ISTP-16, 2005, PRAGUE 16TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA

1

Abstract A numerical calculation was made for the laminar flow and heat transfer characteristics through a porous medium placed in a flow passage with a miter bend for various Prandtl numbers. The flow field remains uniform throughout the vertical flow passage with a porous medium. The local heat transfer tends to be improved at low Prandtl number. The effective region of high thermal performance exists in comparatively high Darcy number for low Prandtl number.

1 Introduction A porous medium is attracted attention to

improve the local heat transfer, mass transfer and flow field in many processes and systems. Concretely speaking, they are heat storage type heat exchangers, regenerators in Stirling engines, filters and catalyzers in heat engines. To advance the application of a porous medium, it is essential to analyze the flow and heat transfer behavior in a porous medium placed in a duct. The author investigated experimentally and numerically the heat transfer characteristics of a channel with a porous medium as a turbulence promoter [1, 2]. In addition, a new method to combine the numerical values with the experimental values on the wall heat transfer of a duct was proposed to estimate the heat transfer between solid material and fluid in a porous medium [3]. However, heat transfer characteristics were not examined for various

kinds of fluids. Georgiadis et al. [4] reported the nonlinear inertia force on heat transport by Benard convection in porous media. It concluded that the lower the inertia drag coefficient including Prandtl number is, the higher the Nusselt number is. Jonsson et al. [5] measured the natural convective heat transfer across a horizontal porous layer. The degree of penetration of fluid into the porous layer depends on the product of Darcy and Rayleigh numbers. The fluid layer affects the Nusselt number.

In the present study, the heat transfer and flow characteristics through a porous medium placed in a flow passage with a miter bend were analyzed numerically for various Prandtl numbers and Darcy numbers. The thermal performance was also evaluated.

2 Description of the Problem Figure 1 shows a schematic diagram of

two-dimensional system. The system is composed of a parallel plates whose upper edge is the origin of x and y axes. The height of the inlet and outlet is yo. A porous medium is placed to cover the duct height h(=2yo) and its length is 5yo. Side walls are heated on both upstream side wall (Au-Ad) and downstream side wall (Bu-Bd) under the uniform heat flux, q. Fluid enters the inlet with a laminar fully developed velocity distribution and flows out after the bend section and 50yo straight section. Non-heated walls are thermally insulated.

EFFECTS OF A POROUS MEDIUM IN A FLOW PASSAGE WITH A MITER BEND

Koichi Ichimiya

Interdisciplinary Graduate School of Medicine and Engineering, Mechanical Systems Engineering Division, University of Yamanashi,

Takeda-4, Kofu, 400-8511 Japan email: ichimiya@yamanashi.ac.jp Tel&Fax: +81-55-220-8434

Keywords: Convective Heat Transfer, Porous Medium, Prandtl Number, Miter Bend,

Numerical Calculation

2

Fig. 1 Coordinate System

In a porous medium, momentum equations

are constructed on the basis of the non-Darcy model in which the Brinkman term and Forchheimer term are included [6]. Thermal radiation is ignored since heat conduction and convection dominate heat transfer field.

3 Numerical Analysis The two-dimensional governing equations

can be written in the dimensionless form by using the following parameters including a stream function ψ and a vorticity ω. Subscripts, f and p present fluid and solid, respectively.

oyxX =

oyyY =

oyhS =

muuU =

muvV =

mouyψ

=Ψ m

o

uyω

=Ω o

opffpf qy

TT )( ,,

−=Θλ

2hDa κ

= f

oAf

yhHλ

2

= )/( pffp HH λλ=

νom yu 2Re =

f

pfCλµ

=Pr )(

2

0TTyqNu

wf

o

−=λ

2

3 )/(ν

λβ foo qyygGr =

where um, h and κ are mean velocity across the section, width of porous medium and permeability, respectively.

Flow equations

02

2

2

2

=Ω+∂Ψ∂

+∂Ψ∂

YX

0Re4

)]]()()[()](

)()[([Re2

)(Re2)(1

2

5.022

5.0225.02

2

2

2

2

2

=∂Θ∂

+

∂Ψ∂

∂Ψ∂

+∂Ψ∂

∂∂

−∂Ψ∂

−

∂Ψ∂

+∂Ψ∂

∂∂

+Ω

+

∂Ω∂

+∂Ω∂

−∂Ω∂

∂Ψ∂

−∂Ω∂

∂Ψ∂

XGr

YYXYX

YXXSDac

SDa

YXYXXY

f

εε

Energy equations For the fluid

0)(RePr

2

)(PrRe

22

2

2

2

=Θ−Θ+

∂Θ∂

+∂Θ∂

−∂Θ∂

∂Ψ∂

−∂Θ∂

∂Ψ∂

pff

ffff

HYXYXXY

ε

For the solid

0)()1( 2

2

2

2

=Θ−Θ−

∂Θ∂

+∂Θ∂

− fpppp H

YXε

where ε is porosity and Da is Darcy number. The value proposed by Ergun [7] is applied for coefficient c in the flow equation in a porous medium. Hf is the dimensionless parameter of the medium expressing the heat transfer between solid and fluid saturated in the porous medium. Gr is the Grashof number.

Dimensionless boundary conditions are as follows : at the entrance, U=(−6)(Y2−Y), V=0, Θf=0 ; at upper and bottom walls, U=0, V=0, ∂Θf/∂Y=∂Θp/∂Y=0 ; at side walls, Au-Ad U=0, V=0, ∂Θf/∂X=RRM∂Θp/∂Y= −1, Bu-Bd U=0, V=0, ∂Θf/∂X=RRM∂Θp/∂Y= 1; at exit, ∂U/∂X=0, V=0, ∂Θf/∂X=0 ; at interface, ∂U/∂Xp=∂U/∂X, (µe/µ)∂V/∂Xp =∂V/∂X; where µe /µ is the ratio of effective viscosity of the saturated porous medium to fluid viscosity in the duct and the value used by Neal et al. [8]. RRM is the thermal conductivity ratio of solid and fluid, λp/λf.

3

EFFECTS OF A POROUS MEDIUM IN A FLOW PASSAGE WITH A MITER BEND

In numerical calculations, the governing equations are integrated and represented as a conservative form. The upwind scheme is employed for discretizing convection terms [9]. The algebraic equations are solved iteratively. Numerical calculations were performed at fixed values Re = 1000, Gr = 100, Hf = 100 and RRM=100, whereas the other parameters were varied, namely, Da = 1.0 ∼ 2.5x10-6 and Pr = 0.01 ∼ 100. The results were evaluated in terms of the dimensionless stream function, the Nusselt number and the thermal performance.

4 Results and Discussions Figure 2(a) and (b) show the dimensionless

stream function Ψ at Re = 1000, Pr=0.7 and

(a) Da=1.0

(b) Da=2.5x10-2 Fig. 2 Dimensionless Stream Function Ψ

(Re=1000, Pr=0.7, Dr=100)

Gr=100 for Da=1.0 and 2.5x10-2, respectively. In large Darcy number [Fig.2(a)], a small vortex and a large recirculating flow are produced on the right upper wall and on the (Au-Ad) side, respectively. The flow on the (Bu-Bd) side is accelerated. The size of recirculating flow decreases and the center of recirculating flow moves to an upstream position. The flow becomes uniform as Da number is lower [10]. These situation are almost same for various Prandtl numbers because the effect of Da number is larger than those of viscous and thermal diffusions.

Figures 3 and 4 are the fluid and solid temperature distributions in the duct at Re=1000, Gr= 100 and Da=2.5x10-2 for Pr=0.01 and 100, respectively. In low Prandtl number [Fig. 3(a)],

(a) Pr=0.01

(b) Pr=100 Fig.3 Dimensionless Fluid Temperature Θf

(Re=1000, Gr=100, Da=2.5x10-2)

4

(a) Pr=0.01

(a) Pr=100 Fig. 4 Dimensionless Solid Temperature Θp

(Re=1000, Gr=100, Da=2.5x10-2)

isothermal lines (Θ=0.3∼0.6) of fluid rise almost linearly downwards along the center of the flow passage due to the porous medium and the high thermal diffusion. In high Prandtl number [Fig. 3(b)], isothermal lines are affected by the porous medium and the high viscous diffusion. The dimensionless temperatures are low and the gradient on the heat transfer surface becomes high. Solid temperatures corresponding to Fig. 3 are presented in Fig. 4. The temperature gradient in the porous medium near the heated surface is not so high as that in the fluid because in the present case the thermal conductivity ratio is high (RRM=100).

In the next stage, the behavior of the local Nusselt number was examined for various

(a) Wall A

(b) Wall B Fig.5 Local Nusselt Number Without a Porous

Medium (Re=1000, Gr=100) Prandtl numbers. Figure 5 (a) and (b) show the local Nusselt number Nu for the case without a porous medium. Generally speaking, Nu number becomes low between Y=1.5∼3.5 on the wall A (Au-Ad) side because of a large recirculating flow. It increases at the end of downstream side because of the flow impingement. In low Prandtl number, the heat transfer characteristics do not change remarkably. On the other hand, Nu number on the wall B (Bu-Bd) side is high at Y=1∼3.5 because of the acceleration by a large recirculating flow along the wall A. Figure 6 (a)

5

EFFECTS OF A POROUS MEDIUM IN A FLOW PASSAGE WITH A MITER BEND

(a) Wall A

(b) Wall B Fig.6 Local Nusselt Number with a Porous

Medium (Re=1000, Gr=100, Da=2.5x10-2) and (b) show the Nu number for the case with a porous medium. The local heat transfer is improved on the wall A because a recirculating flow disappears [Fig.2(b)]. It does not change remarkably on the wall B.

Thermal performance is evaluated in Fig. 7(a) and (b) by the relationship between the mean Nusselt number ratio Num/Nun and the pressure drop ratio ∆P/∆Pn where the subscript n indicates pure fluid flow in the absence of a porous medium. The solid line in the figure shows that the heat transfer improvement is the same as the pressure drop increase. The upper

(a) Wall A

(b) Wall B Fig. 7 Thermal Performance

(Re=1000, Gr=100)

area from the solid line means high heat transfer and low pressure loss. In such a meaning, a porous medium is effective from Da=1.0 to

6

Da=2.5x10-3 for low Prandtl number (Pr=0.01) in the wall A.

5 Conclusions Effects of a porous medium on the heat

transfer and flow were examined numerically using a duct with a miter bend for laminar flow. The flow state remains uniform throughout the flow passage at low Darcy number. Corresponding to the flow, the local heat transfer is improved on the wall A and for low Prandtl number. Therefore, the effective region of high heat transfer and low pressure loss exists in comparatively high Da number for low Prandtl number.

References [1] Ichimiya K and Mitsushiro K. Enhancement of the

Heat Transfer of Wide Temperature Field in a Narrow Flow Passage (Effects of Porous-type Turbulence Promoter in Normal Temperature Field). Proceedings of the 1st World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics (Dubrovnik), Vol.1, pp.659-664, 1988.

[2] Ichimiya K and Kondoh H. Effects of a Single Roughness Element on the Heat Transfer and Flow Situation in a Parallel Plate Duct (Numerical Calculation for a Porous-type and a Solid-type). Advanced Computational Methods in Heat Transfer, Vol. 2, pp.39-49, 1990.

[3] Ichimiya K. A New Method for Evaluation of Heat Transfer Between Solid Material and Fluid in a Porous Medium. Transactions of the ASME Ser. C , Journal of Heat Transfer, Vol. 121, pp.978-983, 1999.

[4] Georgiadis J.G and Catton I. Prandtl Number Effect on Benard Convection in Porous Media. Transactions of the ASME Ser. C , Journal of Heat Transfer, Vol.108, pp.284-290, 1986.

[5] Jonsson T and Catton I. Prandtl Number Dependence of Natural Convection in Porous Media, Transactions of the ASME Ser. C , Journal of Heat Transfer, Vol.109, pp.371-377, 1987.

[6] Vafai K and Tien C.L. Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media. International Journal of Heat and Mass Transfer, Vol.24, pp.195-203, 1981.

[7] Ergun S. Fluid Flow Through Packed Columns. Chemical Engineering Progress, Vol.48, pp.89-94, 1952.

[8] Neal G and Nader W. Practical Significance of Brinkman’s Extension of Darcy’s Law : Coupled

Parallel Flows Within a Channel and a Bounding Porous Medium. The Canadian Journal of Chemical Engineering, Vol.52, pp.475-478, 1974.

[9] Gosman A.D, Pun W.M, Runchal A.K, Spalding D.B and Wolfshtein M. Heat and Mass Transfer in Recirculating Flow, Academic Press, London, 1969.

[10] Ichimiya K and Matsuda T. Effects of a Porous Medium on Local Heat Transfer and Fluid Flow in a Forced Convection Field, International Journal of Heat and Mass Transfer, Vol.40, pp.1567-1576, 1997.