Fluid flow and forced convection heat transfer around

International Journal of Heat and Mass Transfer 63 (2013) 91–100

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Fluid flow and forced convection heat transfer around a solid cylinder wrapped with a porous ring Saman Rashidi a, Ali Tamayol b,c, Mohammad Sadegh Valipour a, Nima Shokri d,⇑ a

Department of Mechanical Engineering, Semnan University, P.O. Box 35196-45399, Semnan, Iran Biomedical Engineering Department, McGill University, Montreal, Canada H3A 0G1 c Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA 02139, USA d School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK b

a r t i c l e

i n f o

Article history: Received 20 September 2012 Received in revised form 4 March 2013 Accepted 5 March 2013

Keywords: Cylinder embedded in porous media Darcy numbers Volume averaged equations Optimization process

a b s t r a c t Convective heat transfer from cylinders embedded in porous media is important for many engineering applications. In the present study, flow-field and heat transfer around a cylinder embedded in a layer of homogenous porous media is investigated numerically. The range of Reynolds and Darcy numbers are chosen to be 1–40 and 1 108–1 101, respectively. Volume averaged equations are used for modeling transport phenomena within the porous layer and conservation laws of mass, momentum, an energy are applied in the clear region. A comprehensive parametric study is carried out and effects of several parameters, such as porous layer thickness and permeability as well as the Darcy and Reynolds numbers on flow-field and heat transfer characteristics are studied. Finally an optimization process is conducted in order to determine the optimal thickness and porosity of the porous layer resulting in the lowest heat transfer from the cylinder. The numerical results indicate that, in the presence of a porous layer around the cylinder, the wake length increases with decreasing the Darcy number while the critical radius of insulation increases. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Fluid flow and heat transfer across a cylinder embedded inside a porous layer has various environmental and engineering applications such as insulation of heat pipes [1], compact heat exchangers design [2], reactor safety analysis and combustion [3] and controlling the spread of petroleum-based products in groundwater and soil. In addition, micro-cylinders covered by a porous structure embedded within channels can be used for designing novel compact micro-heat exchangers for various applications such as electronics cooling [4]. Depending upon the thermophysical properties of the porous medium, the effective solid and fluid interface area and the overall heat and mass transfer rates from the base cylinder may change. Researchers have employed a variety of techniques for characterizing the convection heat transfer across a cylinder embedded in a porous layer [5–7]. Layeghi and Nouri-Borujerdi [7] studied convective heat transfer from an array of circular cylinders in direct contact with passing flow or surrounded by porous media. Their numerical simulations suggested a heat transfer enhancement (over 80%) due to the

⇑ Corresponding author. Tel.: +44 161 3063980. E-mail address: [email protected] (N. Shokri). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.006

presence of a highly conductive porous medium. However, the porous medium layer increased the overall pressure drop of the system. Bhattacharyya et al. [8] presented numerical simulations on the fluid motion around and through a porous cylinder. Their results showed that the drag experienced by the porous cylinder decreases with the increase of the Reynolds number and reduce in the Darcy number. Bhattacharyya and Singh [6] investigated the effect of the porous layer properties on the convective heat transfer augmentation from an isothermal circular tube. The porous layer in their study was of a foam material with high porosity and thermal conductivity. They studied effects of salient parameters such as the permeability and thermal conductivity of the porous layer, flow Reynolds and Grashof numbers, and the porous layer thickness on the mixed convection. For example, they showed that a thin porous layer with high thermal conductivity can significantly enhance the heat transfer even at low permeability. Rong et al. [9] performed numerical simulations of flow around a square crosssection cylinder covered by a porous layer. Their results revealed that the drag and lift coefficient increases with increase of porous layer thickness when Da P 104 and have little changes when Da < 104. In a notable study, Al-Sumaily et al. [10] numerically studied the time-dependent forced convection heat transfer from a single circular cylinder embedded in a horizontal channel filled

92

S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100

Nomenclature cp Cp CF d D Da Dp FD h k K Nu P Pr r R Rc Re T u, v

specific heat (J/kg K) pressure coefficient (–) Forchheimer coefficient (-) porous layer thickness (m) cylinder diameter (m), D = 2R Darcy number (–), Da = K/D2 characteristic diameter of a particle (m) total drag force (N) heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) permeability (m2) Nusselt number (–) pressure (Pa) Prandtel number, (–), t/a radial coordinate (m) cylinder radius (m) thermal conductivity ratio, keff/kf Reynolds number (–), qU1D/l temperature (K) velocity component in r, h direction respectively (m s1)

t q e a h d

x K

fluid kinematic viscosity (m2 s1), t = l/q fluid density (kg m3) porosity (–) thermal diffusivity of fluid (m2/s), k/qcp cross-radial coordinate (m) non-dimensional porous layer thickness (m), d = d/D vorticity (1/s) viscosity ratio (–), K = leff/lf

Subscripts ave average cr critical eff effective f fluid p pressure force s solid v viscous force w wall 1 free stream 1 clear fluid domain 2 porous domain

Greek symbols l dynamic viscosity (kg m1 s1)

by spherical particles under local thermal non-equilibrium condition and constant porosity. They investigated the influence of porous materials and its characteristics on the heat transfer and thermal responses. Similar to Bhattacharyya and Singh [6], they found that the presence of the porous particles enhances the heat transfer rate. Their results suggested that an increase in Biot number reduces fluid Nusselt number while it increases the Nusselt number in the solid phase. In addition, it was observed that increasing the ratio of solid to fluid thermal conductivity reduced the deviations from thermal equilibrium condition in the porous bed. Hooman and his co-workers modeled metalfoam heat exchangers as a solid surface surrounded by a porous material [11,12]. They employed analytical and numerical methods for studying their thermal performance and reported a model for predicting the thermal properties of metalfoam heat exchangers. Recently, they investigated the effect of dust deposition on the hydrodynamic and thermal performance of metalfoam heat exchangers [13,14]. They showed that a thin layer of fouling deposited on the surface of fibers of the metalfoam could significantly affect the thermal performance and pressure drop of the heat exchanger. Valipour and Zare Ghadi [15] numerically studied the forced convective heat transfer around and through a porous circular cylinder. Along with the numerical studies, experiments have been also conducted to study the effects of a porous layer wrapped around a solid cylinder on heat transfer and thermal responses. For example, Ahmed [16] experimentally studied forced convection about a horizontal cylinder embedded in a porous medium for 2000 < Re < 3000. Their experimental results revealed that the average heat transfer increases as the Peclet and Reynolds number increased for steady state condition. Al-Salem et al. [17] experimentally investigated the effects of the porosity and thickness of porous layer on heat transfer enhancement in a cross flow over a heated cylinder. They could experimentally demonstrate a heat transfer augmentation as a result of the porous layer presence. Our literature review showed that the pertinent literature is rich due to diverse applications of flow around a cylinder covered

by a porous layer. However, we noticed a lack of information and data in the targeted range of Reynolds number which is important for designing compact micro heat exchangers. As a result, we focused on this range of Reynolds number. The objective of the present research is to further investigate and provide additional insights about the effects of the presence of porous layers on the thermal insulation or heat transfer augmentation. Besides, we study the potential influences of a porous layer wrapped around a solid cylinder on several hydrodynamics parameters such as drag coefficient, pressure coefficient, length of recirculation region, streamlines and determine the critical radius of porous layer insulation for a cylindrical body in laminar regime. To achieve the objectives of this study, we performed a comprehensive numerical study to investigate the laminar forced convection flow around a solid cylinder wrapped with porous ring and reveal the effects of Darcy and Reynolds numbers, thermal

Fig. 1. Computational domain and geometry of cylinder.


conductivity of porous substrate and porous layer thickness on the flow pattern and heat transfer characteristics. 2. Analysis

Fig. 1 shows the geometry of the studied problem, which comprises a long circular cylinder with the diameter of D embedded within a layer of porous medium of the known thickness (d). The side effects are neglected and the problem is modeled two dimensionally. Laminar flow with the temperature of T1 flowing across the isothermal cylinder at a constant temperature of Tw is considered. In addition, the following assumptions are made: The porous matrix around the cylinder is assumed homogeneous and isotropic with uniform porosity and tortuosity. All fluid properties are considered to be constant. The solid phase temperature is equal to that of the fluid phase (local thermal equilibrium (LTE)). Local thermal equilibrium is often used when studying heat transfer in porous media and this assumption holds when the temperature difference between the solid and fluid phases is not significant. This assumption breaks down during rapid heating or cooling or problems with significant temperature variation across the porous media [18].

Equations that govern the two zones described in the previous section, i.e., the clear fluid and the porous medium zones, are Navier–Stokes and volume averaged equations, respectively. It should be noted that the clear fluid and porous medium domains are indicated by subscripts (1) and (2), respectively. To generalize the analysis, the following dimensionless parameters are substituted in the governing equations [6].

r ; R

h ¼ h ;

u¼

u ; U1

v¼

v U1

;

p¼

p

q

U 21

;

T¼

T T1 Tw T1

The variable with superscript ⁄ denotes dimensional variables. Following the above assumptions and by substituting the dimensionless numbers, the mass conservation equation in region 1 (clear fluid) becomes:

@ @ ðru1 Þ þ ðv 1 Þ ¼ 0 @r @h

ð2Þ

And momentum equations in r and h direction reads: v 1 @u1 @u1 v 21 @p þ u1 ¼ 1 r @h @r r @r

þ

2 @ 2 u1 1 @u1 1 @ 2 u1 2 @ v 1 u1 þ þ Re @r2 r @r r2 @h2 r2 @h r2

!

ð3Þ

v 1 @v 1 r @h

þ u1

And the momentum equations in r and h direction can be written as: 1

v 2 @u2

þ u2

e r @h

@u2 v 22 @p ¼ e 2 @r r @r

! 2e @ 2 u2 1 @u2 1 @ 2 u2 2 @ v 2 u2 þ þ Re @r 2 r @r r 2 @h2 r 2 @h r 2 e eC F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 pffiffiffiffiffiffi u22 þ v 22 u2 2ReDa 2 Da ð7Þ

þK

1

v 2 @v 2

e r @h

þ u2

@ v 2 u2 v 2 e @p2 þ ¼ @r r r @h

! 2e @ 2 v 2 1 @ v 2 1 @ 2 v 2 2 @u2 v 2 þ þ þ r @r r 2 @h2 r 2 @h r2 Re @r2 ffi e eC F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u22 þ v 22 v 2 v 2 pffiffiffiffiffiffi 2ReDa 2 Da ð8Þ þK

@ v 1 u1 v 1 1 @p1 þ ¼ r @h @r r þ

2 @ 2 v 1 1 @ v 1 1 @ 2 v 1 2 @u1 v 1 þ þ þ Re @r 2 r @r r 2 @h2 r2 @h r 2

!

ð4Þ

The energy equation in region 1 can be written as:

! @T 1 v 1 @T 1 2 @ 2 T 1 1 @T 1 1 @ 2 T 1 u1 þ þ ¼ þ 2 RePr @r2 r @r r @h2 @r r @h

! @T 2 v 2 @T 2 2Rc @ 2 T 2 1 @T 2 1 @ 2 T 2 ¼ u2 þ þ þ 2 r @r r @h2 @r r @h RePr @r 2

ð5Þ

ð9Þ

Where Rc, Da, Re and Pr are thermal conductivity ratio, Darcy number, Reynolds number and Prandtel number, respectively expressed as:

Rc ¼

ð1Þ

ð6Þ

The energy equation with the local thermal equilibrium assumption can be written as:

2.2. Governing equations

r¼

In region 2, i.e., the porous medium region, the volume-averaged equations are used. The volume-averaged mass conservation equation becomes:

@ @ ðru2 Þ þ ðv 2 Þ ¼ 0 @r @h

2.1. Problem statement

93

keff ; kf

Da ¼

K D2

;

Re ¼

qU 1 D m ; Pr ¼ l a

ð10Þ

! The volume-averaged fluid velocity V inside the porous region ! with porosity e is related to the fluid velocity v through the ! ! Dupuit–Forchheimer relationship as v ¼ e V . Forchheimer coefficient CF [19] can be calculated depending on the microstructure of the porous medium. Here we used the widely accepted Ergun correlation expressed as [20]:

1:75 C F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 150e3

ð11Þ

There are various models for calculating the permeability of different porous materials [21–24]. Our analysis is generalized and can be applied to any type of porous media. However, as an example, we used the Carman–Kozeny model for calculation of flow coefficients. This model was originally proposed for packed bed of spherical particles (such as sandstone), however, Tamayol and Bahrami [25] have shown that this model can also be applied to various types of fibrous materials (such as foams). As a result we used this relationship for both fibrous and packed bed porous structures. Permeability is calculated from the following relationship [26]: 2

K¼

e3 Dp 1 180 ð1 eÞ2

ð12Þ

where K is the permeability and Dp is the average particle size of the porous bed. In our calculations, we consider (Dp) to be equal to 100 lm.

94


2.3. Boundary conditions The dimensions of the computational domain in the vertical and horizontal directions should be selected in a way that minimizes the effects of the outer boundaries. Based on the results reported by others in the literature, computational domain is defined to be 50 times of the cylinder diameter, i.e., 50D 50D. The governing equations (2)–(9) are subjected to the following boundary conditions: On the surface of the solid cylinder:

u2 ¼ v 2 ¼ 0;

T2 ¼ 1

ð13Þ

Along the upstream boundary (uniform flow):

For 0 < h
rcr . Thus when r < r cr , insulating the cylinder may actually increase the rate of heat transfer from the cylinder instead of decreasing it. Fig. 11 depicts how the porosity and Darcy number affects the critical radius of insulation. As indicated in this figure, the critical radius of insulation decreases with increasing Darcy number in all cases; in particular, the decrease is more obvious at a thick insulation ring. Also shown in this figure is that the rate of heat transfer from the cylinder increases in the domains located above the solid line. Fig. 11 also suggests that for a thick and low permeable insulation ring, insulating the cylinder will increase the rate of heat transfer from the cylinder instead of decreasing it. Fig. 12 illustrates variation of average Nusselt number versus Darcy numbers where the results are reported for Re = 40 and different porous layer thickness. Porous ring is filled with sand stone material, i.e., ks > kf (kf (salt water) = 0.596; ks (sand stone) = 2.327). In this figure, the horizontal line refers to the average Nusselt number for a cylinder without porous layer. It is notable that at non-dimensional porous layer thickness equal to 1 for Da > 2.2 105, the average Nusselt number increases with adding porous layer to cylinder due to relative higher thermal conductivity of porous material. However, for Da < 2.2 105, the average Nusselt number


99

The magnitude of velocity components decreases as Darcy number decreases because with decreasing Darcy number, the no-slip and no-penetration conditions are dominant. For relatively large Darcy numbers (Da = 1 102), the pressure coefficient is significantly larger than that of a single cylinder and these changes are more obvious along the rear part of the cylinder where fluid flows out and adverse pressure gradient decreases . For heat exchanger applications, a porous medium with high permeability (since the fluid flows faster through porous layer with high permeability, so the rate of convection heat transfer are considerably increased with increase of Darcy number) and high thermal conductivity improves the thermal performance. The critical radius of insulation decreases with increasing Darcy number because the convection heat transfer coefficient increases with increase in Darcy number for porous material with low thermal conductivity. Fig. 11. Variation of critical radius of insulation with Darcy numbers and for different porous layer (foam) thickness and Re = 40, ks/kf = 0.15.

References

Fig. 12. Variation of average Nusselt number versus Darcy numbers for different porous layer (sand stone) thickness and Re = 40, ks/kf = 3.9.

decreases with adding porous layer to cylinder because the velocity and rate of convection heat transfer considerably reduces with decrease of permeability. Therefore, for heat transfer enhancement techniques, it is more suitable to choice permeable porous layer with high thermal conductivity. 4. Conclusion In the present work, fluid motion and heat transfer around a solid circular cylinder wrapped with porous layer has been studied. Also, the effects of Darcy and Reynolds numbers and porous layer thickness on the flow patterns and heat transfer characteristics were investigated. The highlights of our parametric study are as follows: The presence of a porous layer around a cylinder increases the wake length. Consequently, a decrease in Darcy number leads to an increase in the wake length. The drag coefficient decreases with increasing Darcy number. Because a highly permeable layer allows the fluid to flow through it with the least resistance and this resistance increases as the permeability decreases.

[1] T. Guldbrandsen, P.W. Karlsson, V. Korsgaard, Analytical model of heat transfer in porous insulation around cold pipes, Int. J. Heat Mass Transfer 54 (1–3) (2011) 288–292. [2] A. Zukauskas, Heat transfer from tubes in cross flow, Adv. Heat Transfer 18 (1987) 87–159. [3] M. Golombok, J. Jariwala, L.C. Shirvill, Gas-solid heat exchange in a fibrous metallic material measured by a heat regenerator technique, Int. J. Heat Mass Transfer 33 (2) (1990) 243–252. [4] A. Tamayol, J. Yeom, M. Akbari, M. Bahrami, Low Reynolds number flows across ordered arrays of micro-cylinders embedded in rectangular micro/ minichannels, Int. J. Heat Mass Transfer 58 (1–2) (2013) 420–426. [5] J. Thevenin, D. Sadaoui, About enhancement of heat transfer over a circular cylinder embedded in a porous medium, Int. Commun. Heat Mass Transfer 22 (2) (1995) 295–304. [6] S. Bhattacharyya, A.K. Singh, Augmentation of heat transfer from a solid cylinder wrapped with a porous layer, Int J. Heat Mass Transfer 52 (7) (2009) 1991–2001. [7] M. Layeghi, A. Nouri-Borujerdi, Fluid flow and heat transfer around circular cylinder in the presence and no-presence of porous media, J. Porous Media 7 (3) (2004) 239–247. [8] S. Bhattacharyya, S. Dhinakaran, A. Khalili, Fluid motion around and through a porous cylinder, Chem. Eng. Sci. 61 (13) (2006) 4451–4461. [9] F.M. Rong, Z.L. Guo, J.H. Lu, B.C. Shi, Numerical simulation of the flow around a porous covering square cylinder in a channel via lattice Boltzmann method, Int. J. Numer. Methods Fluids 65 (10) (2011) 1217–1230. [10] G.F. Al-Sumaily, J. Sheridan, M.C. Thompson, Analysis of forced convection heat transfer from a circular cylinder embedded in a porous medium, Int. J. Therm. Sci. 51 (2012) 121–131. [11] M. Odabaee, K. Hooman, H. Gurgenci, Metal foam heat exchangers for heat transfer augmentation from a cylinder in cross-flow, Transp. Porous Media 86 (3) (2010) 911–923. [12] A. Tamayol, K. Hooman, Thermal assessment of forced convection through metal foam heat exchangers, J. Heat Transfer 133 (2011) 111801. [13] K. Hooman, M. Odabaee, A. Tamayol (Particle deposition effects on heat transfer from a porous heat sink), in: Computational Methods for Thermal Problems, Dalian University of Technology Press, Dalian, China, 2011, pp. 263– 266. [14] K. Hooman, A. Tamayol, M. Malayeri, Impact of particulate deposition on the thermohydraulic performance of metal foam heat exchangers: a simplified theoretical model, J. Heat Transfer 134 (2012) 092601–092607. [15] M.S. Valipour, A. Zare Ghadi, Numerical investigation of forced convective heat transfer around and through a porous circular cylinder with internal heat generation, J. Heat Transfer 134 (2012) 062601. [16] A.H. Ahmed, Forced convection about a horizontal cylinder embedded in a porous medium, J. Kirkuk Univ. Sci. Stud. 5 (1) (2010) 37–52. [17] K. Al-Salem, H.F. Oztop, S. Kiwan, Effects of porosity and thickness of porous sheets on heat transfer enhancement in a cross flow over heated cylinder, Int. Commun. Heat Mass Transfer 38 (9) (2011) 1279–1282. [18] W.J. Minkowycz, A. Haji-Sheikh, K. Vafai, On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the Sparrow number, Int. J. Heat Mass Transfer 42 (18) (1999) 3373–3385. [19] A. Tamayol, K.W. Wong, M. Bahrami, Effects of microstructure on flow properties of fibrous porous media at moderate Reynolds number, Phys. Rev. E 85 (2) (2012) 026318–026324. [20] P.H. Forchheimer, Z.V. Deutsch, Ing 45 (1901) 1782–1788. [21] J. Bear, Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, 1990.

100


[22] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24 (2) (1981) 195–203. [23] K. Vafai, Handbook of Porous Media, Taylor & Francis, 2005. [24] M. Sahimi, Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing, Rev. Mod. Phys. 65 (4) (1993). [25] A. Tamayol, M. Bahrami, Analytical determination of viscous permeability of fibrous porous media, Int. J. Heat Mass Transfer 52 (9–10) (2009) 2407–2414. [26] A.D. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 1998. [27] B. Alazmi, K. Vafai, Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, Int. J. Heat Mass Transfer 44 (9) (2001) 1735–1749. [28] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.

[29] F.M. White, Fluid Mechanics, sixth ed., McGraw Hill Book Company, New York, 2009. [30] L.G. Leal, A. Acrivos, The effect of base bleed on the steady separated flow past bluff objects, J. Fluid Mech. 39 (4) (1970) 735–752. [31] P. Yu, Y. Zeng, Thong S. Lee, Xiao B. Chen, Hong T. Low, Steady flow around and through a permeable circular cylinder, Comput. Fluids 42 (1) (2011) 1–12. [32] M. Zhao, L. Cheng, B. Teng, D. Liang, Numerical simulation of viscous flow past two circular cylinders of different diameters, Appl. Ocean Res. 27 (1) (2005) 39–55. [33] M. Ait Saada, S. Chikh, A. Campo, Natural convection around a horizontal solid cylinder wrapped with a layer of fibrous or porous material, Int. J. Heat Fluid Flow 28 (3) (2007) 483–495. [34] F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, John Wiley and Sons Inc., New York, 1985.