ B0 R0
qffiffiffiffi r f
lf ,
Pr ¼
lf ðcp Þf jf ,
Fig. 10. Streamline plots for different values of Er with t ¼ 10, Re ¼ 10, A1 ¼ 0:2 ¼ A2 , St ¼ 1, /1 ¼ /2 ¼ 0:02, ds1 ¼ ds2 ¼ 50 nm.
l
/1 ; Sc ¼ q Df B , N BT ¼ DTDðTB T11T and St ¼ xUR10 2Þ f
are the Reynolds number, Darcy number, Hartmann number, Prandtl number, Grashof number, dimensionless elastic modulus, Schmidt number, ratio of Brownian and thermophoretic diffusivities, and Strouhal number respectively. In Eqs. (6)–(9), the
subscripts ‘wi’ and ‘wo’ refer to the walls of the inner and outer pipes respectively. Eqs. (1)–(9) are obtained using the nondimensional variables given below:
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Fig. 11. Isotherm plots for different values of Er with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 , St ¼ 1; /1 ¼ /2 ¼ 0:02; ds1 ¼ ds2 ¼ 50 nm.
ð^x; ^r Þ ¼ ^¼ p
p
ðu; v Þ T T2 ^t ¼ U 1 t ; ðu ^; v^ Þ ¼ ; Tb ¼ ; R0 U1 T1 T2 ðn ; n ; g ; g Þ ^ ¼ / ; ð^nwi ; ^nwo ; g ^ wi ; g ^ wo Þ ¼ wi wo wi wo ; / /1 R0
ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þs
ðx; rÞ ; R0 ; 2
qf U 1 q h qr ¼ w ; hr ¼ 2 qf R0
4. Thermal Expansion Coefficient:
ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞs ð10Þ
1. Viscosity (Brinkman):
lf 2:5
ð1 /Þ
jnf ¼ jf 1 þ
!
#
us ds ðqcp Þf df / js 1 þ 25000 jf jf ds 1 /
ð15Þ
2kB T where us ¼ pl 2 is the velocity of nanoparticles due to Brownian f ds
ð11Þ
2. Density:
qnf ¼ ð1 /Þqf þ /qs
ð14Þ
5. Thermal Conductivity (Patel):
"
The thermophysical properties of the nanofluids, base fluid and nanoparticles are related as follows [18,19]:
lnf ¼
ð13Þ
ð12Þ
Since / is small, we approximate the density of the nanofluid by the density of the base fluid; i.e., qnf ’ qf . 3. Heat Capacity:
motion, kB ¼ 1:3806 1023 J K1 is the Boltzmann constant, df is the molecular size of the fluid and ds is the nanoparticle diameter. We note here that for water-based nanofluids,
lf ¼ 8:9 104 Pa s and df ¼ 2 1010 m. 6. Electrical Conductivity (Maxwell):
rnf ¼ rf 1 þ
3ðrs =rf 1Þ/ ðrs =rf þ 2Þ ðrs =rf 1Þ/
ð16Þ
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Fig. 12. Concentration contour plots for different values of Er with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; ds1 ¼ ds2 ¼ 50 nm.
The thermophysical properties of pure water, Cu and CuO [20– 24] at 25 C are given in Table 1 below: The no-slip condition at the inner and outer cylinder walls is assumed, and continuity of the velocity, temperature, heat flux and normal stresses is imposed at the porous-fluid interface. The temperature and heat flux are also assumed to be continuous at the inner cylinder wall. Hence the dimensionless initial and boundary conditions for this problem are as follows: Initial Conditions:
u ¼ v ¼ T ¼ 0 at t ¼ 0
ð17Þ
nwi ¼ nwo ¼ 0;
@nwi @nwo ¼ ¼ 0 at t ¼ 0 @t @t
ð18Þ
gwi ¼ gwo ¼ 0;
@ gwi @ gwo ¼ ¼ 0 at t ¼ 0 @t @t
ð19Þ
/ ¼ 1 at t ¼ 0; / ¼ /r at t ¼ 0;
r
ð20Þ
r>R
ð21Þ
Boundary Conditions:
nwi ¼ 0 ¼ nwo at x ¼ 0
ð22Þ
@nwi @n ¼ 0 ¼ wo at x ¼ L @x @x
ð23Þ
At x ¼ 0;
0 < r < R þ gwi ð0; tÞ :
u ¼ 1 þ A1 sinð2pSt tÞ; At x ¼ L þ nwi ðL; tÞ;
v ¼ 0;
T ¼ 1;
/¼1
ð24Þ
R þ gwi ðL; tÞ < r < 1 þ gwo ðL; tÞ :
u ¼ U r ½1 þ A2 sinð2pSt tÞ;
v ¼ T ¼ 0;
/ ¼ /r
ð25Þ
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@u @T @/ ¼v ¼ ¼ ¼ 0 at r ¼ 0; @n @n @n
0 < x < L þ nwi ðL; tÞ
ð30Þ
where R ¼ RR0 ; L ¼ RL0 ; U r ¼ UU21 and /r ¼ //21 . The streamlines are described using the Stokes streamfunction, which is defined as
u¼
1 @w ; r @r
v¼
1 @w r @x
ð31Þ
Using (31), the streamfunction is obtained by solving the equation
@2w 1 @ @w 2 @w @u @ v þ r ¼ r @x2 r @r @r r @r @r @x
ð32Þ
with boundary conditions
w ¼ r at x ¼ 0;
0 < r < R þ gwi ð0; tÞ
w ¼ r at x ¼ L þ nwi ðL; tÞ; @w ¼ 0 at x ¼ L þ nwi ðL; tÞ; @n @w ¼ 0 at x ¼ 0; @n
ð33Þ
R þ gwi ðL; tÞ < r < 1 þ gwo ðL; tÞ 0 < r < R þ gwi ðL; tÞ
R þ gwi ð0; tÞ < r < 1 þ gwo ð0; tÞ
w ¼ R þ gwi at r ¼ R þ gwi ;
0 < x < L þ nwi ðL; tÞ
w ¼ 1 þ gwo at r ¼ 1 þ gwo ; @w ¼ 0 at r ¼ 0; @n
0 < x < L þ nwi ðL; tÞ
0 < x < L þ nwi ðL; tÞ
ð34Þ ð35Þ ð36Þ ð37Þ ð38Þ ð39Þ
The rate of heat transfer between the inner and outer fluids is described using the cycle-space-averaged Nusselt number Nuav over a period of pulsation St1 at the wall C of the inner pipe. This value is defined by
Nuav ¼ where
Z
1
s
t þ s t
Z Nuav ;t dt ¼ St
1 t þSt
t
ð40Þ
Nuav ;t dt
s ¼ St1 ; t is a reference time and R
Nuav ;t ¼
@T
C @n dl
ð41Þ
lðCÞ
is the space-averaged Nusselt number at the wall of the inner pipe at time t. The space-averaged pressure drops Pi;t and Po;t within the inner and outer tubes (respectively) at time t are given by Fig. 13. Streamline plots for different values of Re with t ¼ 10, Er ¼ 2 104 , A1 ¼ 0:2 ¼ A2 , St ¼ 1, /1 ¼ /2 ¼ 0:02, ds1 ¼ ds2 ¼ 50 nm.
@u @ v @T @/ ¼ ¼ ¼ 0 at x ¼ L þ nwi ðL; tÞ; ¼ @n @n @n @n @u @ v @T @/ ¼ ¼ ¼ 0 at x ¼ 0; ¼ @n @n @n @n
u¼
u¼
v¼
@nwo ; @t
v¼
0
pðL þ nwi ðL; tÞ; r; tÞdr R þ gwi ðL; tÞ
R Rþgwi ð0;tÞ 0
pð0; r; tÞdr R þ gwi ð0; tÞ
ð42Þ
and
0 < r < R þ gwi ðL; tÞ ð26Þ
R 1þgwo ð0;tÞ Po;t ¼
Rþgwi ð0;tÞ
pð0; r; tÞdr
1 R þ gwo ð0; tÞ gwi ð0; tÞ
R 1þgwo ðL;tÞ
Rþgwi ðL;tÞ
pðL þ nwi ðL; tÞ; r; tÞdr
1 R þ gwo ðL; tÞ gwi ðL; tÞ ð43Þ
R þ gwi ð0; tÞ < r < 1 þ gwo ð0; tÞ ð27Þ
@ gwi @/ 1 @T þ ¼ 0 at r ¼ R þ gwi ; 0 < x < L þ nwi ðL; tÞ ; @n NBT @n @t ð28Þ
@nwi ; @t
R Rþgwi ðL;tÞ Pi;t ¼
@ gwo @/ @T ¼0¼ at r ¼ 1 þ gwo ; 0 < x < L þ nwi ðL; tÞ ; @n @n @t ð29Þ
The total pressure drop P drop within the system is taken as the sum of the cycle-space-averaged pressure drops P drop;i and P drop;o within the inner and outer tubes respectively, where
Z Pdrop;i ¼ St
1 t þSt
t
P i;t dt
ð44Þ
P o;t dt
ð45Þ
and
Z Pdrop;o ¼ St
1 t þSt
t
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Fig. 14. Isotherm plots for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
3. Numerical method The mathematical software FreeFem++ (Hecht 2013) is used in the implementation of the mixed finite element method for the present problem. We solve the highly-coupled system of nonlinear Eqs. (1)–(9) using a sequential approach as follows. The flow, heat transfer and concentration fields are solved using equations on an initial undeformed finite element mesh. The resulting fluid stresses are applied to the elastic walls, the pipe deformations are computed and subsequently, the finite element mesh is updated. The flow, heat transfer and concentration fields are re-computed on the updated finite element mesh. This process is repeated with a relative error tolerance of 104 .
4. Grid-independence study and validation of the numerical algorithm In order to determine a suitable mesh for the present problem, we conducted a grid-independence test with six finite element
meshes consisting of 5363, 6823, 7556, 7843, 8813 and 9473 P2 P1 Taylor-Hood elements. The cycle-space-averaged Nusselt number Nuav and total pressure drop P drop were used to represent the finite element solution. The criterion used for grid independence is that the relative errors between Nuav values on successive meshes and between Pdrop values on successive meshes are less than 0.5%. Using this criterion, we determined that a finite element mesh with 7843 elements ensures accuracy and computational efficiency of the solution (see Table 2). We validate the finite element code used in the present study by numerically reproducing results that are available in the literature. The first such result corresponds to the work of Akdag et al. [11]. Comparisons of isotherm plots and the heat transfer enhancement ratio Nuav =Nus are displayed in Figs. 3 and 4 respectively. From these comparisons, it was determined that the results are in good agreement, with a maximum relative error of less than 3%. The present study was also compared with the work of Targui and Kahalerras [7]. The efficiency ratio for different concentrations of Cu-water nanofluid is shown in Fig. 5. In this case, the maximum relative error is less than 0.5%.
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Fig. 15. Concentration contour plots for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
5. Results and discussion The porous medium used in this study is an open-cell aluminium foam that has thermal conductivity jps ¼ 8 W m1 K1 , density qps ¼ 200 kg m3 , specific heat capacity ðcp Þps ¼ 900 J kg
1
K1 and form drag coefficient cF ¼ 0:06 [25–27]. We
assume the values Pr ¼ 6:2; Da ¼ 104 ; e ¼ 0:95; h1 ¼ 0:03;R ¼ 0:5; Ha ¼ 10; Gr ¼ 10; N BT ¼ 1; Sc ¼ 50; U r ¼ 1; qr ¼ 5; hr ¼ 102 ; m ¼ 0:4 and L ¼ 3. The effects of time t, pulsation amplitudes A1 and A2 , pulsation frequency St, dimensionless elastic modulus Er , Reynolds number Re, solid volume fractions /1 (inner fluid) and /2 (outer fluid), and corresponding nanoparticle diameters ds1 and ds2 on fluid flow, heat transfer and nanoparticle concentration are discussed. In Fig. 5 and 6, the axial velocity u is plotted as a function of time at monitoring points ðx; rÞ ¼ ð0:5; 0:25Þ and ðx; rÞ ¼ ð2:5; 0:75Þ for St ¼ 1 and St ¼ 2. We observe that the velocities of the inner and outer nanofluids gradually increase in magnitude as the fluids respond to the imposed pulsation at the inlet of each channel. This response is delayed due to the time required to
transfer kinetic energy throughout the fluid. After 5–7 pulsation cycles, an equilibrium is established between the inlet pulsation and the flow velocity within the system; hence a steady periodic state is achieved. We also note that increasing the pulsation frequency from St ¼ 1 to St ¼ 2 reduces the time required for the fluid to achieve a steady periodic state. The magnitude of flow velocity during a periodic steady state was found to decrease when St is increased. This indicates a reduction in the influence of inlet pulsation on fluid flow with increased pulsation frequency. Figs. 7–9 show the variation of streamlines during a steady periodic cycle for different amplitudes of pulsation. These streamline plots reveal that the porous blocks within the coaxial pipes cause a deflection of fluid away from the adjoining wall. We note that the streamline distortion is more pronounced in the outer channel than it is in the inner channel. This is due to the staggered configuration of the porous blocks in the outer channel, which creates greater flow disturbances than the inline configuration found in the inner channel. It was also observed that the streamline distortion increases as t is increased from t to t þ s =4, and decreases as t is increased from t þ s =4 to t ¼ t þ 3s =4. The observed increase and decrease in streamline distortion corresponds to an
V.M. Job, S.R. Gunakala / International Journal of Heat and Mass Transfer 113 (2017) 1265–1280
Fig. 16. Streamline plots for different values of /1 and /2 with t ¼ 10, Re ¼ 10, A1 ¼ 0:2 ¼ A2 , St ¼ 1, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
acceleration and deceleration of the fluid within the pulsation cycle. In addition, the flow disturbances over the steady periodic cycle increase with an increase in pulsation amplitudes A1 and A2 . Figs. 10–12 reveal a reduction in the deformation of the coaxial pipes with increased Er . This occurs since as Er is increased, the pipe walls become more resistant to elastic deformation in the presence of stresses exerted by the inner and outer fluids. We observe that the streamlines (Fig. 10) move radially inward as Er is increased. Moreover, the temperature of the system (Fig. 11) decreases with increased Er due to a reduction in the radius of the inner channel. A downward movement of the concentration
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contours (Fig. 12) occurs with increased Er as a result of thermophoretic diffusion and a downward movement of the isotherms within the system. Figs. 13–15 display the influence of the Reynolds number Re on fluid flow, heat transfer and concentration distribution. We observe a decrease in the deformation of the pipe walls as Re increases. This is caused by a reduction in the wall stress exerted by the adjacent fluid with increasing Re. Consequently, the streamlines (Fig. 13) are less distorted as Re is increased. A reduction in temperature (Fig. 14) within the pipes occurs with increased Re due to the decreased deformation of the pipe walls. An increase in Re causes a downward movement of concentration contours within the outer channel through an enhancement in mass transport. An upward movement of concentration contours (Fig. 15) is observed within the inner channel as Re increases from 5 to 10. However, this trend is reversed when Re increases from 10 to 25 due to a reduction in the radius of the inner channel and the effects of thermophoresis. The effects of solid volume fractions /1 (CuO-water nanofluid) and /2 (Cu-water nanofluid) on streamline and isotherm plots are showed in Figs. 16 and 17. We observe that the streamlines within the outer channel are increasingly distorted with increased /2 (cf. Fig. 16 a and b, or c and d) due to an increase in flow resistance through the porous blocks within the outer channel, which results from an increase in nanofluid viscosity. However, there is no significant change in the streamlines as /1 varies (cf. Fig. 16 a and c, or b and d) since the impact of increased viscosity on the flow resistance through the porous blocks within the inner channel is very small. A small increase in temperature within the pipes is observed as /1 is increased (cf. Fig. 17 a and b, or c and d), but the reverse occurs when /2 is increased (cf. Fig. 17 a and c, or b and d). This can be explained by an enhancement in the thermal conductivities of the inner and outer nanofluids as /1 and /2 increase. The enhanced thermal conductivity improves the conduction of heat along each channel, which causes an upward movement of the isotherms when /1 is increased and a downward movement of the isotherms when /2 is increased. Tables 3–8 display the effects of pertinent parameters on the maximum deformations nwi;max and nwo;max of the inner and outer pipes, the maximum and minimum radial deformations gwi;max and gwi;min of the inner pipe, the maximum and minimum radial deformations gwo;max and gwo;min of the outer pipe, the cyclespace-averaged Nusselt number Nuav , the pressure drops Pdrop;i and P drop within the inner and outer channel and the total pressure drop P drop within the system. Table 3 reveals that there is a small increase in the inner and outer pipe wall deformations with increased Strouhal number St, which is caused by an increase in flow disturbance as the pulsating frequency increases. The increased flow disturbance leads to a reduction in the cyclespace-averaged Nusselt number Nuav , which indicates an improvement in heat transfer from the inner nanofluid to the outer nanofluid. However, the increased flow disturbance within the coaxial pipes also leads to an increase in the pressure drop in both the inner and outer channels. We observe that the deformations of the pipe walls increase with an increase in the pulsation amplitudes A1 and A2 (Table 4). The cycle-space-averaged Nusselt number Nuav increases as A1 is increased; however, A2 has the opposite effect on Nuav . This suggests that the heat transfer performance is optimal when A1 is small and A2 is large. Additionally, the pressure drops within the inner and outer channels increase with decreased A1 and increased A2 . As seen in Table 5, a reduction in the elastic modulus Er increases the axial and radial deformations of the coaxial pipes. This increase in pipe deformations causes a reduction in pressure
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Fig. 17. Isotherm plots for different values of /1 and /2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
drop within each pipe as Er decreases. Since the width of the inner channel increases as Er decreases, there is an increase in the quantity of heat entering the system. Consequently, Nuav increases with a reduction in Er . Table 6 reveals that the deformations of the inner and outer pipes and the pressure drops within these pipes are reduced as the Reynolds number Re increases. The cycle-space-averaged Nusselt number Nuav decreases with increased Re as a result of the reduction in the pipe wall deformations. We observe in Table 7 that the deformations of the inner and outer pipes increase when /1 decreases and /2 increases. An increase in /1 and /2 reduces the velocity of the inner and outer fluids as a result of an increase in nanofluid viscosity. Due to the counter-current configuration of the inner and outer fluids, the stresses exerted on the pipe walls decrease when /1 is increased, but increase when /2 is increased. Consequently, the pressure drop within each channel decreases with increased /1 , but increases with increased /2 . The cycle-space-averaged Nusselt number Nuav decreases when /1 is increased, but increases when /2 is increased. This is due to the enhancement in the thermal conductivities of the counter-current nanofluids with increased solid volume fraction.
Table 8 shows that the pressure drops within each channel is independent of nanoparticle diameter. However, Nuav increases with an increase in the diameter ds1 of the CuO nanoparticles within the inner fluid, but decreases with an increase in the diameter ds2 of the Cu nanoparticles within the outer fluid. This occurs as a result of an increase in the thermal conductivity of the inner and outer fluids, as well as the counter-current configuration of these fluids. From these results, we infer that the heat transfer performance of the system may be improved by reducing the size of the CuO nanoparticles within the inner fluid and increasing the size of the Cu nanoparticles within the outer fluid.
6. Conclusions In this paper, we studied the mixed convective pulsatile counter-current flow of CuO-water and Cu-water nanofluids through two coaxial elastic pipes in the presence of an applied radial magnetic field. The effects of time, pulsation frequencies and amplitudes of the inner and outer fluids, elastic modulus, Reynolds number, solid volume /1 and /2 of the inner and outer nanofluids, and the diameters of the Cu and CuO nanoparticles
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V.M. Job, S.R. Gunakala / International Journal of Heat and Mass Transfer 113 (2017) 1265–1280 Table 3 Deformation, Nusselt number and pressure drop values for different values of St with Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. St
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.2 0.5 1 2
0.0694 0.0695 0.0696 0.0696
0.1851 0.1852 0.1855 0.1860
0.1608 0.1609 0.1612 0.1617
0.0780 0.0781 0.0782 0.0782
0.3263 0.3265 0.3268 0.3272
0.1447 0.1448 0.1449 0.1454
0.4850 0.4836 0.4831 0.4824
10.8872 11.2877 12.2464 12.8108
24.8771 27.4616 28.0130 28.9293
35.7643 38.7493 40.2594 41.7401
Table 4 Deformation, Nusselt number and pressure drop values for different values of A1 and A2 with t ¼ 5; Re ¼ 10; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. A1
A2
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.2 0.4 0.6 0.8 0.2 0.2 0.2
0.2 0.2 0.2 0.2 0.4 0.6 0.8
0.0696 0.0696 0.0697 0.0697 0.0705 0.0719 0.0739
0.1855 0.1856 0.1857 0.1860 0.1866 0.1884 0.1911
0.1612 0.1612 0.1613 0.1615 0.1618 0.1628 0.1641
0.0782 0.0782 0.0783 0.0774 0.0790 0.0804 0.0825
0.3268 0.3268 0.3268 0.3269 0.3279 0.3297 0.3322
0.1449 0.1450 0.1450 0.1450 0.1454 0.1460 0.1468
0.4831 0.4839 0.4851 0.4872 0.4777 0.4701 0.4598
12.2464 12.4098 12.5637 13.3882 12.3537 12.4701 12.5998
28.0130 28.0557 28.0995 28.1444 28.5687 29.1524 31.2073
40.2594 40.4655 40.6632 41.5326 40.9224 41.6238 43.8071
Table 5 Deformation, Nusselt number and pressure drop values for different values of Er with t ¼ 5; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
gwi;max
nwi;max
Er
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
0.0109
0.0049
0.0951
0.0424
6
1 10
0.0018
0.0071
0.0005
0.0020
1 105
0.0158
0.0597
0.0469
0.0175
5 104
0.0294
0.1025
0.0850
0.0329
0.1691
0.0749
2 104
0.0696
0.1855
0.1612
0.0782
0.3268
0.1449
P drop;i
P drop;o
P drop
0.3963
13.5824
32.6541
46.2365
0.4230
13.2042
31.0090
44.2132
0.4457
12.9266
29.8891
42.8157
0.4831
12.2464
28.0130
40.2594
Table 6 Deformation, Nusselt number and pressure drop values for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. Re
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
5 10 15 25
0.1479 0.0696 0.0505 0.0373
0.2834 0.1855 0.1529 0.1198
0.2344 0.1612 0.1290 0.0932
0.1658 0.0782 0.0562 0.0408
0.4689 0.3268 0.2580 0.1919
0.2234 0.1449 0.1129 0.0833
0.5410 0.4831 0.4618 0.3664
22.3220 12.2464 8.5408 5.6575
52.9754 28.0130 19.6879 13.0530
65.2974 40.2594 28.2287 18.7105
Table 7 Deformation, Nusselt number and pressure drop values for different values of /1 and /2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. /1
/2
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.01 0.02 0.03 0.05 0.02 0.02 0.02
0.02 0.02 0.02 0.02 0.01 0.03 0.05
0.0760 0.0696 0.0674 0.0653 0.0658 0.0769 0.1072
0.1872 0.1855 0.1850 0.1847 0.1837 0.1882 0.1970
0.1613 0.1612 0.1611 0.1606 0.1600 0.1623 0.1641
0.0850 0.0782 0.0759 0.0737 0.0741 0.0860 0.1184
0.3341 0.3268 0.3244 0.3226 0.3210 0.3365 0.3706
0.1453 0.1449 0.1442 0.1437 0.1440 0.1446 0.1455
0.4899 0.4831 0.4823 0.4815 0.4751 0.4947 0.5262
12.3215 12.2464 12.2058 12.1464 12.0494 12.4734 13.0274
29.3404 28.0130 27.5901 27.2587 27.0070 29.7618 36.5284
41.6619 40.2594 39.7959 39.4051 39.0564 42.2352 49.5558
Table 8 Deformation, Nusselt number and pressure drop values for different values of ds1 and ds2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 . ds1
ds2
Nuav
P drop;i
P drop;o
P drop
50 75 100 150 50 50 50
50 50 50 50 75 100 150
0.4831 0.4832 0.4834 0.4835 0.4787 0.4764 0.4741
12.2464 12.2464 12.2464 12.2464 12.2464 12.2464 12.2464
28.0130 28.0130 28.0130 28.0130 28.0130 28.0130 28.0130
40.2594 40.2594 40.2594 40.2594 40.2594 40.2594 40.2594
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on fluid flow, heat transfer and concentration distribution within the coaxial system were examined. The results show that the velocities of the inner and outer nanofluids increase in magnitude as time progresses and approach a steady periodic state. It was also observed that over a steady periodic cycle, the flow disturbances (streamline distortions) increase as the amplitude of pulsation is increased. We found that the flow disturbances are significantly affected by an increase in the elastic modulus of the pipes. An increase in streamline distortions occurs when the Reynolds number decreases. Moreover, the streamline distortions increase with an increase in the solid volume fraction of nanoparticles within the outer channel. The temperature of the system decreases with an increase in the elastic modulus of the pipes and the Reynolds number. An increase in the solid volume fraction of nanoparticles within the inner channel causes an increase in temperature, but the trend is reversed when the solid volume fraction of nanoparticles within the outer channel increases. The parameters considered in this study influence the concentration distribution within the coaxial pipes through variations in mass transport and thermophoresis. The pipe wall deformations are significantly affected by each parameter except the variation of the diameters of the Cu and CuO nanoparticles. It was discovered that the heat transfer performance within the system may be improved by increasing the pulsation frequency, pulsation amplitude of the outer channel, elastic modulus of the pipes, Reynolds number, solid volume fraction of nanoparticles in the inner fluid and nanoparticle size in the outer fluid. Similarly, we may improve the heat transfer performance within the system by decreasing the pulsation amplitude of the inner pipe, solid volume fraction of nanoparticles in the outer fluid and nanoparticle size in the inner fluid. Moreover, with the exception of the Reynolds number and the size of nanoparticles, the enhancement of heat transfer performance achieved by varying the pertinent parameters is accompanied by an increase in pressure drop within the system.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study of pulsatile MHD counter-current nanofluid flows through two elastic coaxial pipes containing porous blocks Victor M. Job, Sreedhara Rao Gunakala ⇑ Department of Mathematics and Statistics, The University of the West Indies, St. Augustine, Trinidad and Tobago
a r t i c l e
i n f o
Article history: Received 22 March 2017 Received in revised form 8 June 2017 Accepted 14 June 2017
Keywords: Two-phase nanofluid MHD Mixed convection Elastic pipes Pulsatile flow Porous blocks
a b s t r a c t We investigate the MHD convective pulsatile flow through a counter-current two-nanofluid (Cu-water and CuO-water) system. This system consists of two elastic coaxial pipes, with porous blocks mounted on the walls of each channel. Hot CuO-water nanofluid flows upwards in the inner channel, while cold Cu-water nanofluid flows downwards in the outer channel. A two-phase model that considers the effects of thermophoresis and brownian motion on the concentration of nanoparticles within the fluid is used to describe heat and mass transfer within the system. The governing equations with the associated initial and boundary conditions are solved using the mixed finite element method with P2 P 1 Taylor-Hood elements. The influence of time, pulsation frequencies and amplitudes of the inner and outer fluids, elastic modulus, Reynolds number, solid volume fractions of the inner and outer nanofluids, and the diameters of the Cu and CuO nanoparticles on fluid flow, heat transfer, nanoparticle concentration and pressure drop is studied. We found that heat transfer may be enhanced by increasing parameters such as the elastic modulus of the pipes, Reynolds number, solid volume fraction of nanoparticles in the inner fluid and nanoparticle size in the outer fluid. Similarly, heat transfer is enhanced by reducing the solid volume fraction of nanoparticles in the outer fluid and nanoparticle size in the inner fluid. In most cases, the heat transfer enhancement observed in the study is accompanied by increases in pressure drop. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Convective fluid flows through channels containing porous blocks have been studied by many authors. This is due to their importance in heat transfer systems such as heat exchangers, electronic cooling devices and thermal regenerators [1]. Ko and Anand [1] conducted an experimental study on heat transfer enhancement with or without the use of wall-mounted porous baffles in a uniformly heated rectangular channel. In their study, the authors found that when porous baffles are used the heat transfer is enhanced by 300 percent, as compared to the case of no baffles. Furthermore, an increase in the thickness of the porous baffles results in enhanced heat transfer. However, a significant increase in pumping power was observed with the use of porous baffles. Forced convective heat transfer in a horizontal channel containing staggered wall-mounted porous baffles was investigated in 2004 by Miranda and Anand [2]. In this connection, the heat transfer enhancement ratio for solid baffles was found to be greater than that of porous baffles. However, porous baffles show greater over-
⇑ Corresponding author. E-mail address: [email protected] (S.R. Gunakala). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.062 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
all heat transfer performance than solid baffles since the Darcy friction factor ratio is smaller in the case of porous baffles. The flow of fluids through coaxial pipes has wide applications to heat exchange in refrigerators, automobile engine cooling, solar heating and air conditioning systems [3,4]. Furthermore, due to the growing need for the development of fluids with better heat transfer properties than conventional coolants, the study of nanofluids in heat exchangers is increasingly important. The heat transfer performance of TiO2 -water nanofluid in a double tube counter flow heat exchanger was investigated experimentally by Duangthongsuk and Wongwises [5]. The study showed that the use of this nanofluid gives significant heat transfer enhancement compared to water. Also, no additional pumping power is required, since the pressure drop and friction factor as the same as those of water. Moreover, an increase in the rate of heat transfer was observed as the Reynolds number is increased. The effects of an applied radial magnetic field on water based Mn-Zn ferrite nanofluid in a counter-flow double-pipe heat exchanger was examined by Bahiraei and Hangi [6]. The authors of this study observed that the pressure drop and heat transfer increase with an increase in the magnetic field strength and nanoparticle concentration. Targui and Kahalerras [7] analyzed the heat transfer performance of forced convective nanofluid flows through a double pipe parallel flow
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Nomenclature A1 A2 cp cF Da DB DT ds1 ds2 E Er g Gr h1 h2 Ha K kB Nu NBT L p Pr R R0 Re Sc St t T
inner pipe pulsation amplitude outer pipe pulsation amplitude 1 specific heat capacity, J kg K1 form drag coefficient Darcy number Brownian diffusivity thermophoretic diffusivity nanoparticle diameter in inner nanofluid nanoparticle diameter in outer nanofluid elastic modulus, kg m1 s2 dimensionless elastic modulus gravitational acceleration Grashof number porous block width, m pipe wall thickness, m Hartmann number permeability, m2 Boltzmann constant Nusselt number Browniannthermophoretic diffusivity ratio initial length of elastic pipe pressure, kg m1 s2 Prandtl number initial radius of the inner pipe initial radius of the outer pipe Reynolds number Schmidt number Strouhal number time, s temperature, K
heat exchanger with porous baffles. In their study, the authors examined water-based nanofluids that contain Cu, Ag, Al2 O3 , CuO or TiO2 nanoparticles. It was found that heat transfer is enhanced with increased solid volume fraction of these nanofluids. The highest rate of heat transfer was obtained with the Ag-water nanofluid, whereas the TiO2 -water nanofluid gave the lowest heat transfer rates. In each case, the addition of nanoparticles to the base fluid increases the pressure drop within the heat exchanger, and the highest values are obtained for the Ag-water nanofluid. Finally, it was concluded that the best heat transfer performance is achieved when nanoparticles are added only to the cold fluid in the annular gap. The use of pulsating flow is a useful method of heat transfer enhancement in many engineering systems such as heat exchangers in stirling or pulse tube cryocoolers. Pulsating flow phenomena may also be found in human respiratory and circulatory systems [4,8,9]. The effects of pulsation on forced convection within a channel with two wall-mounted porous blocks was considered by Huang and Yang [9]. An increase in the rate of heat transfer was observed with an increase in pulsating amplitude. It was also shown that critical values of the permeability of the porous blocks and the pulsation frequency exist at which the heat transfer is minimum and maximum respectively. Targui and Kahalerras [4] examined the use of porous baffles and pulsating flow for heat transfer enhancement in double pipe parallel flow heat exchangers. The results of this study showed that the greatest heat transfer enhancement occurs when pulsation is applied only to the hot fluid. It was observed that an increase in the amplitude and frequency of pulsation cause an enhancement in heat transfer performance. Furthermore, the heat transfer performance is enhanced by a decrease in the permeability of the porous baffles. A numerical
u,v U1 U2 x,r b
e g j l m n
q r /
x
velocity components, m s1 cycle-averaged inlet velocity of inner channel, m s1 cycle-averaged inlet velocity of outer channel, m s1 space coordinates, m coefficient of thermal expansion, K1 porosity radial deformation, m thermal conductivity, W m1 K1 viscosity, kg m1 s1 Poisson’s ratio axial deformation, m density, kg m3 electrical conductivity, X1 m1 nanoparticle solid volume fraction pulsation frequency, s1
Subscripts 1 inner channel 2 outer channel eff effective f fluid m nanofluid in porous layer nf nanofluid in free-fluid region ps porous solid av average w pipe wall wi inner pipe wall wo outer pipe wall
Fig. 1. Schematic diagram of the problem.
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Fig. 2. Schematic diagram of the problem (axisymmetric view).
study was conducted by Nandi and Chattopadhyay [10] on unsteady forced convective flow through a wavy channel with pulsation at the inlet. This investigation determined that the inlet pulsation enhances heat transfer and causes a reduction in the pressure drop within the channel. Akdag et al. [11] numerically investigated the heat transfer performance of pulsating Al2 O3 water nanofluid flow in a wavy channel. It was observed in this study that the average Nusselt number is enhanced by increases in nanoparticle concentration, pulsation frequency and pulsation amplitude. Moreover, the pressure drop in the channel increases with increased nanoparticle concentration and pulsation amplitude.
Table 1 Thermophysical properties of pure water, Cu and CuO at 25 C. Quantity
Pure water
Cu
CuO
q (kg m )
997.1 4179
8933 385
6510 540
3
1
cp (J kg
j (W m
1
K1 ) K1 )
b (K1 )
r (X1 m1 )
0.613
401
18
21 105 0.05
1:67 105
8:5 106
5:96 107
1:1 103
Extensive research has been conducted on fluid flows through rigid tubes; however, only a few studies exist that consider the flow of fluids through flexible or elastic tubes. In particular, the study of convective heat transfer through flexible or elastic tubes has potential applications in refrigeration systems and in bioheat transfer [12,13]. The pressure-driven pulsatile flow through microtubes having thin elastic walls was investigated analytically by San and Staples [14]. It was observed that the shear stress at the walls of elastic tubes is greater than it is at the walls of rigid tubes for small values of the Womersley number Wo. Furthermore, the shear stress decreases with increased Wo. De Andrade et al. [15] studied three-dimensional pulsatile flows in rigid and elastic curved pipes using the CFD commercial software STAR-CCM+. In the case of the elastic curved pipe, the wall deformation increases with a decrease in Young’s modulus and an increase in Womersley number. Coccarelli and Nithiarasu [13] used the finite element method to analyze pulsatile fluid flow and heat transfer in rigid and elastic tubes and tube networks with convective thermal boundary conditions imposed at the elastic walls. The authors found that a decrease in the material parameter of the wall increases the average flow speed via a reduction in flow resistance. The average temperature of the fluid also increases as the material parameter of the wall is increased. In this paper, we consider the mixed convective counter-current flow of CuO-water (inner pipe) and Cu-water (outer pipe) nanofluids in elastic coaxial pipes containing wall-mounted porous blocks in the presence of a uniform radial magnetic field. The nanofluid heat and mass transfer is described using a two-phase model that incorporates the effects of thermophoresis and brownian motion on the concentration of nanoparticles within the fluid. Using the mixed finite element method, the governing system of equations is solved and the effects of pertinent parameters on fluid flow, heat transfer, nanoparticle concentration distribution and pressure drop within the system are investigated. 2. Problem formulation We consider a system that is composed of two vertical nanofluid-filled circular coaxial elastic pipes with initial length L and initial radii R and R0 (0 < R < R0 ) as shown in Figs. 1 and 2. CuO-water nanofluid enters the inner channel at x ¼ 0 with pulsating velocity U 1 ½1 þ A1 sinð2px1 tÞ in the positive x-direction and concentration /1 . Cu-water nanofluid enters the outer channel at x ¼ L with velocity U 2 ½1 þ A2 sinð2px2 tÞ in the opposite direction of the flow in the inner channel and concentration /2 . In this study, A1 ; x1 and A2 ; x2 denote the amplitude and frequency of pulsation in the inner and outer pipe respectively, and U i (i ¼ 1; 2) denotes the cycle-averaged velocity at the inlet of the inner and outer channels respectively. Furthermore, the pulsation frequency is taken to be equal for both nanofluids (x1 ¼ x2 ¼ x) [4]. We consider the nanoparticles in the Cu-water and CuO-water nanofluids to be spherical in shape with diameters ds1 and ds2 respectively. The inner pipe undergoes time-dependent axial and radial deformations nwi
Table 2 1 Grid-independence test values at Pr ¼ 6:2; jps ¼ 8 W m1 K1 ; qps ¼ 200 kg m3 ; ðcp Þps ¼ 900 J kg K1 ; cF ¼ 0:06; t ¼ 10; Re ¼ 10; U r ¼ 1; Ha ¼ 10; Gr ¼ 10; Da ¼ 104 ; h1 ¼ 0:03; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; N BT ¼ 1; ds1 ¼ ds2 ¼ 50 nm; Sc ¼ 50; Er ¼ 2 104 ; qr ¼ 5; hr ¼ 102 ; m ¼ 0:4 and L ¼ 3.
e ¼ 0:95;
Number of elements
Nuav
P drop
Time (s)
5363 6823 7556 7843 8813 9273
0.4282 0.4647 0.4814 0.4831 0.4846 0.4855
32.9062 37.7851 37.8102 40.2594 40.3884 40.4781
6552 8297 10127 12011 12309 12958
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Fig. 3. Comparison of heat transfer enhancement ratio between present code and Akdag et al. [11].
nel. The blocks in the inner and outer channels have lengths R =2 and ðR0 R Þ=2 respectively. The nanofluid flows are influenced by the presence of a uniform radial magnetic field with strength B0 , and free convection due to the presence of buoyancy forces. In order to analyze the present problem, we make the following assumptions: 1. The fluid flow is incompressible and axially symmetric. 2. The volume charge density is very small compared to the Lorentz force and the magnetic Reynolds number is very small. 3. The nanofluids are modelled as two-phase Newtonian fluids which have low nanoparticle concentrations. 4. The Cu and CuO nanoparticles have uniform size and are spherical in shape. 5. Brownian diffusivity and thermophoresis are the only important nanoparticle-fluid slip mechanisms in nanofluids. 6. Heat transfer associated with nanoparticle dispersion, thermal radiation and viscous and Joule dissipation are negligible. 7. The Boussinesq approximation is applicable. 8. The fluid and porous medium are in local thermal equilibrium. 9. The thickness of each elastic pipe is very small compared to its radius. 10. Axial and radial deformations are sufficiently small so that the inner and outer pipes exhibit linear elastic behaviour. Using these assumptions, we obtain the following governing equations for the problem in dimensionless form. Continuity Equation for Nanofluids Fig. 4. Comparison between isotherm plots Akdag et al. [11] present code.
@u 1 @ þ ðrv Þ ¼ 0 @x r @r
ð1Þ
Continuity Equation for Nanoparticles
and gwi , while the outer pipe undergoes axial and radial deformations nwo and gwo . The inlet temperature of the inner and outer channels are T 1 and T 2 respectively (T 1 > T 2 ) and the outer channel wall is thermally insulated. Porous blocks with porosity e, permeability K and width h1 are inserted in a staggered configuration within the annular gap and on the inside surface of the inner chan-
" # @/ 1 @/ @/ 1 @2/ 1 @ @/ ¼ þ þv r u þ @t e @x @r ReSc @x2 r @r @r " # 2 1 @ T 1 @ @T þ r þ ReScNBT @x2 r @r @r
ð2Þ
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Fig. 5. Comparison of efficiency ratio between present code and Targui and Kahalerras [7].
Fig. 6. Plot of axial velocity u over time at monitoring points ðx; rÞ ¼ ð0:5; 0:25Þ and ðx; rÞ ¼ ð2:5; 0:75Þ.
X-Momentum Equation
" 1 @u 1 @u @u @p 1 @ u þv þ ¼ þ @r @x eRe @x e @t e2 @x
lnf lf rnf rf
R-Momentum Equation
! !# lnf @u 1 @ lnf @u þ r r @r lf @x lf @r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p 1 cF u pffiffiffiffiffiffi u2 þ v 2 u ReDa Da ðqbÞnf Gr Ha2 T ð3Þ uþ Re ðqbÞf Re2
" 1 @v 1 @v @v @p 1 @ ¼ þ u þv þ @r eRe @x e @t e2 @x @r
!
lnf @ v 1 @ lnf @ v þ r r @r lf @x lf @r
1 lnf v lnf 1 v eRe lf r2 lf ReDa cF pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi u2 þ v 2 v Da
!#
ð4Þ
Energy Equation
ðqcp Þeff @T ðqcp Þnf @T @T þ þv u @x @r ðqcp Þf @t ðqcp Þf jeff @T 1 @ jeff @T 1 @ þ r ¼ RePr @x jf @x r @r jf @r
ð5Þ
The effective heat capacity ðqcp Þeff and effective thermal conductivity
jeff are given by [16]:
ðqcp Þeff ¼ ð1 eÞðqcp Þps þ eðqcp Þnf
jeff ¼ ð1 eÞjps þ ejnf The governing equations in the free-fluid region are obtained from Eqs. (1)–(5) when e ¼ 1 and K ! 1. The axial and radial deformations nwi ,nwo , gwi and gwo of the inner and outer pipe are described by the following dimensionless equations [17].
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Fig. 7. Streamline plots with A1 ¼ 0:2 ¼ A2 , t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
Axial Deformation of the Inner Pipe
@ nwi @ 2 nwi @g ¼ E þ m wi r @x2 @x @t2 2
!
1 þ qr hr Re
lnf @u @ v þ lf @r @x wi
Axial Deformation of the Outer Pipe
ð6Þ
Radial Deformation of the Inner Pipe
@ 2 gwi pwi @nwi ¼ E g þ m r wi qr hr @x @t 2
Fig. 8. Streamline plots with A1 ¼ 0:2, A2 ¼ 0:8, t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
@ 2 nwo @ 2 nwo @g ¼ Er þ m wo 2 @x2 @x @t
!
þ
1 qr hr Re
lnf @u @ v þ lf @r @x wo
ð8Þ
Radial Deformation of the Outer Pipe
ð7Þ
@ 2 gwo pwo @nwo ¼ E g þ m r wo qr hr @x @t2
ð9Þ
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Fig. 9. Streamline plots with A1 ¼ 0:8, A2 ¼ 0:2, t ¼ 10, Re ¼ 10, St ¼ 1, /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
where Gr ¼
Re ¼
q2f bf gðT 1 T 2 Þh3 , l2f
qf U 1 R0 lf
; Da ¼ RK2 , 0
Er ¼ q
E
2 w U 1 ð1mÞ
Ha ¼ B0 R0
qffiffiffiffi r f
lf ,
Pr ¼
lf ðcp Þf jf ,
Fig. 10. Streamline plots for different values of Er with t ¼ 10, Re ¼ 10, A1 ¼ 0:2 ¼ A2 , St ¼ 1, /1 ¼ /2 ¼ 0:02, ds1 ¼ ds2 ¼ 50 nm.
l
/1 ; Sc ¼ q Df B , N BT ¼ DTDðTB T11T and St ¼ xUR10 2Þ f
are the Reynolds number, Darcy number, Hartmann number, Prandtl number, Grashof number, dimensionless elastic modulus, Schmidt number, ratio of Brownian and thermophoretic diffusivities, and Strouhal number respectively. In Eqs. (6)–(9), the
subscripts ‘wi’ and ‘wo’ refer to the walls of the inner and outer pipes respectively. Eqs. (1)–(9) are obtained using the nondimensional variables given below:
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Fig. 11. Isotherm plots for different values of Er with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 , St ¼ 1; /1 ¼ /2 ¼ 0:02; ds1 ¼ ds2 ¼ 50 nm.
ð^x; ^r Þ ¼ ^¼ p
p
ðu; v Þ T T2 ^t ¼ U 1 t ; ðu ^; v^ Þ ¼ ; Tb ¼ ; R0 U1 T1 T2 ðn ; n ; g ; g Þ ^ ¼ / ; ð^nwi ; ^nwo ; g ^ wi ; g ^ wo Þ ¼ wi wo wi wo ; / /1 R0
ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þs
ðx; rÞ ; R0 ; 2
qf U 1 q h qr ¼ w ; hr ¼ 2 qf R0
4. Thermal Expansion Coefficient:
ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞs ð10Þ
1. Viscosity (Brinkman):
lf 2:5
ð1 /Þ
jnf ¼ jf 1 þ
!
#
us ds ðqcp Þf df / js 1 þ 25000 jf jf ds 1 /
ð15Þ
2kB T where us ¼ pl 2 is the velocity of nanoparticles due to Brownian f ds
ð11Þ
2. Density:
qnf ¼ ð1 /Þqf þ /qs
ð14Þ
5. Thermal Conductivity (Patel):
"
The thermophysical properties of the nanofluids, base fluid and nanoparticles are related as follows [18,19]:
lnf ¼
ð13Þ
ð12Þ
Since / is small, we approximate the density of the nanofluid by the density of the base fluid; i.e., qnf ’ qf . 3. Heat Capacity:
motion, kB ¼ 1:3806 1023 J K1 is the Boltzmann constant, df is the molecular size of the fluid and ds is the nanoparticle diameter. We note here that for water-based nanofluids,
lf ¼ 8:9 104 Pa s and df ¼ 2 1010 m. 6. Electrical Conductivity (Maxwell):
rnf ¼ rf 1 þ
3ðrs =rf 1Þ/ ðrs =rf þ 2Þ ðrs =rf 1Þ/
ð16Þ
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Fig. 12. Concentration contour plots for different values of Er with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; ds1 ¼ ds2 ¼ 50 nm.
The thermophysical properties of pure water, Cu and CuO [20– 24] at 25 C are given in Table 1 below: The no-slip condition at the inner and outer cylinder walls is assumed, and continuity of the velocity, temperature, heat flux and normal stresses is imposed at the porous-fluid interface. The temperature and heat flux are also assumed to be continuous at the inner cylinder wall. Hence the dimensionless initial and boundary conditions for this problem are as follows: Initial Conditions:
u ¼ v ¼ T ¼ 0 at t ¼ 0
ð17Þ
nwi ¼ nwo ¼ 0;
@nwi @nwo ¼ ¼ 0 at t ¼ 0 @t @t
ð18Þ
gwi ¼ gwo ¼ 0;
@ gwi @ gwo ¼ ¼ 0 at t ¼ 0 @t @t
ð19Þ
/ ¼ 1 at t ¼ 0; / ¼ /r at t ¼ 0;
r
ð20Þ
r>R
ð21Þ
Boundary Conditions:
nwi ¼ 0 ¼ nwo at x ¼ 0
ð22Þ
@nwi @n ¼ 0 ¼ wo at x ¼ L @x @x
ð23Þ
At x ¼ 0;
0 < r < R þ gwi ð0; tÞ :
u ¼ 1 þ A1 sinð2pSt tÞ; At x ¼ L þ nwi ðL; tÞ;
v ¼ 0;
T ¼ 1;
/¼1
ð24Þ
R þ gwi ðL; tÞ < r < 1 þ gwo ðL; tÞ :
u ¼ U r ½1 þ A2 sinð2pSt tÞ;
v ¼ T ¼ 0;
/ ¼ /r
ð25Þ
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@u @T @/ ¼v ¼ ¼ ¼ 0 at r ¼ 0; @n @n @n
0 < x < L þ nwi ðL; tÞ
ð30Þ
where R ¼ RR0 ; L ¼ RL0 ; U r ¼ UU21 and /r ¼ //21 . The streamlines are described using the Stokes streamfunction, which is defined as
u¼
1 @w ; r @r
v¼
1 @w r @x
ð31Þ
Using (31), the streamfunction is obtained by solving the equation
@2w 1 @ @w 2 @w @u @ v þ r ¼ r @x2 r @r @r r @r @r @x
ð32Þ
with boundary conditions
w ¼ r at x ¼ 0;
0 < r < R þ gwi ð0; tÞ
w ¼ r at x ¼ L þ nwi ðL; tÞ; @w ¼ 0 at x ¼ L þ nwi ðL; tÞ; @n @w ¼ 0 at x ¼ 0; @n
ð33Þ
R þ gwi ðL; tÞ < r < 1 þ gwo ðL; tÞ 0 < r < R þ gwi ðL; tÞ
R þ gwi ð0; tÞ < r < 1 þ gwo ð0; tÞ
w ¼ R þ gwi at r ¼ R þ gwi ;
0 < x < L þ nwi ðL; tÞ
w ¼ 1 þ gwo at r ¼ 1 þ gwo ; @w ¼ 0 at r ¼ 0; @n
0 < x < L þ nwi ðL; tÞ
0 < x < L þ nwi ðL; tÞ
ð34Þ ð35Þ ð36Þ ð37Þ ð38Þ ð39Þ
The rate of heat transfer between the inner and outer fluids is described using the cycle-space-averaged Nusselt number Nuav over a period of pulsation St1 at the wall C of the inner pipe. This value is defined by
Nuav ¼ where
Z
1
s
t þ s t
Z Nuav ;t dt ¼ St
1 t þSt
t
ð40Þ
Nuav ;t dt
s ¼ St1 ; t is a reference time and R
Nuav ;t ¼
@T
C @n dl
ð41Þ
lðCÞ
is the space-averaged Nusselt number at the wall of the inner pipe at time t. The space-averaged pressure drops Pi;t and Po;t within the inner and outer tubes (respectively) at time t are given by Fig. 13. Streamline plots for different values of Re with t ¼ 10, Er ¼ 2 104 , A1 ¼ 0:2 ¼ A2 , St ¼ 1, /1 ¼ /2 ¼ 0:02, ds1 ¼ ds2 ¼ 50 nm.
@u @ v @T @/ ¼ ¼ ¼ 0 at x ¼ L þ nwi ðL; tÞ; ¼ @n @n @n @n @u @ v @T @/ ¼ ¼ ¼ 0 at x ¼ 0; ¼ @n @n @n @n
u¼
u¼
v¼
@nwo ; @t
v¼
0
pðL þ nwi ðL; tÞ; r; tÞdr R þ gwi ðL; tÞ
R Rþgwi ð0;tÞ 0
pð0; r; tÞdr R þ gwi ð0; tÞ
ð42Þ
and
0 < r < R þ gwi ðL; tÞ ð26Þ
R 1þgwo ð0;tÞ Po;t ¼
Rþgwi ð0;tÞ
pð0; r; tÞdr
1 R þ gwo ð0; tÞ gwi ð0; tÞ
R 1þgwo ðL;tÞ
Rþgwi ðL;tÞ
pðL þ nwi ðL; tÞ; r; tÞdr
1 R þ gwo ðL; tÞ gwi ðL; tÞ ð43Þ
R þ gwi ð0; tÞ < r < 1 þ gwo ð0; tÞ ð27Þ
@ gwi @/ 1 @T þ ¼ 0 at r ¼ R þ gwi ; 0 < x < L þ nwi ðL; tÞ ; @n NBT @n @t ð28Þ
@nwi ; @t
R Rþgwi ðL;tÞ Pi;t ¼
@ gwo @/ @T ¼0¼ at r ¼ 1 þ gwo ; 0 < x < L þ nwi ðL; tÞ ; @n @n @t ð29Þ
The total pressure drop P drop within the system is taken as the sum of the cycle-space-averaged pressure drops P drop;i and P drop;o within the inner and outer tubes respectively, where
Z Pdrop;i ¼ St
1 t þSt
t
P i;t dt
ð44Þ
P o;t dt
ð45Þ
and
Z Pdrop;o ¼ St
1 t þSt
t
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Fig. 14. Isotherm plots for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
3. Numerical method The mathematical software FreeFem++ (Hecht 2013) is used in the implementation of the mixed finite element method for the present problem. We solve the highly-coupled system of nonlinear Eqs. (1)–(9) using a sequential approach as follows. The flow, heat transfer and concentration fields are solved using equations on an initial undeformed finite element mesh. The resulting fluid stresses are applied to the elastic walls, the pipe deformations are computed and subsequently, the finite element mesh is updated. The flow, heat transfer and concentration fields are re-computed on the updated finite element mesh. This process is repeated with a relative error tolerance of 104 .
4. Grid-independence study and validation of the numerical algorithm In order to determine a suitable mesh for the present problem, we conducted a grid-independence test with six finite element
meshes consisting of 5363, 6823, 7556, 7843, 8813 and 9473 P2 P1 Taylor-Hood elements. The cycle-space-averaged Nusselt number Nuav and total pressure drop P drop were used to represent the finite element solution. The criterion used for grid independence is that the relative errors between Nuav values on successive meshes and between Pdrop values on successive meshes are less than 0.5%. Using this criterion, we determined that a finite element mesh with 7843 elements ensures accuracy and computational efficiency of the solution (see Table 2). We validate the finite element code used in the present study by numerically reproducing results that are available in the literature. The first such result corresponds to the work of Akdag et al. [11]. Comparisons of isotherm plots and the heat transfer enhancement ratio Nuav =Nus are displayed in Figs. 3 and 4 respectively. From these comparisons, it was determined that the results are in good agreement, with a maximum relative error of less than 3%. The present study was also compared with the work of Targui and Kahalerras [7]. The efficiency ratio for different concentrations of Cu-water nanofluid is shown in Fig. 5. In this case, the maximum relative error is less than 0.5%.
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Fig. 15. Concentration contour plots for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
5. Results and discussion The porous medium used in this study is an open-cell aluminium foam that has thermal conductivity jps ¼ 8 W m1 K1 , density qps ¼ 200 kg m3 , specific heat capacity ðcp Þps ¼ 900 J kg
1
K1 and form drag coefficient cF ¼ 0:06 [25–27]. We
assume the values Pr ¼ 6:2; Da ¼ 104 ; e ¼ 0:95; h1 ¼ 0:03;R ¼ 0:5; Ha ¼ 10; Gr ¼ 10; N BT ¼ 1; Sc ¼ 50; U r ¼ 1; qr ¼ 5; hr ¼ 102 ; m ¼ 0:4 and L ¼ 3. The effects of time t, pulsation amplitudes A1 and A2 , pulsation frequency St, dimensionless elastic modulus Er , Reynolds number Re, solid volume fractions /1 (inner fluid) and /2 (outer fluid), and corresponding nanoparticle diameters ds1 and ds2 on fluid flow, heat transfer and nanoparticle concentration are discussed. In Fig. 5 and 6, the axial velocity u is plotted as a function of time at monitoring points ðx; rÞ ¼ ð0:5; 0:25Þ and ðx; rÞ ¼ ð2:5; 0:75Þ for St ¼ 1 and St ¼ 2. We observe that the velocities of the inner and outer nanofluids gradually increase in magnitude as the fluids respond to the imposed pulsation at the inlet of each channel. This response is delayed due to the time required to
transfer kinetic energy throughout the fluid. After 5–7 pulsation cycles, an equilibrium is established between the inlet pulsation and the flow velocity within the system; hence a steady periodic state is achieved. We also note that increasing the pulsation frequency from St ¼ 1 to St ¼ 2 reduces the time required for the fluid to achieve a steady periodic state. The magnitude of flow velocity during a periodic steady state was found to decrease when St is increased. This indicates a reduction in the influence of inlet pulsation on fluid flow with increased pulsation frequency. Figs. 7–9 show the variation of streamlines during a steady periodic cycle for different amplitudes of pulsation. These streamline plots reveal that the porous blocks within the coaxial pipes cause a deflection of fluid away from the adjoining wall. We note that the streamline distortion is more pronounced in the outer channel than it is in the inner channel. This is due to the staggered configuration of the porous blocks in the outer channel, which creates greater flow disturbances than the inline configuration found in the inner channel. It was also observed that the streamline distortion increases as t is increased from t to t þ s =4, and decreases as t is increased from t þ s =4 to t ¼ t þ 3s =4. The observed increase and decrease in streamline distortion corresponds to an
V.M. Job, S.R. Gunakala / International Journal of Heat and Mass Transfer 113 (2017) 1265–1280
Fig. 16. Streamline plots for different values of /1 and /2 with t ¼ 10, Re ¼ 10, A1 ¼ 0:2 ¼ A2 , St ¼ 1, Er ¼ 2 104 , ds1 ¼ ds2 ¼ 50 nm.
acceleration and deceleration of the fluid within the pulsation cycle. In addition, the flow disturbances over the steady periodic cycle increase with an increase in pulsation amplitudes A1 and A2 . Figs. 10–12 reveal a reduction in the deformation of the coaxial pipes with increased Er . This occurs since as Er is increased, the pipe walls become more resistant to elastic deformation in the presence of stresses exerted by the inner and outer fluids. We observe that the streamlines (Fig. 10) move radially inward as Er is increased. Moreover, the temperature of the system (Fig. 11) decreases with increased Er due to a reduction in the radius of the inner channel. A downward movement of the concentration
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contours (Fig. 12) occurs with increased Er as a result of thermophoretic diffusion and a downward movement of the isotherms within the system. Figs. 13–15 display the influence of the Reynolds number Re on fluid flow, heat transfer and concentration distribution. We observe a decrease in the deformation of the pipe walls as Re increases. This is caused by a reduction in the wall stress exerted by the adjacent fluid with increasing Re. Consequently, the streamlines (Fig. 13) are less distorted as Re is increased. A reduction in temperature (Fig. 14) within the pipes occurs with increased Re due to the decreased deformation of the pipe walls. An increase in Re causes a downward movement of concentration contours within the outer channel through an enhancement in mass transport. An upward movement of concentration contours (Fig. 15) is observed within the inner channel as Re increases from 5 to 10. However, this trend is reversed when Re increases from 10 to 25 due to a reduction in the radius of the inner channel and the effects of thermophoresis. The effects of solid volume fractions /1 (CuO-water nanofluid) and /2 (Cu-water nanofluid) on streamline and isotherm plots are showed in Figs. 16 and 17. We observe that the streamlines within the outer channel are increasingly distorted with increased /2 (cf. Fig. 16 a and b, or c and d) due to an increase in flow resistance through the porous blocks within the outer channel, which results from an increase in nanofluid viscosity. However, there is no significant change in the streamlines as /1 varies (cf. Fig. 16 a and c, or b and d) since the impact of increased viscosity on the flow resistance through the porous blocks within the inner channel is very small. A small increase in temperature within the pipes is observed as /1 is increased (cf. Fig. 17 a and b, or c and d), but the reverse occurs when /2 is increased (cf. Fig. 17 a and c, or b and d). This can be explained by an enhancement in the thermal conductivities of the inner and outer nanofluids as /1 and /2 increase. The enhanced thermal conductivity improves the conduction of heat along each channel, which causes an upward movement of the isotherms when /1 is increased and a downward movement of the isotherms when /2 is increased. Tables 3–8 display the effects of pertinent parameters on the maximum deformations nwi;max and nwo;max of the inner and outer pipes, the maximum and minimum radial deformations gwi;max and gwi;min of the inner pipe, the maximum and minimum radial deformations gwo;max and gwo;min of the outer pipe, the cyclespace-averaged Nusselt number Nuav , the pressure drops Pdrop;i and P drop within the inner and outer channel and the total pressure drop P drop within the system. Table 3 reveals that there is a small increase in the inner and outer pipe wall deformations with increased Strouhal number St, which is caused by an increase in flow disturbance as the pulsating frequency increases. The increased flow disturbance leads to a reduction in the cyclespace-averaged Nusselt number Nuav , which indicates an improvement in heat transfer from the inner nanofluid to the outer nanofluid. However, the increased flow disturbance within the coaxial pipes also leads to an increase in the pressure drop in both the inner and outer channels. We observe that the deformations of the pipe walls increase with an increase in the pulsation amplitudes A1 and A2 (Table 4). The cycle-space-averaged Nusselt number Nuav increases as A1 is increased; however, A2 has the opposite effect on Nuav . This suggests that the heat transfer performance is optimal when A1 is small and A2 is large. Additionally, the pressure drops within the inner and outer channels increase with decreased A1 and increased A2 . As seen in Table 5, a reduction in the elastic modulus Er increases the axial and radial deformations of the coaxial pipes. This increase in pipe deformations causes a reduction in pressure
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Fig. 17. Isotherm plots for different values of /1 and /2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
drop within each pipe as Er decreases. Since the width of the inner channel increases as Er decreases, there is an increase in the quantity of heat entering the system. Consequently, Nuav increases with a reduction in Er . Table 6 reveals that the deformations of the inner and outer pipes and the pressure drops within these pipes are reduced as the Reynolds number Re increases. The cycle-space-averaged Nusselt number Nuav decreases with increased Re as a result of the reduction in the pipe wall deformations. We observe in Table 7 that the deformations of the inner and outer pipes increase when /1 decreases and /2 increases. An increase in /1 and /2 reduces the velocity of the inner and outer fluids as a result of an increase in nanofluid viscosity. Due to the counter-current configuration of the inner and outer fluids, the stresses exerted on the pipe walls decrease when /1 is increased, but increase when /2 is increased. Consequently, the pressure drop within each channel decreases with increased /1 , but increases with increased /2 . The cycle-space-averaged Nusselt number Nuav decreases when /1 is increased, but increases when /2 is increased. This is due to the enhancement in the thermal conductivities of the counter-current nanofluids with increased solid volume fraction.
Table 8 shows that the pressure drops within each channel is independent of nanoparticle diameter. However, Nuav increases with an increase in the diameter ds1 of the CuO nanoparticles within the inner fluid, but decreases with an increase in the diameter ds2 of the Cu nanoparticles within the outer fluid. This occurs as a result of an increase in the thermal conductivity of the inner and outer fluids, as well as the counter-current configuration of these fluids. From these results, we infer that the heat transfer performance of the system may be improved by reducing the size of the CuO nanoparticles within the inner fluid and increasing the size of the Cu nanoparticles within the outer fluid.
6. Conclusions In this paper, we studied the mixed convective pulsatile counter-current flow of CuO-water and Cu-water nanofluids through two coaxial elastic pipes in the presence of an applied radial magnetic field. The effects of time, pulsation frequencies and amplitudes of the inner and outer fluids, elastic modulus, Reynolds number, solid volume /1 and /2 of the inner and outer nanofluids, and the diameters of the Cu and CuO nanoparticles
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V.M. Job, S.R. Gunakala / International Journal of Heat and Mass Transfer 113 (2017) 1265–1280 Table 3 Deformation, Nusselt number and pressure drop values for different values of St with Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. St
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.2 0.5 1 2
0.0694 0.0695 0.0696 0.0696
0.1851 0.1852 0.1855 0.1860
0.1608 0.1609 0.1612 0.1617
0.0780 0.0781 0.0782 0.0782
0.3263 0.3265 0.3268 0.3272
0.1447 0.1448 0.1449 0.1454
0.4850 0.4836 0.4831 0.4824
10.8872 11.2877 12.2464 12.8108
24.8771 27.4616 28.0130 28.9293
35.7643 38.7493 40.2594 41.7401
Table 4 Deformation, Nusselt number and pressure drop values for different values of A1 and A2 with t ¼ 5; Re ¼ 10; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. A1
A2
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.2 0.4 0.6 0.8 0.2 0.2 0.2
0.2 0.2 0.2 0.2 0.4 0.6 0.8
0.0696 0.0696 0.0697 0.0697 0.0705 0.0719 0.0739
0.1855 0.1856 0.1857 0.1860 0.1866 0.1884 0.1911
0.1612 0.1612 0.1613 0.1615 0.1618 0.1628 0.1641
0.0782 0.0782 0.0783 0.0774 0.0790 0.0804 0.0825
0.3268 0.3268 0.3268 0.3269 0.3279 0.3297 0.3322
0.1449 0.1450 0.1450 0.1450 0.1454 0.1460 0.1468
0.4831 0.4839 0.4851 0.4872 0.4777 0.4701 0.4598
12.2464 12.4098 12.5637 13.3882 12.3537 12.4701 12.5998
28.0130 28.0557 28.0995 28.1444 28.5687 29.1524 31.2073
40.2594 40.4655 40.6632 41.5326 40.9224 41.6238 43.8071
Table 5 Deformation, Nusselt number and pressure drop values for different values of Er with t ¼ 5; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm.
gwi;max
nwi;max
Er
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
0.0109
0.0049
0.0951
0.0424
6
1 10
0.0018
0.0071
0.0005
0.0020
1 105
0.0158
0.0597
0.0469
0.0175
5 104
0.0294
0.1025
0.0850
0.0329
0.1691
0.0749
2 104
0.0696
0.1855
0.1612
0.0782
0.3268
0.1449
P drop;i
P drop;o
P drop
0.3963
13.5824
32.6541
46.2365
0.4230
13.2042
31.0090
44.2132
0.4457
12.9266
29.8891
42.8157
0.4831
12.2464
28.0130
40.2594
Table 6 Deformation, Nusselt number and pressure drop values for different values of Re with t ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02, Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. Re
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
5 10 15 25
0.1479 0.0696 0.0505 0.0373
0.2834 0.1855 0.1529 0.1198
0.2344 0.1612 0.1290 0.0932
0.1658 0.0782 0.0562 0.0408
0.4689 0.3268 0.2580 0.1919
0.2234 0.1449 0.1129 0.0833
0.5410 0.4831 0.4618 0.3664
22.3220 12.2464 8.5408 5.6575
52.9754 28.0130 19.6879 13.0530
65.2974 40.2594 28.2287 18.7105
Table 7 Deformation, Nusselt number and pressure drop values for different values of /1 and /2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 ; ds1 ¼ ds2 ¼ 50 nm. /1
/2
nwi;max
gwi;max
gwi;min
nwo;max
gwo;max
gwo;min
Nuav
P drop;i
P drop;o
P drop
0.01 0.02 0.03 0.05 0.02 0.02 0.02
0.02 0.02 0.02 0.02 0.01 0.03 0.05
0.0760 0.0696 0.0674 0.0653 0.0658 0.0769 0.1072
0.1872 0.1855 0.1850 0.1847 0.1837 0.1882 0.1970
0.1613 0.1612 0.1611 0.1606 0.1600 0.1623 0.1641
0.0850 0.0782 0.0759 0.0737 0.0741 0.0860 0.1184
0.3341 0.3268 0.3244 0.3226 0.3210 0.3365 0.3706
0.1453 0.1449 0.1442 0.1437 0.1440 0.1446 0.1455
0.4899 0.4831 0.4823 0.4815 0.4751 0.4947 0.5262
12.3215 12.2464 12.2058 12.1464 12.0494 12.4734 13.0274
29.3404 28.0130 27.5901 27.2587 27.0070 29.7618 36.5284
41.6619 40.2594 39.7959 39.4051 39.0564 42.2352 49.5558
Table 8 Deformation, Nusselt number and pressure drop values for different values of ds1 and ds2 with t ¼ 10; Re ¼ 10; A1 ¼ 0:2 ¼ A2 ; St ¼ 1; /1 ¼ /2 ¼ 0:02; Er ¼ 2 104 . ds1
ds2
Nuav
P drop;i
P drop;o
P drop
50 75 100 150 50 50 50
50 50 50 50 75 100 150
0.4831 0.4832 0.4834 0.4835 0.4787 0.4764 0.4741
12.2464 12.2464 12.2464 12.2464 12.2464 12.2464 12.2464
28.0130 28.0130 28.0130 28.0130 28.0130 28.0130 28.0130
40.2594 40.2594 40.2594 40.2594 40.2594 40.2594 40.2594
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on fluid flow, heat transfer and concentration distribution within the coaxial system were examined. The results show that the velocities of the inner and outer nanofluids increase in magnitude as time progresses and approach a steady periodic state. It was also observed that over a steady periodic cycle, the flow disturbances (streamline distortions) increase as the amplitude of pulsation is increased. We found that the flow disturbances are significantly affected by an increase in the elastic modulus of the pipes. An increase in streamline distortions occurs when the Reynolds number decreases. Moreover, the streamline distortions increase with an increase in the solid volume fraction of nanoparticles within the outer channel. The temperature of the system decreases with an increase in the elastic modulus of the pipes and the Reynolds number. An increase in the solid volume fraction of nanoparticles within the inner channel causes an increase in temperature, but the trend is reversed when the solid volume fraction of nanoparticles within the outer channel increases. The parameters considered in this study influence the concentration distribution within the coaxial pipes through variations in mass transport and thermophoresis. The pipe wall deformations are significantly affected by each parameter except the variation of the diameters of the Cu and CuO nanoparticles. It was discovered that the heat transfer performance within the system may be improved by increasing the pulsation frequency, pulsation amplitude of the outer channel, elastic modulus of the pipes, Reynolds number, solid volume fraction of nanoparticles in the inner fluid and nanoparticle size in the outer fluid. Similarly, we may improve the heat transfer performance within the system by decreasing the pulsation amplitude of the inner pipe, solid volume fraction of nanoparticles in the outer fluid and nanoparticle size in the inner fluid. Moreover, with the exception of the Reynolds number and the size of nanoparticles, the enhancement of heat transfer performance achieved by varying the pertinent parameters is accompanied by an increase in pressure drop within the system.
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