- Email: [email protected]
Effect of rotating cylinder on heat transfer in a square enclosure filled with nanofluids
International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of rotating cylinder on heat transfer in a square enclosure filled with nanofluids R. Roslan a, H. Saleh b, I. Hashim b,⇑ a b
Faculty of Science, Technology & Human Development, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia School of Mathematical Sciences & Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 5 January 2012 Received in revised form 11 July 2012 Accepted 18 July 2012 Available online 11 August 2012 Keywords: Mixed convection Nanofluids Rotating cylinder
a b s t r a c t Convective heat transfer in a differentially heated square enclosure with an inner rotating cylinder is studied theoretically. The free space between the cylinder and the enclosure walls is filled with water– Ag, water–Cu, water–Al2O3 or water–TiO2 nanofluids. The governing equations are formulated for velocity, pressure and temperature formulation and are modeled in COMSOL, a partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). The governing parameters considered are the solid volume fraction, 0.0 6 / 6 0.05, the cylinder radius, 0 6 R 6 0.3 and the angular rotational velocity, 1000 6 X 6 1000. The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the strength of the flow circulation is much stronger for a higher nanoparticle concentration, a better thermal conductivity value and a smaller cylinder with a faster, negative rotation. The maximum heat transfer are obtained at a high nanoparticle concentration with a good conductivity value, a slow positive rotation and a moderate cylinder size located in the center of the enclosure. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The fluid flow and heat transfer around a rotating circular cylinder are considered to be fundamental fluid mechanics problems with a huge number of practical applications such as rotating-tube heat exchangers, rotating shafts, drilling of oil wells, nuclear reactor fuel rods and steel suspension bridge cables. Some theoretical and experimental studies can be found in the literature related to convective heat transfer with a rotating cylinder, but not for configurations similar to the present work. Hayase et al. [1] analyzed the forced convective heat transfer between rotating coaxial cylinders with periodically embedded cavities. They concluded that the transport of momentum and heat is raised by a factor of 1.1 when cavities are embedded in the outer cylinder. Fu et al. [2] numerically studied the natural convection of an enclosure by rotating a small circular cylinder near a hot wall and showed that the rotating cylinder’s direction had an important effect on enhancing the natural convective heat transfer crossing the enclosure. The effect of the rotating cylinder on the heat transfer in a square enclosure filled with a clear fluid or porous medium was numerically investigated by Misirlioglu [3]. He concluded that the rotation is more effective in the forced convection regime than
⇑ Corresponding author. Tel.: +603 8921 5758; fax: +603 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.051
in the mixed and natural convection regimes. Ghazanfarian and Nobari [4] studied a rotating cylinder with cross-flow oscillation and found that the Nusselt number and the drag coefficient decreased rapidly on increasing the rotation. Shih et al. [5] investigated the periodic rotating cylinder and concluded that for high Reynolds number cases, heat transfer was independent of the shape of the object. Convective heat transfer across a circular cylinder rotating with a constant rate varying between 0 and 6 was presented by Paramane and Sharma [6]. They concluded that the average Nusselt number decreased with increasing rotation rate and increased with increasing Reynolds number. Costa and Raimundo [7] considered an active rotating cylinder inside a differentially heated square enclosure. They concluded that, for high values of the cylinder radius, the overall Nusselt number was small if the rotation velocity was low. Hussain and Hussein [8] analyzed a conductive rotating cylinder at different locations and found that the heat transfer rate increases as the Reynolds and Richardson numbers increase. Depending on the angular velocity of the cylinder, the convection phenomenon inside the cavity becomes natural, mixed, and forced. Some works dealing with the stationary cylinder case are reported in [9–14]. Classical studies on the natural convection in an enclosure without a cylinder were reviewed by Ostrach [15]. Natural convective heat transfer in an enclosure filled with water can be enhanced by adding copper, Cu [16– 18], copper oxide, CuO [19], alumina, Al2O3, titania, TiO2 [20]
7248
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Nomenclature Cp g k ‘ Nu p&P Pr r&R
specific heat capacity gravitational acceleration thermal conductivity width and height of cavity average Nusselt number pressure & dimensionless pressure Prandtl number radius of rotating cylinder & dimensionless radius of rotating cylinder Ra Rayleigh number Re Reynolds number Ri Richardson number T temperature u, v velocity components in the x- and y-directions U, V dimensionless velocity components in the X- and Ydirections x, y & X, Y space coordinates & dimensionless space coordinates
and argentum, Ag [21] nano-scale particles. In general, solid nano-scale particles with range of 10–50 nm in a base fluid are known as nanofluids [22]. Differentially-heated moving sidewalls for the nanofluid problem was studied by Tiwari and Das [23]. They concluded that when the Richardson number equal unity, the average Nusselt number increases substantially with the increase in the volume fraction of the nanoparticles. A moving top wall was investigated by Talebi et al. [24]. Their results showed that, for given Reynolds and Rayleigh numbers, an increase in the volume fraction of the nanoparticles enhanced the heat transfer inside the enclosure. To the best of our knowledge, a study on the effect of a rotating cylinder on the heat transfer in an enclosure filled with nanofluids has not yet been undertaken. This is the topic of the present work. The flow fields, temperature distributions and overall heat transfer rate will be presented graphically.
Greek symbols a thermal diffusivity b thermal expansion coefficient m kinematic viscosity / solid volume fraction W stream function H dimensionless temperature x, X angular rotational velocity, dimensionless angular rotational velocity q density l dynamic viscosity subscript 0 c bf h nf sp
qbf @V @V @P U þV ¼ þ Pr bf @X @Y @Y qnf "
!
lnf @ 2 V @ 2 V þ Rabf Prbf þ lbf @X 2 @Y 2 #
qbf qsp bsp 1/þ/ H qnf qbf bbf
U
@H @ H knf ðqCpÞbf þV ¼ @X @Y kbf ðqCpÞnf
@2H @X 2
þ
@2H
ð3Þ
!
@Y 2
ð4Þ
The effective density of the nanofluids, qnf, is given as
qnf ¼ ð1 /Þqbf þ /qsp
ð5Þ
and / is the solid volume fraction of nanoparticles. Thermal diffusivity of the nanofluids is
anf ¼ 2. Mathematical formulation
reference value cold base fluid hot nanofluid solid nanoparticles
knf ðqCpÞnf
ð6Þ
where, the heat capacitance of the nanofluids given is A schematic diagram of a square enclosure with an inner rotating circular cylinder is shown in Fig. 1(a). The left enclosure has constant hot temperature (Th) and the right enclosure has a constant cold temperature (Tc). The top and bottom horizontal straight walls are kept in adiabatic conditions. Under the influence of the vertical gravitational field, the vertical walls at different levels of temperature lead to a natural convection problem. Due to the non-slip boundary condition for velocity on its surface, the rotating cylinder, radius r, induces a forced flow. The overall situation results in a mixed convection problem. The fluid between the enclosure walls and the cylinder is a water-based nanofluids containing Ag, Cu, Al2O3 or TiO2 nanoparticles. Thermo-physical properties of water with Cu, Al2O3, Ag and TiO2 are given in Table 1. The continuity, momentum under the Boussinesq approximation, and energy equations for the Newtonian fluid, laminar and steady state flow can be written in their dimensionless form as follows [7,18]:
@U @V þ ¼0 @X @Y qbf @U @U @P þV ¼ þ Prbf U @X @Y @X qnf
ð1Þ
lnf @ 2 U @ 2 U þ lbf @X 2 @Y 2
! ð2Þ
ðqCpÞnf ¼ ð1 /ÞðqCpÞbf þ /ðqCpÞsp
ð7Þ
The thermal expansion coefficient of the nanofluids can be determined by
bnf ¼ ð1 /Þbbf þ /bsp
ð8Þ
The ratio of dynamic viscosity of the nanofluids given by Brinkman [26] is
lnf 1 ¼ lbf ð1 /Þ2:5
ð9Þ
The ratio of thermal conductivity of nanofluids restricted to spherical nanoparticles is approximated by the Maxwell–Garnetss (MG) [21] model (see Fig. 2):
knf ksp þ 2kbf 2/ðkbf ksp Þ ¼ kbf ksp þ 2kbf þ /ðkbf ksp Þ
ð10Þ
The nondimensional Navier–Stokes (1)–(3) and energy Eq. (4) are nondimensionalized using the following dimensionless quantities:
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7249
(a)
Fig. 2. The ratio of thermal conductivity of water–Ag, water–Cu, water–Al2O3 and water–TiO2 using Maxwell–Garnetts (MG) model and the ratio dynamical viscosity using Brinkman model versus volume fraction.
Over the rotating cylinder, velocity components are specified as
(b)
u ¼ xðy y0 Þ
ð16Þ
v ¼ xðx x0 Þ
ð17Þ
or, in the dimensionless form can be written as
U ¼ XðY Y 0 Þ
ð18Þ
V ¼ XðX X 0 Þ
ð19Þ
Over the surface of the cylinder, the absolute value of velocity can be evaluated as [7]
jVj ¼ jXjR
ð20Þ
The fluid motion is displayed using the stream function W obtained from velocity components U and V. The relationships between the stream function and the velocity components are: U = @ W/@Y and V = @ W/@X, which yield a single equation,
@2W @X Fig. 1. (a) Schematic representation of the model and (b) mesh distribution.
Table 1 Thermo-physical properties of water with Cu [16], Al2O3 [25], Ag and TiO2 [21]. Physical properties
Water
Ag
Cu
Al2O3
TiO2
Cp (J/kg K) q (kg/m3) k (W m1 K1) b 105 (1/K)
4179 997.1 0.6 21
235 10,500 429 5.4
383 8954 400 1.67
765 3600 46 0.63
686.2 4250 8.954 2.4
x X¼ ; ‘ Prbf ¼
y Y¼ ; ‘
U¼
mbf ; Rabf ¼ abf
u‘
abf
;
V¼
v‘ abf
gbbf ðT h T c Þ‘3
mbf abf
;
;
H¼ P¼
T Tc ; Th Tc
p‘
qnf a
2
2 bf
;
X¼
x‘ abf
ð11Þ
U = V = 0 on the walls and the boundary conditions for the nondimensional temperatures are:
H ¼ 1 at X ¼ 0 H ¼ 0 at X ¼ 1 @H
¼ 0 at Y ¼ 0 and Y ¼ 1 @Y @H ¼ 0 at the cylinder surface @g
þ
@2W @Y 2
¼
@U @V @Y @X
ð21Þ
where W = 0 at all walls of the enclosure and the value of W at the surface of the cylinder can be evaluated from (20). The typical dimensionless parameter used to evaluate the relative importance of the natural and forced convection is a modified form of the Richardson number, defined elsewhere as Ribf ¼ ðRabf =Pr bf Þ=Re2bf , where Re is the Reynolds number. For the present problem and dimensionless strategy, this parameter becomes [7]
Ribf ¼
Rabf Prbf 4X2 R4
ð22Þ
for X – 0 and R – 0. The physical quantities of interest in this problem are the average Nusselt number, representation of overall heat transfer performance crossing the enclosure that defined by:
r R¼ ; ‘
2
2
ð12Þ ð13Þ ð14Þ ð15Þ
Nu ¼
Z 0
1
knf @ H dY kbf @X
ð23Þ
3. Computational methodology The governing equations along with the boundary condition are modeled and solved numerically using COMSOL. COMSOL is a general-purpose interlinked partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). This program contains state-of-the-art numerical algorithms and visualization tools bundled with an easy to use interface. We consider the following application modes in COMSOL. We use in COMSOL the
7250
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
incompressible, laminar flow (spf) for Eqs. (1)–(3) and the heat transfer in fluids (ht) for Eq. (4). P2-P1 Lagrange elements and the Galerkin least-square method are used to assure stability. The discretizations of the governing PDE from Eqs. (1)–(4) using the GFEM scheme results in a set of nonlinear equations which are solved using the damped Newton method. With this method, the Jacobian is evaluated only during the first iteration. Consequently, the Jacobian is factorized only once while for all subsequent iterations, the same Jacobian and hence its LU factors are used repeatedly. Since factorization is the most expensive part of the computations, by using the damped Newton algorithm, the expensive factorization step can be skipped after the first iteration. The core of the resulting nonlinear equations is the solution of a sparse linear system, which is the most computationally intensive part of the solver both in terms of CPU time and memory requirement. A parallel direct solver (PARDISO) is then implemented. Details of the implementation of the Newton method and PARDISO to solve the Navier–Stokes equations were given by Raju and Khaitan [27]. In this study, a triangular mesh, shown in Fig. 1(b), is generated on a square enclosure with a central cylinder. Several grid sensitivity tests were conducted to determine whether the mesh scheme is sufficient and to ensure that the results are grid independent. We use the COMSOL default settings for predefined mesh sizes, i.e. extremely coarse, extra coarse, coarser, coarse, normal, fine, finer, extra fine and extremely fine. In the tests, we consider the parameters / = 0.05, R = 0.25 and X = 500 as shown in Table 2. The relative change of the average Nusselt number Nu is defined as the ratio of the Nu difference at each evaluation point with the current mesh condition to the Nu at the same evaluation point with the previous mesh condition. Therefore, the relative change for the first mesh condition is not defined. We can see that the relative change tends to decrease when refining the mesh. The simulation of the model with extremely fine mesh has the best accuracy, but considering the CPU time, a finer mesh size was selected for all the computations done in this paper. To validate the computational code, the previously published problems for natural convection in a differentially heated enclosure filled with nanofluids were solved. Table 3 shows the comparison of the average Nusselt number between the present result and the available results found in the literature for various Rayleigh numbers and concentration levels. The comparison was in good agreement with the results reported in the literature. An additional verification of accuracy for the present code is shown in Table 4, which gives a comparison of the minimum values of the stream function between the present and the literature results for convective heat transfer in a differentially heated square enclosure filled with air [28]. Next, an active rotating cylinder was placed in the center of the enclosure [7]. We see that the deviations vary from 0.2% at X = 500, R = 0.2–5.7% at X = 500, R = 0.4. We conclude that the present computations are acceptable for the considered R range.
Table 2 Grid sensitivity check at / = 0.05, R = 0.25 and X = 500. Predefined mesh size
Mesh elements
Nu
Relative change (%)
CPU time (s)
Extremely coarse Extra coarse Coarser Coarse Normal Fine Finer Extra fine Extremely fine
286 550 760 1276 1938 2734 9132 25,684 33,720
4.0807 4.1253 4.1484 4.1660 4.1722 4.1779 4.1856 4.1897 4.1906
– 1.08 0.56 0.42 0.15 0.14 0.18 0.10 0.02
3 4 5 5 6 8 17 47 54
Table 3 Comparison of the average Nusselt number between the present and the [16] result for a different Rabf and / at Prbf = 6.2 and R = 0. /
Rabf = 104, Cu
Rabf = 105, Cu
Rabf = 104, Al2O3
Rabf = 105, Al2O3
Present
[16]
Present
[16]
Present
[25]
Present
[25]
0 0.01 0.02 0.03 0.04 0.05
2.272 2.304 2.326 2.357 2.388 2.409
2.299 2.335 2.375 2.417 2.459 2.503
4.716 4.771 4.825 4.877 4.949 5.079
4.720 4.793 4.875 4.960 5.048 5.137
2.272 2.299 2.311 2.337 2.361 2.399
2.297 2.335 2.372 2.410 2.448 2.487
4.716 4.757 4.796 4.834 4.871 4.936
4.552 4.626 4.701 4.776 4.851 4.927
Table 4 Comparison of the minimum values of the stream function between the present and the literatures result for a different R and X at Pr = 0.71 and / = 0.0. Literatures R = 0
R = 0.2
R = 0.4
X = 500 X = 0 Present [7] [28]
9.64 19.61 – 19.65 9.61 –
X = 500 X = 500 X = 0
8.99 9.05 8.95 9.11 – –
13.77 13.80 –
X = 500
3.25 1.92 3.21 1.95 – –
4. Results and discussion The analyses in the numerical investigation are performed in the following ranges of the associated dimensionless groups: the solid volume fraction, 0.0 6 / 6 0.05, the radius of the rotating cylinder, 0 6 R 6 0.3 and the angular rotational velocity, 1000 6 X 6 1000. The Prandtl number and the Rayleigh number are fixed at Prbf = 6.2 and Rabf = 105, respectively. Following theses dimensionless parameters, the Richardson number interval is, 19.1358 < Ribf < 1. Fig. 3(a)–(c) shows the effects of Cu nanoparticle concentration, /, on the flow and thermal fields in the square enclosure with constant values of R = 0.2 and X = 100. Fluid flow takes place in the free space between the cylinder and the enclosure. The flow rotates in the clockwise direction due to the natural convection effects. It indicates that the fluid filling the enclosure is moving up along both the left heated wall and the top insulated wall, down along the cooled right wall, and horizontally to the left along the insulated base enclosure. The cylinder rotates in the counter-clockwise direction in the center of the enclosure. This motion creates some recirculating cells between the natural convective flow and the cylinder. The size of the recirculating cells can be widened by increasing the nanoparticle concentration. As can be seen in Fig. 3, the strength of the flow circulation and the average Nusselt number also increase with increasing nanoparticle concentration. This is due to the fact that a high concentration of solid nanoparticles yields the higher thermal conductivity that eventually leads to higher energy, which accelerates the flow. Actually, this phenomenon is also accompanied by an undesirable effect promoted by the viscosity that suppresses the flow, but is small compared to the favorable effect driven by the presence of the high thermal conductivity. Fig. 4 shows the effects of nanoparticle type, on the flow and temperature structures in the enclosure with constant values of / = 0.05, R = 0.2 and X = 100. Streamlines and isotherms show essentially the same behavior as for the previous case, except that the recirculating cells merge with the main cells as presented in Fig. 4(a) for the Ag nanoparticles. This is due to the convection velocity significantly increasing with the added high concentration of Ag nanoparticles. The recirculating cells appear again when the
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7251
Fig. 3. Streamlines (left), isotherms (right) at R = 0.2, X = 100 and water–Cu.
enclosure filled with water–Al2O3 or water–TiO2 as displayed in Fig. 4(b) and (c), respectively. Water–Ag has a faster flow circulation and a better heat transfer performance than the other nanofluid types. This is related to the variation in the thermal conductivity values of the nanoparticle types as shown in Table 1.
Fig. 5 illustrates the effects of the cylinder radius, R, for water– Cu nanofluids with volume fraction 5% and X = 100 on the flow and thermal fields in the enclosure. As can be seen in Fig. 5, R affects the fluid temperatures and the flow characteristics. The strength of the flow circulation is much lower for a larger cylinder. This is because
7252
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Fig. 4. Streamlines (left), isotherms (right) at / = 0.05, R = 0.2 and X = 100.
the radius of the cylinder is large and the space for the fluid flow between the cylinder and the enclosure walls is narrow, restraining the natural convective flow. The average temperature gradient at the hot wall increases with increasing R, but increasing R further reduces the Nu. This is due to the cylinder pushing the fluid in the
center toward the walls which leads to the formation of a thinner thermal boundary layer at the heated and cooled walls. However, further increasing the cylinder radius restrains the convective flow. The effect of varying the rotational velocity of the cylinder on the fluid pattern in the enclosure is presented in Fig. 6 at R = 0.2,
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7253
Fig. 5. Streamlines (left), isotherms (right) at X = 100, / = 0.05 for water–Cu.
/ = 0.05 for water–Cu in the form of streamlines and isotherms. Fig. 6(a) demonstrates that the streamline shape is mainly conditioned by the enclosure walls, and is denser close to the cylinder, thus indicating an intense flow there. In the case X = 500, the clockwise rotation of the cylinder induces an overall convective
flow in the clockwise direction. The case X = 0 corresponds to a motionless cylinder which was presented in Fig. 6(b). In this case, the heat transfer is due to natural convection only. A kind of symmetry was observed in the temperature field which is typical of a buoyancy-driven heat transfer problem in a square enclosure.
7254
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Fig. 6. Streamlines (left), isotherms (right) at R = 0.2, / = 0.05 for water–Cu.
The isotherms are essentially near the hot and cold walls and are horizontal at the center of the enclosure since the stationary adiabatic cylinder does not interact with the heat transferred by the fluid from the hot to the cold walls. Fig. 6(c) shows the angular velocities in the positive direction, X = 500. The main flow in the
clockwise direction develops naturally within the enclosure due to density differences in the fluid by temperature gradients. The fluid movement close to the cylinder surface, promoted by the anti-clockwise rotation of the cylinder, is however in anticlockwise direction. Some recirculating cells appear in the flow
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7255
(a)
Fig. 7. Variation of Nu with X for different R and fixed / = 0.05.
(b)
Fig. 9. Variation of Nu with R for different (a) types of solid nanoparticles and (b) cylinder locations. Fig. 8. Variation of Nu with R for different / of Cu nanoparticles.
between them. As can be seen in Fig. 6, the strength of the flow circulation is much higher for the negative rotation because the forced and free convection flows combined. Variations of the average Nusselt number with the rotational velocities are presented in Fig. 7 with different values of R for water–Cu nanofluids with / = 0.05. For rotational velocities close to zero (almost motionless cylinder), the maximum overall Nusselt number occurs at R between R = 0.1 and R = 0.2. This is because there should be an optimum cylinder radius R (or an optimum space between the cylinder and the walls) for a particular nanofluid to be effective in transferring heat. Fig. 7 shows that the average Nusselt number varies slightly, with a maximum of about 2% for the motionless cylinder for any cylinder radius considered here. Since the cylinder body is insulated and kept silent, this would have a small impact on the heat transfer performance. Further, the average Nusselt number does not change significantly when varying the angular velocities of the negative or positive rotation for R 6 0.1. This phenomenon occurs since the forced convection effect, which either aids (negative rotation) or opposes (positive rotation) the natural buoyancy at the center enclosure, is very small. The fastest negative rotation, X = 1000 gives a remarkable increase of the Nu for the biggest cylinder studied in this work. This is because the cylinder velocities for negative rotation accelerates the main flow which leads to a better heat transfer performance when crossing the enclosure. The minimum Nu was observed at X = 600 for the biggest cylinder, R = 0.3. In this case, the cylinder rotation maximally blocks the natural buoyancy force which reduces the overall heat transfer performance substantially. The average Nusselt number is plotted against the cylinder radius in Fig. 8 for different / of Cu nanoparticles at X = 500. This figure shows that the Nu increases with increasing / for any R value
considered here. This is due to the higher conductivity of the fluids when increasing / which accelerates the convective flow on the free space between the cylinder and the enclosure. Heat transfer performance at the hot wall increases with increasing R up to a certain maximum value of R. However, the heat transfer performance decreases when the cylinder radius is larger than the maximum value. Fig. 8 also shows that the maximum average Nusselt number is at R = 0.18, R = 0.19, R = 0.195 and R = 0.2 for / = 0.05, / = 0.03, / = 0.01 and / = 0.0, respectively. This suggests that higher nanoparticle concentration needs more extra space between the cylinder and the wall which is achievable by decreasing the cylinder radius for accelerating the flow and hence enhancing heat transfer. Fig. 9 depicts the variation of the average Nusselt number, Nu, versus the solid volume fraction of the argentum (Ag), copper (Cu), alumina (Al2O3) and titania (TiO2) for different cylinder locations at R = 0.2 and X = 50. The results in Fig. 9(a) show the dependence of heat transfer rate on the different types of nanoparticles. The lowest heat transfer was obtained for TiO2 since it has a lower thermal conductivity (see Fig. 2) which leads to a lower temperature gradient and smaller enhancements in the heat transfer rate. It can be seen in Fig. 2 that increasing / will increase the ratio of thermal conductivity of the various types of nanofluids monotonically that eventually leads to a monotone increase in the heat transfer rate as shown in Fig. 9(a). Fig. 2 also showed that Ag and Cu have the same thermal conductivity property but give different results for the heat transfer performance. This means that the heat transfer rate does not only depend on the thermal conductivity value as concluded by Saleh et al. [29]. Fig. 9(b) shows that the Nu is almost constant when increasing the Cu concentrations with the cylinders located at the lower or upper portion of the enclosure. It is also observed the Nu has the lowest value when the cylinder is located in the upper portion of the enclosure at any Cu concentration. Since the stronger convective flow is confined near the
7256
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
upper half, a stagnated region of a cold heavy fluid is formed in the bottom half which yields a poor heat transfer rate. 5. Conclusions The present numerical simulations studied the effect of a rotating cylinder on heat transfer in a square enclosure filled with nanofluids. The dimensionless forms of the governing equations are modeled and solved using the COMSOL program. Detailed computational results for flow and temperature fields, and the overall heat transfer performance crossing the enclosure have been presented graphically. The results show that the thermal and fluid dynamics are highly influenced by the size of the cylinder, rotational speed and direction, nanoparticle concentration and its thermal conductivity values. The most important results of the present analysis are: 1. The recirculating cells emerge between the natural convective flow and the positive cylinder rotation. The size of the recirculating cells widen with increasing nanoparticle concentration. 2. The strength of the flow circulation is much stronger for a higher nanoparticle concentration with better thermal conductivity and a smaller cylinder with a faster, negative rotation. 3. The heat transfer rate increases monotonically by increasing the nanoparticle concentration for the fixed size cylinder located in the center with constant rotation. The maximum heat transfer performance for the cylinder with moderate radius located in the center is obtained at slow positive rotation, high nanoparticle concentration and good thermal conductivity values. The theoretical prediction of this paper is expected to be a useful guide for experimentalists studying nanofluids and rotating cylinders. The factors solid nanoparticle size, nanolayer, shape, temperature, pH of nanofluid solution, presence of surfactants, nature of the nanoparticle component and cylinder rotation with a variable rate will be the focus of our future research. Acknowledgments The authors would like to acknowledge the financial support received from the Grants FRGS/1/2011/SG/UKM/01/13 and FRGS/1/ 2009/SG/UTHM/02/4/0734. References [1] T. Hayase, J.A.C. Humphrey, R. Greif, Numerical calculation of convective heat transfer between rotating coaxial cylinders with periodically embedded cavities, J. Heat Transfer 114 (1992) 589–597. [2] W. Fu, C. Cheng, W. Shieh, Enhancement of natural convection heat transfer of an enclosure by a rotating circular cylinder, Int. J. Heat Mass Transfer 37 (1994) 1885–1897. [3] A. Misirlioglu, The effect of rotating cylinder on the heat transfer in a square cavity filled with porous medium, Int. J. Eng. Sci. 44 (2006) 1173–1187. [4] J. Ghazanfarian, M. Nobari, A numerical study of convective heat transfer from a rotating cylinder with cross-flow oscillation, Int. J. Heat Mass Transfer 52 (2009) 5402–5411.
[5] Y. Shih, J. Khodadadi, K. Weng, A. Ahmed, Periodic fluid flow and heat transfer in a square cavity due to an insulated or isothermal rotating cylinder, J. Heat Transfer 131 (2009) 1–11. [6] S. Paramane, A. Sharma, Numerical investigation of heat and fluid flow across a rotating circular cylinder maintained at constant temperature in 2-D laminar flow regime, Int. J. Heat Mass Transfer 52 (2009) 3205–3216. [7] V. Costa, A. Raimundo, Steady mixed convection in a differentially heated square enclosure with an active rotating circular cylinder, Int. J. Heat Mass Transfer 53 (2010) 1208–1219. [8] S.H. Hussain, A.K. Hussein, Mixed convection heat transfer in a differentially heated square enclosure with a conductive rotating circular cylinder at different vertical locations, Int. Commun. Heat Mass Transfer 38 (2011) 263– 274. [9] N.K. Ghaddar, Natural convection heat transfer between a uniformly heated cylindrical element and its rectangular enclosure, Int. J. Heat Mass Transfer 35 (1992) 2327–2334. [10] G. Cesini, M. Paroncini, G. Cortella, M. Manzan, Natural convection from a horizontal cylinder in a rectangular cavity, Int. J. Heat Mass Transfer 42 (1999) 1801–1811. [11] D. Roychowdhury, S. Das, T. Sundararajan, Numerical simulation of natural convective heat transfer and fluid flow around a heated cylinder inside an enclosure, Heat Mass Transfer 38 (2002) 565–576. [12] B.S. Kim, D.S. Lee, M.Y. Ha, H.S. Yoon, A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transfer 51 (2008) 1888–1906. [13] S.H. Hussain, A.K. Hussein, Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations, Int. Commun. Heat Mass Transfer 37 (2010) 1115–1126. [14] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass Transfer 53 (2010) 5905–5919. [15] S. Ostrach, Natural convection in enclosures, J. Heat Transfer 110 (1988) 1175– 1190. [16] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy–driven heat transfer enhancement in a two–dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [17] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Commun. Heat Mass Transfer 33 (2006) 727–736. [18] M.K. Das, P.S. Ohal, Natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids, Int. J. Numer. Methods Heat Fluid Flow 19 (2009) 411–431. [19] B. Ghasemi, S.M. Aminossadati, Natural convection heat transfer in an inclined enclosure filled with a water–CuO nanofluid, Numer. Heat Transfer Part A 55 (2009) 807–823. [20] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [21] E.B. Ögüt, Natural convection of water-based nanofluids in an inclined enclosure with a heat source, Int. J. Thermal Sci. 48 (2009) 2063–2073. [22] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–105. [23] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [24] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, Int. Commun. Heat Mass Transfer 37 (2010) 79–90. [25] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) 4506–4516. [26] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [27] M.P. Raju, S.K. Khaitan, High performance computing of three-dimensional finite element codes on a 64-bit machine, J. Appl. Fluid Mech. 5 (2012) 123– 132. [28] G. De Vahl Davis, I. P Jones, Natural convection in a square cavity: a bench mark numerical solution, Int. J. Numer. Methods Fluids 3 (1983) 227–248. [29] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluidfilled trapezoidal enclosure, Int. J. Heat Mass Transfer 54 (2011) 194–201.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of rotating cylinder on heat transfer in a square enclosure filled with nanofluids R. Roslan a, H. Saleh b, I. Hashim b,⇑ a b
Faculty of Science, Technology & Human Development, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia School of Mathematical Sciences & Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 5 January 2012 Received in revised form 11 July 2012 Accepted 18 July 2012 Available online 11 August 2012 Keywords: Mixed convection Nanofluids Rotating cylinder
a b s t r a c t Convective heat transfer in a differentially heated square enclosure with an inner rotating cylinder is studied theoretically. The free space between the cylinder and the enclosure walls is filled with water– Ag, water–Cu, water–Al2O3 or water–TiO2 nanofluids. The governing equations are formulated for velocity, pressure and temperature formulation and are modeled in COMSOL, a partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). The governing parameters considered are the solid volume fraction, 0.0 6 / 6 0.05, the cylinder radius, 0 6 R 6 0.3 and the angular rotational velocity, 1000 6 X 6 1000. The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the strength of the flow circulation is much stronger for a higher nanoparticle concentration, a better thermal conductivity value and a smaller cylinder with a faster, negative rotation. The maximum heat transfer are obtained at a high nanoparticle concentration with a good conductivity value, a slow positive rotation and a moderate cylinder size located in the center of the enclosure. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The fluid flow and heat transfer around a rotating circular cylinder are considered to be fundamental fluid mechanics problems with a huge number of practical applications such as rotating-tube heat exchangers, rotating shafts, drilling of oil wells, nuclear reactor fuel rods and steel suspension bridge cables. Some theoretical and experimental studies can be found in the literature related to convective heat transfer with a rotating cylinder, but not for configurations similar to the present work. Hayase et al. [1] analyzed the forced convective heat transfer between rotating coaxial cylinders with periodically embedded cavities. They concluded that the transport of momentum and heat is raised by a factor of 1.1 when cavities are embedded in the outer cylinder. Fu et al. [2] numerically studied the natural convection of an enclosure by rotating a small circular cylinder near a hot wall and showed that the rotating cylinder’s direction had an important effect on enhancing the natural convective heat transfer crossing the enclosure. The effect of the rotating cylinder on the heat transfer in a square enclosure filled with a clear fluid or porous medium was numerically investigated by Misirlioglu [3]. He concluded that the rotation is more effective in the forced convection regime than
⇑ Corresponding author. Tel.: +603 8921 5758; fax: +603 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.051
in the mixed and natural convection regimes. Ghazanfarian and Nobari [4] studied a rotating cylinder with cross-flow oscillation and found that the Nusselt number and the drag coefficient decreased rapidly on increasing the rotation. Shih et al. [5] investigated the periodic rotating cylinder and concluded that for high Reynolds number cases, heat transfer was independent of the shape of the object. Convective heat transfer across a circular cylinder rotating with a constant rate varying between 0 and 6 was presented by Paramane and Sharma [6]. They concluded that the average Nusselt number decreased with increasing rotation rate and increased with increasing Reynolds number. Costa and Raimundo [7] considered an active rotating cylinder inside a differentially heated square enclosure. They concluded that, for high values of the cylinder radius, the overall Nusselt number was small if the rotation velocity was low. Hussain and Hussein [8] analyzed a conductive rotating cylinder at different locations and found that the heat transfer rate increases as the Reynolds and Richardson numbers increase. Depending on the angular velocity of the cylinder, the convection phenomenon inside the cavity becomes natural, mixed, and forced. Some works dealing with the stationary cylinder case are reported in [9–14]. Classical studies on the natural convection in an enclosure without a cylinder were reviewed by Ostrach [15]. Natural convective heat transfer in an enclosure filled with water can be enhanced by adding copper, Cu [16– 18], copper oxide, CuO [19], alumina, Al2O3, titania, TiO2 [20]
7248
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Nomenclature Cp g k ‘ Nu p&P Pr r&R
specific heat capacity gravitational acceleration thermal conductivity width and height of cavity average Nusselt number pressure & dimensionless pressure Prandtl number radius of rotating cylinder & dimensionless radius of rotating cylinder Ra Rayleigh number Re Reynolds number Ri Richardson number T temperature u, v velocity components in the x- and y-directions U, V dimensionless velocity components in the X- and Ydirections x, y & X, Y space coordinates & dimensionless space coordinates
and argentum, Ag [21] nano-scale particles. In general, solid nano-scale particles with range of 10–50 nm in a base fluid are known as nanofluids [22]. Differentially-heated moving sidewalls for the nanofluid problem was studied by Tiwari and Das [23]. They concluded that when the Richardson number equal unity, the average Nusselt number increases substantially with the increase in the volume fraction of the nanoparticles. A moving top wall was investigated by Talebi et al. [24]. Their results showed that, for given Reynolds and Rayleigh numbers, an increase in the volume fraction of the nanoparticles enhanced the heat transfer inside the enclosure. To the best of our knowledge, a study on the effect of a rotating cylinder on the heat transfer in an enclosure filled with nanofluids has not yet been undertaken. This is the topic of the present work. The flow fields, temperature distributions and overall heat transfer rate will be presented graphically.
Greek symbols a thermal diffusivity b thermal expansion coefficient m kinematic viscosity / solid volume fraction W stream function H dimensionless temperature x, X angular rotational velocity, dimensionless angular rotational velocity q density l dynamic viscosity subscript 0 c bf h nf sp
qbf @V @V @P U þV ¼ þ Pr bf @X @Y @Y qnf "
!
lnf @ 2 V @ 2 V þ Rabf Prbf þ lbf @X 2 @Y 2 #
qbf qsp bsp 1/þ/ H qnf qbf bbf
U
@H @ H knf ðqCpÞbf þV ¼ @X @Y kbf ðqCpÞnf
@2H @X 2
þ
@2H
ð3Þ
!
@Y 2
ð4Þ
The effective density of the nanofluids, qnf, is given as
qnf ¼ ð1 /Þqbf þ /qsp
ð5Þ
and / is the solid volume fraction of nanoparticles. Thermal diffusivity of the nanofluids is
anf ¼ 2. Mathematical formulation
reference value cold base fluid hot nanofluid solid nanoparticles
knf ðqCpÞnf
ð6Þ
where, the heat capacitance of the nanofluids given is A schematic diagram of a square enclosure with an inner rotating circular cylinder is shown in Fig. 1(a). The left enclosure has constant hot temperature (Th) and the right enclosure has a constant cold temperature (Tc). The top and bottom horizontal straight walls are kept in adiabatic conditions. Under the influence of the vertical gravitational field, the vertical walls at different levels of temperature lead to a natural convection problem. Due to the non-slip boundary condition for velocity on its surface, the rotating cylinder, radius r, induces a forced flow. The overall situation results in a mixed convection problem. The fluid between the enclosure walls and the cylinder is a water-based nanofluids containing Ag, Cu, Al2O3 or TiO2 nanoparticles. Thermo-physical properties of water with Cu, Al2O3, Ag and TiO2 are given in Table 1. The continuity, momentum under the Boussinesq approximation, and energy equations for the Newtonian fluid, laminar and steady state flow can be written in their dimensionless form as follows [7,18]:
@U @V þ ¼0 @X @Y qbf @U @U @P þV ¼ þ Prbf U @X @Y @X qnf
ð1Þ
lnf @ 2 U @ 2 U þ lbf @X 2 @Y 2
! ð2Þ
ðqCpÞnf ¼ ð1 /ÞðqCpÞbf þ /ðqCpÞsp
ð7Þ
The thermal expansion coefficient of the nanofluids can be determined by
bnf ¼ ð1 /Þbbf þ /bsp
ð8Þ
The ratio of dynamic viscosity of the nanofluids given by Brinkman [26] is
lnf 1 ¼ lbf ð1 /Þ2:5
ð9Þ
The ratio of thermal conductivity of nanofluids restricted to spherical nanoparticles is approximated by the Maxwell–Garnetss (MG) [21] model (see Fig. 2):
knf ksp þ 2kbf 2/ðkbf ksp Þ ¼ kbf ksp þ 2kbf þ /ðkbf ksp Þ
ð10Þ
The nondimensional Navier–Stokes (1)–(3) and energy Eq. (4) are nondimensionalized using the following dimensionless quantities:
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7249
(a)
Fig. 2. The ratio of thermal conductivity of water–Ag, water–Cu, water–Al2O3 and water–TiO2 using Maxwell–Garnetts (MG) model and the ratio dynamical viscosity using Brinkman model versus volume fraction.
Over the rotating cylinder, velocity components are specified as
(b)
u ¼ xðy y0 Þ
ð16Þ
v ¼ xðx x0 Þ
ð17Þ
or, in the dimensionless form can be written as
U ¼ XðY Y 0 Þ
ð18Þ
V ¼ XðX X 0 Þ
ð19Þ
Over the surface of the cylinder, the absolute value of velocity can be evaluated as [7]
jVj ¼ jXjR
ð20Þ
The fluid motion is displayed using the stream function W obtained from velocity components U and V. The relationships between the stream function and the velocity components are: U = @ W/@Y and V = @ W/@X, which yield a single equation,
@2W @X Fig. 1. (a) Schematic representation of the model and (b) mesh distribution.
Table 1 Thermo-physical properties of water with Cu [16], Al2O3 [25], Ag and TiO2 [21]. Physical properties
Water
Ag
Cu
Al2O3
TiO2
Cp (J/kg K) q (kg/m3) k (W m1 K1) b 105 (1/K)
4179 997.1 0.6 21
235 10,500 429 5.4
383 8954 400 1.67
765 3600 46 0.63
686.2 4250 8.954 2.4
x X¼ ; ‘ Prbf ¼
y Y¼ ; ‘
U¼
mbf ; Rabf ¼ abf
u‘
abf
;
V¼
v‘ abf
gbbf ðT h T c Þ‘3
mbf abf
;
;
H¼ P¼
T Tc ; Th Tc
p‘
qnf a
2
2 bf
;
X¼
x‘ abf
ð11Þ
U = V = 0 on the walls and the boundary conditions for the nondimensional temperatures are:
H ¼ 1 at X ¼ 0 H ¼ 0 at X ¼ 1 @H
¼ 0 at Y ¼ 0 and Y ¼ 1 @Y @H ¼ 0 at the cylinder surface @g
þ
@2W @Y 2
¼
@U @V @Y @X
ð21Þ
where W = 0 at all walls of the enclosure and the value of W at the surface of the cylinder can be evaluated from (20). The typical dimensionless parameter used to evaluate the relative importance of the natural and forced convection is a modified form of the Richardson number, defined elsewhere as Ribf ¼ ðRabf =Pr bf Þ=Re2bf , where Re is the Reynolds number. For the present problem and dimensionless strategy, this parameter becomes [7]
Ribf ¼
Rabf Prbf 4X2 R4
ð22Þ
for X – 0 and R – 0. The physical quantities of interest in this problem are the average Nusselt number, representation of overall heat transfer performance crossing the enclosure that defined by:
r R¼ ; ‘
2
2
ð12Þ ð13Þ ð14Þ ð15Þ
Nu ¼
Z 0
1
knf @ H dY kbf @X
ð23Þ
3. Computational methodology The governing equations along with the boundary condition are modeled and solved numerically using COMSOL. COMSOL is a general-purpose interlinked partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). This program contains state-of-the-art numerical algorithms and visualization tools bundled with an easy to use interface. We consider the following application modes in COMSOL. We use in COMSOL the
7250
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
incompressible, laminar flow (spf) for Eqs. (1)–(3) and the heat transfer in fluids (ht) for Eq. (4). P2-P1 Lagrange elements and the Galerkin least-square method are used to assure stability. The discretizations of the governing PDE from Eqs. (1)–(4) using the GFEM scheme results in a set of nonlinear equations which are solved using the damped Newton method. With this method, the Jacobian is evaluated only during the first iteration. Consequently, the Jacobian is factorized only once while for all subsequent iterations, the same Jacobian and hence its LU factors are used repeatedly. Since factorization is the most expensive part of the computations, by using the damped Newton algorithm, the expensive factorization step can be skipped after the first iteration. The core of the resulting nonlinear equations is the solution of a sparse linear system, which is the most computationally intensive part of the solver both in terms of CPU time and memory requirement. A parallel direct solver (PARDISO) is then implemented. Details of the implementation of the Newton method and PARDISO to solve the Navier–Stokes equations were given by Raju and Khaitan [27]. In this study, a triangular mesh, shown in Fig. 1(b), is generated on a square enclosure with a central cylinder. Several grid sensitivity tests were conducted to determine whether the mesh scheme is sufficient and to ensure that the results are grid independent. We use the COMSOL default settings for predefined mesh sizes, i.e. extremely coarse, extra coarse, coarser, coarse, normal, fine, finer, extra fine and extremely fine. In the tests, we consider the parameters / = 0.05, R = 0.25 and X = 500 as shown in Table 2. The relative change of the average Nusselt number Nu is defined as the ratio of the Nu difference at each evaluation point with the current mesh condition to the Nu at the same evaluation point with the previous mesh condition. Therefore, the relative change for the first mesh condition is not defined. We can see that the relative change tends to decrease when refining the mesh. The simulation of the model with extremely fine mesh has the best accuracy, but considering the CPU time, a finer mesh size was selected for all the computations done in this paper. To validate the computational code, the previously published problems for natural convection in a differentially heated enclosure filled with nanofluids were solved. Table 3 shows the comparison of the average Nusselt number between the present result and the available results found in the literature for various Rayleigh numbers and concentration levels. The comparison was in good agreement with the results reported in the literature. An additional verification of accuracy for the present code is shown in Table 4, which gives a comparison of the minimum values of the stream function between the present and the literature results for convective heat transfer in a differentially heated square enclosure filled with air [28]. Next, an active rotating cylinder was placed in the center of the enclosure [7]. We see that the deviations vary from 0.2% at X = 500, R = 0.2–5.7% at X = 500, R = 0.4. We conclude that the present computations are acceptable for the considered R range.
Table 2 Grid sensitivity check at / = 0.05, R = 0.25 and X = 500. Predefined mesh size
Mesh elements
Nu
Relative change (%)
CPU time (s)
Extremely coarse Extra coarse Coarser Coarse Normal Fine Finer Extra fine Extremely fine
286 550 760 1276 1938 2734 9132 25,684 33,720
4.0807 4.1253 4.1484 4.1660 4.1722 4.1779 4.1856 4.1897 4.1906
– 1.08 0.56 0.42 0.15 0.14 0.18 0.10 0.02
3 4 5 5 6 8 17 47 54
Table 3 Comparison of the average Nusselt number between the present and the [16] result for a different Rabf and / at Prbf = 6.2 and R = 0. /
Rabf = 104, Cu
Rabf = 105, Cu
Rabf = 104, Al2O3
Rabf = 105, Al2O3
Present
[16]
Present
[16]
Present
[25]
Present
[25]
0 0.01 0.02 0.03 0.04 0.05
2.272 2.304 2.326 2.357 2.388 2.409
2.299 2.335 2.375 2.417 2.459 2.503
4.716 4.771 4.825 4.877 4.949 5.079
4.720 4.793 4.875 4.960 5.048 5.137
2.272 2.299 2.311 2.337 2.361 2.399
2.297 2.335 2.372 2.410 2.448 2.487
4.716 4.757 4.796 4.834 4.871 4.936
4.552 4.626 4.701 4.776 4.851 4.927
Table 4 Comparison of the minimum values of the stream function between the present and the literatures result for a different R and X at Pr = 0.71 and / = 0.0. Literatures R = 0
R = 0.2
R = 0.4
X = 500 X = 0 Present [7] [28]
9.64 19.61 – 19.65 9.61 –
X = 500 X = 500 X = 0
8.99 9.05 8.95 9.11 – –
13.77 13.80 –
X = 500
3.25 1.92 3.21 1.95 – –
4. Results and discussion The analyses in the numerical investigation are performed in the following ranges of the associated dimensionless groups: the solid volume fraction, 0.0 6 / 6 0.05, the radius of the rotating cylinder, 0 6 R 6 0.3 and the angular rotational velocity, 1000 6 X 6 1000. The Prandtl number and the Rayleigh number are fixed at Prbf = 6.2 and Rabf = 105, respectively. Following theses dimensionless parameters, the Richardson number interval is, 19.1358 < Ribf < 1. Fig. 3(a)–(c) shows the effects of Cu nanoparticle concentration, /, on the flow and thermal fields in the square enclosure with constant values of R = 0.2 and X = 100. Fluid flow takes place in the free space between the cylinder and the enclosure. The flow rotates in the clockwise direction due to the natural convection effects. It indicates that the fluid filling the enclosure is moving up along both the left heated wall and the top insulated wall, down along the cooled right wall, and horizontally to the left along the insulated base enclosure. The cylinder rotates in the counter-clockwise direction in the center of the enclosure. This motion creates some recirculating cells between the natural convective flow and the cylinder. The size of the recirculating cells can be widened by increasing the nanoparticle concentration. As can be seen in Fig. 3, the strength of the flow circulation and the average Nusselt number also increase with increasing nanoparticle concentration. This is due to the fact that a high concentration of solid nanoparticles yields the higher thermal conductivity that eventually leads to higher energy, which accelerates the flow. Actually, this phenomenon is also accompanied by an undesirable effect promoted by the viscosity that suppresses the flow, but is small compared to the favorable effect driven by the presence of the high thermal conductivity. Fig. 4 shows the effects of nanoparticle type, on the flow and temperature structures in the enclosure with constant values of / = 0.05, R = 0.2 and X = 100. Streamlines and isotherms show essentially the same behavior as for the previous case, except that the recirculating cells merge with the main cells as presented in Fig. 4(a) for the Ag nanoparticles. This is due to the convection velocity significantly increasing with the added high concentration of Ag nanoparticles. The recirculating cells appear again when the
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7251
Fig. 3. Streamlines (left), isotherms (right) at R = 0.2, X = 100 and water–Cu.
enclosure filled with water–Al2O3 or water–TiO2 as displayed in Fig. 4(b) and (c), respectively. Water–Ag has a faster flow circulation and a better heat transfer performance than the other nanofluid types. This is related to the variation in the thermal conductivity values of the nanoparticle types as shown in Table 1.
Fig. 5 illustrates the effects of the cylinder radius, R, for water– Cu nanofluids with volume fraction 5% and X = 100 on the flow and thermal fields in the enclosure. As can be seen in Fig. 5, R affects the fluid temperatures and the flow characteristics. The strength of the flow circulation is much lower for a larger cylinder. This is because
7252
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Fig. 4. Streamlines (left), isotherms (right) at / = 0.05, R = 0.2 and X = 100.
the radius of the cylinder is large and the space for the fluid flow between the cylinder and the enclosure walls is narrow, restraining the natural convective flow. The average temperature gradient at the hot wall increases with increasing R, but increasing R further reduces the Nu. This is due to the cylinder pushing the fluid in the
center toward the walls which leads to the formation of a thinner thermal boundary layer at the heated and cooled walls. However, further increasing the cylinder radius restrains the convective flow. The effect of varying the rotational velocity of the cylinder on the fluid pattern in the enclosure is presented in Fig. 6 at R = 0.2,
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7253
Fig. 5. Streamlines (left), isotherms (right) at X = 100, / = 0.05 for water–Cu.
/ = 0.05 for water–Cu in the form of streamlines and isotherms. Fig. 6(a) demonstrates that the streamline shape is mainly conditioned by the enclosure walls, and is denser close to the cylinder, thus indicating an intense flow there. In the case X = 500, the clockwise rotation of the cylinder induces an overall convective
flow in the clockwise direction. The case X = 0 corresponds to a motionless cylinder which was presented in Fig. 6(b). In this case, the heat transfer is due to natural convection only. A kind of symmetry was observed in the temperature field which is typical of a buoyancy-driven heat transfer problem in a square enclosure.
7254
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
Fig. 6. Streamlines (left), isotherms (right) at R = 0.2, / = 0.05 for water–Cu.
The isotherms are essentially near the hot and cold walls and are horizontal at the center of the enclosure since the stationary adiabatic cylinder does not interact with the heat transferred by the fluid from the hot to the cold walls. Fig. 6(c) shows the angular velocities in the positive direction, X = 500. The main flow in the
clockwise direction develops naturally within the enclosure due to density differences in the fluid by temperature gradients. The fluid movement close to the cylinder surface, promoted by the anti-clockwise rotation of the cylinder, is however in anticlockwise direction. Some recirculating cells appear in the flow
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
7255
(a)
Fig. 7. Variation of Nu with X for different R and fixed / = 0.05.
(b)
Fig. 9. Variation of Nu with R for different (a) types of solid nanoparticles and (b) cylinder locations. Fig. 8. Variation of Nu with R for different / of Cu nanoparticles.
between them. As can be seen in Fig. 6, the strength of the flow circulation is much higher for the negative rotation because the forced and free convection flows combined. Variations of the average Nusselt number with the rotational velocities are presented in Fig. 7 with different values of R for water–Cu nanofluids with / = 0.05. For rotational velocities close to zero (almost motionless cylinder), the maximum overall Nusselt number occurs at R between R = 0.1 and R = 0.2. This is because there should be an optimum cylinder radius R (or an optimum space between the cylinder and the walls) for a particular nanofluid to be effective in transferring heat. Fig. 7 shows that the average Nusselt number varies slightly, with a maximum of about 2% for the motionless cylinder for any cylinder radius considered here. Since the cylinder body is insulated and kept silent, this would have a small impact on the heat transfer performance. Further, the average Nusselt number does not change significantly when varying the angular velocities of the negative or positive rotation for R 6 0.1. This phenomenon occurs since the forced convection effect, which either aids (negative rotation) or opposes (positive rotation) the natural buoyancy at the center enclosure, is very small. The fastest negative rotation, X = 1000 gives a remarkable increase of the Nu for the biggest cylinder studied in this work. This is because the cylinder velocities for negative rotation accelerates the main flow which leads to a better heat transfer performance when crossing the enclosure. The minimum Nu was observed at X = 600 for the biggest cylinder, R = 0.3. In this case, the cylinder rotation maximally blocks the natural buoyancy force which reduces the overall heat transfer performance substantially. The average Nusselt number is plotted against the cylinder radius in Fig. 8 for different / of Cu nanoparticles at X = 500. This figure shows that the Nu increases with increasing / for any R value
considered here. This is due to the higher conductivity of the fluids when increasing / which accelerates the convective flow on the free space between the cylinder and the enclosure. Heat transfer performance at the hot wall increases with increasing R up to a certain maximum value of R. However, the heat transfer performance decreases when the cylinder radius is larger than the maximum value. Fig. 8 also shows that the maximum average Nusselt number is at R = 0.18, R = 0.19, R = 0.195 and R = 0.2 for / = 0.05, / = 0.03, / = 0.01 and / = 0.0, respectively. This suggests that higher nanoparticle concentration needs more extra space between the cylinder and the wall which is achievable by decreasing the cylinder radius for accelerating the flow and hence enhancing heat transfer. Fig. 9 depicts the variation of the average Nusselt number, Nu, versus the solid volume fraction of the argentum (Ag), copper (Cu), alumina (Al2O3) and titania (TiO2) for different cylinder locations at R = 0.2 and X = 50. The results in Fig. 9(a) show the dependence of heat transfer rate on the different types of nanoparticles. The lowest heat transfer was obtained for TiO2 since it has a lower thermal conductivity (see Fig. 2) which leads to a lower temperature gradient and smaller enhancements in the heat transfer rate. It can be seen in Fig. 2 that increasing / will increase the ratio of thermal conductivity of the various types of nanofluids monotonically that eventually leads to a monotone increase in the heat transfer rate as shown in Fig. 9(a). Fig. 2 also showed that Ag and Cu have the same thermal conductivity property but give different results for the heat transfer performance. This means that the heat transfer rate does not only depend on the thermal conductivity value as concluded by Saleh et al. [29]. Fig. 9(b) shows that the Nu is almost constant when increasing the Cu concentrations with the cylinders located at the lower or upper portion of the enclosure. It is also observed the Nu has the lowest value when the cylinder is located in the upper portion of the enclosure at any Cu concentration. Since the stronger convective flow is confined near the
7256
R. Roslan et al. / International Journal of Heat and Mass Transfer 55 (2012) 7247–7256
upper half, a stagnated region of a cold heavy fluid is formed in the bottom half which yields a poor heat transfer rate. 5. Conclusions The present numerical simulations studied the effect of a rotating cylinder on heat transfer in a square enclosure filled with nanofluids. The dimensionless forms of the governing equations are modeled and solved using the COMSOL program. Detailed computational results for flow and temperature fields, and the overall heat transfer performance crossing the enclosure have been presented graphically. The results show that the thermal and fluid dynamics are highly influenced by the size of the cylinder, rotational speed and direction, nanoparticle concentration and its thermal conductivity values. The most important results of the present analysis are: 1. The recirculating cells emerge between the natural convective flow and the positive cylinder rotation. The size of the recirculating cells widen with increasing nanoparticle concentration. 2. The strength of the flow circulation is much stronger for a higher nanoparticle concentration with better thermal conductivity and a smaller cylinder with a faster, negative rotation. 3. The heat transfer rate increases monotonically by increasing the nanoparticle concentration for the fixed size cylinder located in the center with constant rotation. The maximum heat transfer performance for the cylinder with moderate radius located in the center is obtained at slow positive rotation, high nanoparticle concentration and good thermal conductivity values. The theoretical prediction of this paper is expected to be a useful guide for experimentalists studying nanofluids and rotating cylinders. The factors solid nanoparticle size, nanolayer, shape, temperature, pH of nanofluid solution, presence of surfactants, nature of the nanoparticle component and cylinder rotation with a variable rate will be the focus of our future research. Acknowledgments The authors would like to acknowledge the financial support received from the Grants FRGS/1/2011/SG/UKM/01/13 and FRGS/1/ 2009/SG/UTHM/02/4/0734. References [1] T. Hayase, J.A.C. Humphrey, R. Greif, Numerical calculation of convective heat transfer between rotating coaxial cylinders with periodically embedded cavities, J. Heat Transfer 114 (1992) 589–597. [2] W. Fu, C. Cheng, W. Shieh, Enhancement of natural convection heat transfer of an enclosure by a rotating circular cylinder, Int. J. Heat Mass Transfer 37 (1994) 1885–1897. [3] A. Misirlioglu, The effect of rotating cylinder on the heat transfer in a square cavity filled with porous medium, Int. J. Eng. Sci. 44 (2006) 1173–1187. [4] J. Ghazanfarian, M. Nobari, A numerical study of convective heat transfer from a rotating cylinder with cross-flow oscillation, Int. J. Heat Mass Transfer 52 (2009) 5402–5411.
[5] Y. Shih, J. Khodadadi, K. Weng, A. Ahmed, Periodic fluid flow and heat transfer in a square cavity due to an insulated or isothermal rotating cylinder, J. Heat Transfer 131 (2009) 1–11. [6] S. Paramane, A. Sharma, Numerical investigation of heat and fluid flow across a rotating circular cylinder maintained at constant temperature in 2-D laminar flow regime, Int. J. Heat Mass Transfer 52 (2009) 3205–3216. [7] V. Costa, A. Raimundo, Steady mixed convection in a differentially heated square enclosure with an active rotating circular cylinder, Int. J. Heat Mass Transfer 53 (2010) 1208–1219. [8] S.H. Hussain, A.K. Hussein, Mixed convection heat transfer in a differentially heated square enclosure with a conductive rotating circular cylinder at different vertical locations, Int. Commun. Heat Mass Transfer 38 (2011) 263– 274. [9] N.K. Ghaddar, Natural convection heat transfer between a uniformly heated cylindrical element and its rectangular enclosure, Int. J. Heat Mass Transfer 35 (1992) 2327–2334. [10] G. Cesini, M. Paroncini, G. Cortella, M. Manzan, Natural convection from a horizontal cylinder in a rectangular cavity, Int. J. Heat Mass Transfer 42 (1999) 1801–1811. [11] D. Roychowdhury, S. Das, T. Sundararajan, Numerical simulation of natural convective heat transfer and fluid flow around a heated cylinder inside an enclosure, Heat Mass Transfer 38 (2002) 565–576. [12] B.S. Kim, D.S. Lee, M.Y. Ha, H.S. Yoon, A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transfer 51 (2008) 1888–1906. [13] S.H. Hussain, A.K. Hussein, Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations, Int. Commun. Heat Mass Transfer 37 (2010) 1115–1126. [14] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass Transfer 53 (2010) 5905–5919. [15] S. Ostrach, Natural convection in enclosures, J. Heat Transfer 110 (1988) 1175– 1190. [16] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy–driven heat transfer enhancement in a two–dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [17] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Commun. Heat Mass Transfer 33 (2006) 727–736. [18] M.K. Das, P.S. Ohal, Natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids, Int. J. Numer. Methods Heat Fluid Flow 19 (2009) 411–431. [19] B. Ghasemi, S.M. Aminossadati, Natural convection heat transfer in an inclined enclosure filled with a water–CuO nanofluid, Numer. Heat Transfer Part A 55 (2009) 807–823. [20] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [21] E.B. Ögüt, Natural convection of water-based nanofluids in an inclined enclosure with a heat source, Int. J. Thermal Sci. 48 (2009) 2063–2073. [22] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–105. [23] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [24] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, Int. Commun. Heat Mass Transfer 37 (2010) 79–90. [25] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) 4506–4516. [26] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [27] M.P. Raju, S.K. Khaitan, High performance computing of three-dimensional finite element codes on a 64-bit machine, J. Appl. Fluid Mech. 5 (2012) 123– 132. [28] G. De Vahl Davis, I. P Jones, Natural convection in a square cavity: a bench mark numerical solution, Int. J. Numer. Methods Fluids 3 (1983) 227–248. [29] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluidfilled trapezoidal enclosure, Int. J. Heat Mass Transfer 54 (2011) 194–201.