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Procedía Engineering 121 (2015) 1171 - 1178

Procedía Engineering

www.elsevier.com/locate/procedia

9th International Symposium on Heating, Ventilation and Air Conditioning (ISHVAC) and the 3rd International Conference on Building Energy and Environment (COBEE)

Effect of Rayleigh numbers on natural convection and heat transfer with thermal radiation in a cavity partially filled with porous

medium

Jun Yanga*, Yuancheng Wanga, Xiaojing Zhanga ,Yu Pana

aCollege of Thermal Energy Engineering, Shandong Jianzhu University, Jinan 250101, China

Abstract

In this paper, the effects of Rayleigh numbers on natural convection and heat transfer with thermal radiation in a cavity partially filled with a porous medium have been studied numerically. The governing equations for the momentum and heat transfer in both free fluid and porous medium were solved by the finite element method. The radiative heat transfer is calculated by making use of the radiosity of the surfaces that assumed to be grey. Comparisons with experimental and numerical results in the literature have been carried out. Effects of Rayleigh number on natural convection and heat transfer in both free fluid and porous medium were analyzed. It was found that Rayleigh numbers can significantly change the temperature fields in both the regions of free flow and porous medium. The mean temperature at the interface decreases and the temperature gradients are created on the upper two corners of the porous medium region as Ra increases.

© 2015TheAuthors.Publishedby ElsevierLtd.Thisis an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ISHVAC-COBEE 2015 Keywords: natural convection; heat transfer; Rayleigh numbers; cavity; porous medium;

1. Introduction

Natural convection and heat transfer in composite porous/fluid domain exist in many natural phenomena and engineering applications, such as grain storage, air conditioning systems and insulation used in buildings. During the past several decades, several experiments and numerical simulations have been presented to describe the phenomena of natural convection and heat transfer with and without thermal radiation in a cavity [1-6]. The published literature also contains several studies of natural convection in cavities that are partially filled with porous medium, however,

* Jun Yang.. E-mail address: 18254117196@163.com

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ISHVAC-COBEE 2015 doi:10.1016/j.proeng.2015.09.131

none appears to include the interaction of thermal radiation and natural convection [7-10]. Rayleigh numbers and thermal radiation between the walls and roof of the buildings and grain storage silos affects the natural convection flow and heat transfer, and influence the moisture transportation. In this paper, we focus on natural convection and heat transfer with thermal radiation in a cavity partially filled with a porous medium in which a differentially heated cavity in which the two side walls are held at constant temperatures, and the upper and lower walls are deemed to be adiabatic, and a classical investigation of Rayleigh numbers that affect the natural convection and temperature fields is conducted.

2. Mathematical formulation

The system studied in this work consists of a differentially heated cavity of width W and height H, and the porous medium is half of the whole cavity.

2.1 Governing equations

In this analysis we shall consider the fluid to be air, which are incompressible and laminar. The thermophysical properties of the fluid are assumed constant, except for the density in the buoyancy term in the momentum equations. The porous medium is considered as bulk wheat, which are homogeneous, isotropic and in local thermodynamic equilibrium with air. The governing conservation equation for the free fluid and the porous medium will be written separately.

For the fluid region, we have

Continuity: v - 0 (1)

Momentum: PV-(w) = ^V2v-Vp - pogP{T - Ta ) ^ TV, 1 Pocav -VT = kV2T

Thermal energy: ' o a (3)

in which v is the velocity vector, p is the pressure, T is the temperature of the fluid and porous medium, p is the density of the fluid, po is the density of the fluid at the temperature To, g is the gravity vector, /3 is the coefficient of volumetric expansion of the fluid, c, k and jU are the specific heat, thermal conductivity and viscosity of the fluid, respectively. The third term on the right hand side of the equation (2) arises as a result of Boussinesq's approximation.

For the porous region, the governing equations are

Continuity: VD = 0 (4)

Momentum: " (VD VD 2 vD ~ PogP{T ~ To • vd (5)

pcnvD - VT = keffV2T

Thermal energy: ^p p D eff (6)

in which vd is the volume averaged Darcian or superficial velocity of the fluid through the porous medium, K is the permeability tensor, which in this work is taken to be isotropic, Pp is the density of porous medium, cp is specific heat of porous medium, kejj is the effective thermal conductivity of the porous medium. The boundary conditions for the equations (1) to (6) are

v = v D = 0 when X = 0 W and ^ = 0, H ;T = T when X = 0 ;T = Tc when X = W. ny • VT = 0 when y = 0;Q + kny • VT = 0 when > = H; V = Vd when Y = Y-^;

Q + kefn-VT = kn-VTl Y - Y T = T\„ Y - Y

^ ef I porous 1 fluid when ~ interface . 1 porous I fluid when ~ interface

Where y is unit normal vector, the argument of which is positive in the direction leaving the surface. Q is the net radiation flux on the surface.

There is an exchange of radiant energy between the two side walls and the roof of the cavity and the interface of the saturated porous medium and fluid. The fluid above the porous region is deemed to be transparent to thermal radiation and not participating in radiative heat transfer. The porous medium is opaque to thermal radiation at its upper surface which can absorb and emit radiation.

2.2. Validation of the model

The model was validated by comparing simulation results with two sets of published data. One set, that of Beckermann et al.[8] deals with heat transfer in a cavity partially filled with a porous medium, but that excludes the effects of thermal radiation. Beckermann et al.[8] presented both experimental and numerical studies on natural convection in a rectangular cavity filled to half of its width with a vertical layer of saturated porous medium, namely glass beads.The aspect ratio, H/W, of the cavity is 1.0, s is the surface emissivity, Ra is the Rayleigh

number,Ra Ca^ g^(Th Tc )H /(k^) , Pr is Prandtl number, Da is the Darcy number, Da K / H and K is the permeability of the porous medium. Rk is the ratio of thermal conductivities between the porous medium

and fluid and k eff ^ , k is the thermal conductivity of fluid. The surface emissivity, Rayleigh, Darcy and Prandtl numbers are summarized in Table 1. In this paper, the results of experiment 2 and experiment 4 of the experiments were compared with our numerical results.

Table 1. Summary of experimental conditions from Beckermann et al. [8]

Test number Test fluid Ra Pr Da Rt S

Experiment 2 water 3.208 > <107 6.97 1.296 > <10"5 1.383 0.38

Experiment 4 glycerin 3.471 > <106 12630 7.112 > <10"7 2.259 0.38

A comparison between measured and predicted dimensionless temperatures by the present numerical model is shown in Fig. 1. It can be seen that agreement between the two sets of values is good, especially in the Fig.la. The discrepancies occur in Fig.1b, which is the comparison between measured and predicted results for glycerin. This is possibly due to the inaccuracies in determining the exact position of the movable thermocouple probe, and nonuniformities in the porosity at the walls, which is expected to produce a considerable difference between the

numerical models. In addition, the viscosity of glycerin varies by almost an order of magnitude over the temperature range in the experiments [8].

-Analysis,y/L=0.847 \

-----Analysis,y/L=0.496

----------Analysis,y/L=0.055 "

▼ Experiment,y/L=0.847 O Experiment,y/L=0.496 * Experiment,y/L=0.055

—I—|—i—|—i—|—i—|—i—|—i—|—i—|—i—|—i—[—

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(a) experiment 2

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

-Analysis,y/L=0.875

----Analysis,y/L=0.500

-------Analysis,y/L=0.141

▼ Experiment,y/L=0.875 o Experiment,y/L=0.500 ★ Experiment,y/L=0.141

l—1—i—1—i—1—i—1—i—1—i—1—i—1—i—1—i—1—i—1—r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(b) experiment 4

Fig.1. The dimensionless temperature comparison between the experimental and predicted results for experiment 2 and experiment 4

3. Results and discussion

A cavity with an air fluid overlying a porous medium, which occupies half of the whole cavity, was chosen as prototype system for investigation. The system comprises the classical differentially heated cavity, in which the isothermal vertical side-walls are maintained at constant but different temperatures, the upper and lower surfaces of the cavity are adiabatic. The three cases are investigated with the following parameters: Tc = 288.5K, Th = 298.5K, T0=(Th+Tc)/2=293.5K, Rk = 5.31 ,s =0.8, Da= 5.78e-7 and Ra=104,105,106.

Fig. 2. shows the variations of streamlines and temperature fields with s =0.8 at different Rayleigh numbers. It can be seen that the fluid rises along the hot side wall and falls along the cold side wall, leading to a circulating currency in the whole region besides a circulation in the free flow. With the increase of Ra, more and more intensive natural convection in the free flow results in the isotherms on the upper two corners of the porous medium region to be more distorted, which means the heat transfer in the porous medium has changed from that dominated by thermal conduction to that dominated by the conduction together with natural convection, although the natural convection is much weaker than the conduction in the porous medium.

From Fig. 2. one can see that the mean temperature at the interface decreases and the temperature gradients is created on the upper two corners in porous medium region with the increase of Ra because of the natural convection in the free flow. This is very important to the cereal grain storage. For example, if the upper surface of grains in a silo becomes warmer, the moisture of the stored cereal grains can transport from warm regions to cooler regions, resulting in the local higher moisture content in a silo and the deterioration of cereal grains.

(a) Ra =104. Rt=531, Da= 5.78e-7

(b)Ra =105. i?j-=5.31. Da= 5.78e-7

(c) Ra =106, Rk=5.31, Da= 5.78e-7 Fig. 2. The streamlines and isotherm fields with several Rayleigh numbers at £ =0.8

The effect on average convective Nusselt number, Nuc, due to changing Ra is shown in Figs. 3a-3d. It can be seen that the Nu on the side walls in the free flow and porous medium regions increase with increase of Ra. This means that, as Ra is increased, the thermal boundary near the side wall appears, and convection plays an important role in heat transfer.

3.5 3.02.0' 1.5 ■ 1.0-

—I—

—I—

—I—

—I—

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

'°9i0Ra

4.0 4.5 5.0 5.5 '°910Ra

(a) Nu c on the hot side of the free flow region

(b) Nuc on the cold side of the free flow region

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7

0.0 1 07;

0.3 0.6-

0.8 0.5-

5.0 log10Ra

—I—

—I—

log10Ra

(d) Nuc on the cold side of the porous region

(c) Nuc On the hot side of the porous region Fig. 3 Effect of Ra on the mean convective Nusselt numbers at £ =0.0, 0.1, 0.3, 0.5, 0.8. (Rk=5.31, Da= 5.78e-7)

4. Conclusions

The combined radiation-natural convection heat transfer in a differentially heated cavity partially filled with a porous medium has been investigated numerically. Comparisons with experimental and numerical simulation results from the literature have been carried out to check the accuracy of the present numerical method. The investigation shows the effects of Rayleigh numbers on natural convection and heat transfer in both the free fluid and porous medium.

The Nuc on the side walls in the free flow and porous medium regions increase with the increase of Ra, and

convection plays an important role in the heat transfer with the increase of Ra. The mean temperature at the interface decreases and temperature gradients are created on the upper two corners in porous medium region with the increase of Ra.

Acknowledgements

This study is supported by the National Natural Science Foundation of China (No. 51276102), Shandong Provincial Natural Science Foundation of China (ZR2011EEM011) and the Key Project of Chinese Ministry of Education (No.211096). The authors also wish to express their appreciation to Shandong Provincial Key Laboratory of Building Energy-Saving Technique and Key Laboratory of Renewable Energy Utilization Technologies in Buildings of the National Education Ministry for providing the support for this study.

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