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Engineering

Procedía Engineering 121 (2015) 1193 - 1199 =

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 surface thermal radiation on natural convection and heat transfer in a cavity containing a horizontal porous layer

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

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

Abstract

In this paper, the effects of surface radiation on heat transfer and natural convection in a cavity containing a horizontal porous layer 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 thermal radiation on natural convection and heat transfer in both free fluid and porous medium were analyzed. It was found that surface thermal radiation 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. PublishedbyElsevierLtd.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; surface thermal radiation; cavity; porous medium

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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 [1-8]. The published literature also contains several studies of natural convection in cavities that are partially filled with porous medium, however, none appears to include the interaction of thermal radiation and natural convection [9-12]. Thermal radiation

E-mail address: wycj n1@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.2015.09.137

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, so it is important to investigate the interaction of surface thermal radiation with natural convection in a cavity. 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 the thermal physical parameters 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-(vv) = ^V2v-Vp - PogP{T - T0 ) ^

Th . Poca v -VT = kV T

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, P 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 "Vp " To • vd ^

Thermal energy: p c vD • VT = keffV2T (6)

in which v d 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, pis 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 = VD = 0 when X = 0, Wandy = 0, H .T = Th whgn x = q.T = Tc whgn x = w.

"y * VT = 0 when y = 0;Q + hly * VT = 0 when y = H . v = V d when 7 = Y-erface.

Q + kefn-Vt| = kn-VTl 7 = Y A = T\„ .. 7 = Y

^ eJJ I porous 1 fluid when interface. \ porous \ fluid when interface

Where ny 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.[10] deals with heat transfer in a cavity partially filled with a porous medium, but that excludes the effects of thermal radiation. Beckermann et al.[10] 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 = Cap2gfi(Th ~ Tc )H3 /(kp) , Pr is Prandtl number, Da is the Darcy number, Da = K /H2 and

K is the permeability of the porous medium. Rk is the ratio of thermal conductivities between the porous medium

and fluid and Rk = kf / k, 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. [10]

Test number Test fluid Ra Pr Da Rk S

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

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

A comparison between measured and predicted dimensionless temperatures by the present numerical model is shown in Figs. 1. It can be seen that agreement between the two sets of values is good, especially in the Fig. 1 a. 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 [10].

1.0-, 0.9 0.8 0.70.6 0.5-

0.40.3 0.2 -0.1 -0.0

-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—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

(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

i—|—.—|—.—|—.—|—.—|—.—|—.—|—.—|—.—|—.—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, To=(Th+Tc)/2, Rk = 5.31, Da= 5.78e-7 and S =0.0, 0.1,0.5,1.0.

In order to demonstrate the effect of surface radiation on the flow fields, the isotherm plots from the simulations for cases with and without radiation at several emissivities are presented in figure 2. Comparing with results without radiation, the results with radiation show that temperature rises at the interface and decreases at the top wall in the free flow region. Because of the natural convection and thermal radiation at the interface between the free flow and porous medium, the mean temperature in porous medium is higher than that without thermal radiation, and the mean temperature in the porous medium increases slightly with the increase of emissivity on the surfaces. The prediction in figure 2 also indicates that the appearance of surface thermal radiation can significantly shifts the line of the mean temperature (T=293.5K) towards the right in the porous medium region and change the temperature fields even if the given emissivity is quite small. However, the emissivity from 0.5 to 1.0, the changes are not as significant as that as emissivity increases from 0.0 to 0.5.

(a) S =0.0 (b) S =0.1

(c) S =0.5 (d) S =1.0

Fig. 2. the isotherms for the nonradiation and radiation with several emissivities (Ra =106, Rk=5.31, Da= 5.78e-7)

The effects of emissivity on mean convective Nusselt number are shown in Figs 3(a)-(c) for Ra=106,105and104, respectively. The general effect of surface emissivity on the mean convective Nusselt number, Nu„ , on the both hot

and cold side wall in the cavity is different between the free flow region and porous medium region. The Nuc on

the side wall in the porous medium region is much lower than that in the free flow region, and changes very little because of the weaker natural convection and lack of thermal radiation in the porous medium. The thermal radiation on the surfaces makes the temperature gradient near the hot side wall increase slightly in the free flow region, and thus rises the Nuc on the hot side slightly with the increase of emissivity. However, as radiative energy is

proportional to the fourth power of absolute temperature, the radiative heat exchange between the hot wall and the insulated (top) wall becomes stronger than that between the cold and insulated walls, the temperature gradients are

weakened near the hot wall but strengthened near the cold wall. Thus Nuc at the hot wall is lower than that at the

cold wall, and the temperature gradient near the cold side increase slightly in the free flow region with further increase of emissivity.

Effect on radiative Nusselt number due to variation of emissivity is shown in Fig. 3(d). As expected, Nur increases rapidly with the increase of emissivity. This indicates that surface radiation plays an important part in heat transfer in the cavity, especially at higher emissivity.

1.91.81.7 1.6 1.5 1.4 1.3 1.21.1 -1.00.90.80.7 0.6 0.5 0.4

- hot side wall in the free flow region

- hot side wall in the porous medium region cold side wall in the free flow

cold side wall in the porous medium region

—I-1-1-1-1-1-1-1-1-1-1-1

0.0 0.2 0.4 0.6 0.8 1.0

(a) Ra =10", R=5.31, Da= 5.78e-7

4.4 4.2 4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 ' 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

- hot side wall in the free flow region

- hot side wall in the porous medium region cold side wall in the free flow

cold side wall in the porous medium region

—I— 0.0

—I— 0.2

—I—

—I— 0.6

(b) Ra =105, R=5.31, Da= 5.78e-7

5.0-, 4.54.0 3.5 3.0 2.5-1

2.0 1.5 1.0 0.5-1

- hot side wall in the free flow region

- hot side wall in the porous medium region cold side wall in the free flow

cold side wall in the porous medium region

—I— 0.2

—I—

24 22 20 18 16 14 12 Î10 8 6 4 2 0

- hot side wall in free flow region,Ra=1e6

- hot side wall in free flow region,Ra=1e5 hot side wall in free flow region,Ra=1e4 cold side wall in free flow region,Ra=1e6

- cold side wall in free flow region,Ra=1e5 cold side wall in free flow region,Ra=1e4

—I-1-1—

0.4 0.6

(c)Ra =106, Ri=5.31, Da= 5.78e-7

(d) Ra=10", 105,106, R=5.31, Da= 5.78e-7

Fig. 3 The effect of surface emissivity on the mean convective and radiative Nusselt numbers with several Rayleigh numbers

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 thermal radiation on natural convection and heat transfer in both the free fluid and porous medium.

The surface thermal radiation alters the distribution of the temperature in the free flow and porous medium regions even if the given emissivity is quite small, but there are no remarkable changes with the increase of emissivity from 0.5 to 1.0. Comparing with nonradiation, the temperature rises at the interface and decreases at the top insulated wall, resulting in the temperature in porous medium being higher than that without thermal radiation, and the temperature in the porous medium increases slightly with the increase of emissivity on the surfaces.

Acknowledgements

This study is supported by the National Natural Science Foundation of China (No. 51276102), Shandong Provincial Natural Science Foundation of China (ZR2011EEM011). 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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