combined mixed convection and thermal radiation in a ... nadjib hamici.pdf · inclination angle and...

CFD & Tech 2016 02 – 03 Mai 2016, CRND-Draria, Alger

COMBINED MIXED CONVECTION AND THERMAL RADIATION IN

A SQUARE CAVITY WITH AN INSIDE INCLINED HEATER

N. Hamici, D. Sadaoui,A. Sahi

Laboratoire de Mécanique, Matériaux et Energétique (L2ME), Université de Bejaia, Algérie [email protected]

Abstract: The present paper analyses the inclination effect of a thin heated plate located

inside a square cavity on thermal field, fluid flow and heat transfer under mixed convection

conditions and with taking into count of thermal surface radiation. The finite volume method

is adopted to solve the mixed convection governing equations, and the surface to surface

method (S2S) is used to model radiative transfer in the grey, diffuse and opaque cavity walls

with no-participating medium. Flow parameters such as Grashof Number Gr and Richardson

number (Ri), and Geometrical parameters like heated plate inclination are varied to show their

effect on heat transfer enhancement, while the walls emissivity was kept constant.The

obtained results show that there is not a limited angle which boost heat transfer for all the

considered cases, but for every Gr and Ri value there is a specific inclination which gives the

more efficient rate of heat transfer, regardless from wall emissivity.

Keywords: Heat transfer, Mixed convection, Cavity, Thermal radiation, inclined plate.

I. INTRODUCTION

Mixed and natural convection in cavities filled with air, has been the subject of many

studies due to the importance of this phenomenon for facilitating effective heat transfer in

order to obtain a specified temperature, or a more efficient heat removal in many engineering

fields, such as cooling electronic modules encapsulated inside a cabinet, solar collectors,

thermal design of buildings, air conditioning and many others disciplines. The aim of heat

transfer studies is to choose the geometrical and flow parameters that give maximum

effectiveness for each specific case. A literature review on the subject shows that many

authors have considered numerous theoretical and experimental studies related to natural

convection and mixed convection in cavities containing internal partitions or heated plates.

One of the systematic numerical investigation in natural convection was directed by

Abdul hakeem (2011)-(2008), where the authors studied the effect of isothermal temperature

or isoflux boundary condition, and the effect of different locations of the heat generating

baffles inside a square cavity. They found that the type of boundary conditions and the

location of the baffles affects sorely the fluid flow, thermal field and the overall heat transfer

inside the cavity. They also concluded that baffle location depend on the applied thermal

boundary condition and for each case, there is an optimal baffle location. Kumar De (2006)

likewise studied effect of boundary conditions and the effect of heated cylinder location. He

observed that for the case where the heated body is placed at the center of the cavity with

isolated top and bottom walls, and cold sidewalls, the isothermal boundary condition gives a

higher heat transfer while the cylinder location does not play an important role.

Mahmoodi(2011) studied the length and orientation effect of a thin heater inside a cavity with

nanofluid as working fluid. His results showed that for low Rayleigh number heat transfer is

better for the horizontal heater while at high Rayleigh number the positions does not affect

heat transfer rate. He also found that increasing fin length leads to increase of heat transfer


rate. Waheb(2015) considered the effect of inclined partitions inside a rectangular closed

enclosure. They found that the maximum heat transfer rate occurs for an angle of 45°, which

is better than the non-partitioned case. Ben-Nakhi (2007) studied the case of a finned pipe

placed in the center of a square enclosure with constant internal heat generation; he concluded

that the temperature and the stream function can be controlled through the finned pipe

inclination angle and fins length.The effect of combined natural convection and thermal

radiation in a square cavity with heated plate was studied by Sarvanan(2010) .It was found

that radiation homogenize the temperature fields and makes the whole cavity thermally active.

He also found that the effect of emissivity is more perceptible ah high Rayleigh numbers. In

the case of mixed convection with an inside heat source, many studies directed by Rahman

(2008), Saeidi (2006) and Shahi (2010), considered the effect of cavity inlet and outlet

position on heat transfer rate and concluded that these last depend of the heat source

emplacement, dimension and orientation. They also conclude that the fluid flow and thermal

field is still dependent of Reynolds number, Prandtl number and other adimensional

governing numbers. For the case of combined mixed convection and thermal radiation in

rectangular cavity, Bahlaoui (2009) studied several effects for thermal enhancing in

rectangular cavity with isolated partition at the bottom. They conclude that the presence of

radiation reduces the quality of convective heat transfer (Convective Nusselt number) and

increases radiative heat transfer (radiative Nusselt) .He also found that as emissivity of heat

source increased, the cavity average temperature decreased. Balaji (2006)studied mixed

convection with surface radiation from a horizontal channel with protruding heat sources. The

results showed that the mean temperature inside the channel decreased as the Reynolds

number increased. He also notices that the effect of radiation interaction decreased as

Reynolds number increased.

Natural and mixed convection, both conjugated with surface radiation, for multiple

geometrical parameters and flow parameters, have been investigated by many researchers.

Despite the number of investigations carried out, informations about conjugated mixed

convection and surface radiation in vented square cavity containing an inside thin heater, are

still scarce. Therefore, the essential objective of this work is to examine the effects of thin

orientation and flow parameters on the overall heat transfer and fluid flow inside the

cavity.The results of this study are appropriate for geometric optimization and for advancing

practical architecture of cooling enhancement technique of ventilated enclosures using finned

surfaces.

II. PROBLEM STATEMENT A schematic view of the square cavity with boundary conditions, together with the

system of coordinates is depicted in Figure. 1. The height and the width of the cavity are

denoted H. The length of the cavity perpendicular to its plane is assumed to be long. Hence,

the problem is considered to be two-dimensional. The top and bottom wall are kept insulated

whereas the sidewalls are maintained at a cold temperature Tc. A thin finned plate with a

length h, an inclination φ and an isothermal temperature hot temperature TH>Tc is located at

the center of the cavity. The system is submitted to an incoming flow with a uniform velocity

U0, with an ambient temperature Tc. The inlet opening is located on the bottom of the left

vertical wall, whereas the outlet opening is located on the top of the opposite side wall, the

both with a size w. Air is considered as the working fluid and is assumed to be a non-

participating medium. The cavity walls and plate surfaces are assumed opaque, grey and

diffuse emitters and reflectors of radiation with an emissivity ɛ. The flowis assumedto be

laminar, incompressible and steady state. The thermophysical proprieties of the fluid at a


mean temperature is assumed to be constant, except for the density variation in the buoyancy

force terms which is determined using the Boussinesq approximation.

Figure 1. Schematic of the studied configuration

In the light of assumptionsmentioned above, the non-dimensional continuity, Navier-Stokes

and energy equations are written as follows:

𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦=0 (1)

𝑢𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= −

𝜕𝑝

𝜕𝑥+

1

Re 𝜕²𝑢

𝜕𝑥 ²+

𝜕²𝑢

𝜕𝑦 ² (2)

𝑢𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦= −

𝜕𝑝

𝜕𝑦+

1

Re 𝜕2𝑣

𝜕𝑥2+

𝜕2𝑣

𝜕𝑦2 + 𝑅𝑖𝜃 (3)

𝑢𝜕𝜃

𝜕𝑥+ 𝑣

𝜕𝜃

𝜕𝑦=

1

Re Pr 𝜕²𝜃

𝜕𝑥 ²+

𝜕²𝜃

𝜕𝑦 ² (4)

The dimensionless variable in the above equations (1-4) are defined as:

𝑥 =𝑋

𝐻 , 𝑦 =

𝑌

𝐻 , 𝑢 =

𝑈

𝑢0 , 𝑣 =

𝑉

𝑢0 , 𝑝 =

𝑃

𝜌𝑢02 , 𝜃 =

𝑇−𝑇𝑐

𝑇𝐻−𝑇𝑐 ,

𝑃𝑟 =𝜐

𝛼

The variable mentioned in the previous equations (1-4), represents: (X,Y) Dimensional

Cartesian coordinates (m), (x,y)Dimensionless Cartesian coordinates,(U,V) Dimensional X

and Y velocity respectively (m.s-1

), (u,v)Dimensionless xand y velocity component

respectively,P dimensional pressure,pdimensionless pressure, θ dimensionless temperature, ρ

the density of the fluid (kg.m-3

), α thermal diffusivity of the fluid, β Thermal expansion

(5)


coefficient (K-1

), υ Kinematic viscosity diffusivity of the fluid (m².s-1

) and Prthe Prandtl

number which is dimensionless.

Mixed convection is identified by three dimensionless parameters, namely the Grashof

number Gr, the Reynolds number Re, and the Richardson number Riexpressed as:

𝐺𝑟 =𝑔𝛽𝛥𝑇 𝐻3

𝜐2, 𝑅𝑒 =

𝑢0𝐻

𝜐 ,𝑅𝑖 =

𝐺𝑟

𝑅𝑒 2 (6)

The Richardson number Ri is used to express the importance of natural convection relative to

forced convection. It may be noted that the flow and heat transfer is dominated by forced

convection when Ri≤1, it is dominated by natural convection for Ri≥1, and it is mixed regime

as Ri is of the order of 1.

II.1 Boundary conditions

The boundary conditions applied to the computational domain are as follows:

at the inlet of the cavity

x=0 and 0 ≤ y ≤ w : u=1, v=0, θ=0

at the left side wall and right side wall respectively

(x=0 ,w ≤ y ≤ 1) and ( x=1, 0 ≤ y ≤ 1-w) :u=0, v=0, θ=0

at the bottom wall

y=0 and 0 < x <1: u=v=0 and𝜕𝜃

𝜕𝑦= 𝑁𝑟𝑄𝑟

at the top wall

y=1 and 0 < x <1: u=v=0 and𝜕𝜃

𝜕𝑦= −𝑁𝑟𝑄𝑟

at the heated plate

u=0, v=0, θ=1

at the outlet

x=1 and (1-w < y < 1):𝜕𝑢

𝜕𝑥= 0 ,

𝜕𝜃

𝜕𝑥= 0

II.2 Radiation formulation

In the present work, there is interaction between convection and radiation in an

enclosure containing transparent medium, calculation of radiative heat transfer is carried out

using the radiosity formulation. The walls of the cavity and the plate are divided into N

number of radiative surface elements. Each segment is sufficiently short to be considered

isothermal. The view factor Fij is defined as the fraction radiation leaving the area ithat

reaches the area j which is determined by Hottel’s(1967) crossed string method. The

dimensionless radiosity Ri for a surface element Aiis obtained by resolving the system:

𝛿𝑖𝑗 − 1 − ɛ𝑖 𝐹𝑖𝑗 𝑅𝑗 = ɛ𝑖𝑁𝑗=1 𝑖

4 (7)

The dimensionless net radiative heat flux along Siis given by:

𝑄𝑟 ,𝑖 = 𝑅𝑖 − 𝑅𝑗𝐹𝑖𝑗𝑁𝑗=1 (8)

The variable used in the previous equations are defined by: j dimensional radiosity (W.m-2

),

R =j/σT4Dimensionless radiosity,NrDimensionless net radiation number 4

r hN T ( T / H)


qrDimensional net radiative flux density (W.m-2

), Qr=qr/σT4 Dimensionless net radiative heat

flux,σStefan-Boltzmann constant, σ=5.67x10-8

(W.m-2

.K-4

), δijdelta Kronecker andΘ = T/TH

the dimensionless temperature for radiation formulation.

II.3. Heat transfer

To identify the most efficient cooling in the different studied configurations, it is

necessary to define Nusselt number on the heated plate. For the present study, contributions of

both convection and radiation to heat transfer should be considered. Hence, the mean total

Nusselt number (Nu) at the walls, is defined as the sum of the corresponding convective and

radiative Nusselt numbers, respectively (Nuc) and (Nur).

𝑁𝑢𝑇 = 𝑁𝑢𝐶 + 𝑁𝑢𝑅 = 𝜕𝜃

𝜕𝑋𝑑𝑌

𝑊𝑎𝑙𝑙+ 𝑁𝑟𝑄𝑟𝑑𝑌𝑊𝑎𝑙𝑙

(9)

III.Method of solution

The numerical solution of the continuity, momentum and energy equations (1)-(4) with

the corresponding boundary conditions was obtained by using the finite volume method.The

computational domain has been discretized with a regular and uniform space grid in the

Cartesian directions. The SIMPLE algorithm was used to couple momentum and continuity

equations with Power Law discretization for the convective-diffusive terms, while the

advective terms are discretized using second-order central difference scheme. The

convergence criterion was set as 10-6

for all dependent variables.

III.1 Code validation

In order to verify the accuracy of the present work, it is crucial to compare our results

with those previously reported in the literature. The numerical code was validated against the

results of A. Bahlaoui et al. (2009) obtained in the case of mixed convection and surface

radiation in a horizontal ventilated heated cavity with an adiabatic thin partition on the heated

surface. Comparisons, made in terms of total Nusselt number, evaluated at the heated wall for

various wall emissivity and various Reynolds number, are presented in Figure. 2. As shown, a

good agreement was obtained with a relative maximum deviation limited to 1,43 % for an

emissivity ɛ=0.85 and Re=3000, providing sufficient confidence in the present computations.


Figure 2. Total Nusselt number comparison with A. Bahlaoui et al. (2009)

III.2 Grid independence study

In order to have a grid independent solution and a good compromise between accuracy

and computational time, a grid testing was performed using different grid density (45 x 45 to

245 x 245) and for different flow configurations. The variation of the average total Nusselt

number on the heated plate is selected as a sensitivity measure for the grid independence test.

Figure. 3 illustrate grid dependence study for Gr =105, Re =750, ɛ = 0,8 and φ = 0° . It can be

clearly seen that Nusselt number remained almost the same for grid finer than 165x165. Thus,

the numerical results presented in this study were conducted with the 165x165 grid system.


Figure3. Grid dependency (Gr =105, Re =750, ɛ = 0,8 , φ = 0° )

IV. RESULTS AND DISCUSSION

As stated earlier, the main objective of the current study is to investigate the

characteristics of flow and heat transfer in open cavity submitted to the interaction of mixed

convection and radiation in presence of an inclined thin plate. The computations were carried

out for Pr=0.71. The characteristic temperature difference ΔT and the average temperature of

air T0were chosen to be 20K and 303.15K respectively. The results have been obtained for

Grashof number ranging from 103 to 10

6, various Reynolds numbers and plate inclination

from 0 to 160°.All the cavity walls were considered to have the same emissivity ɛ that was

fixed to ɛ=0,7 for our case.The results are discussed in term of isotherms and streamlines plots

and Nusselt numbers.

The effect of thin heater tilt angle on isotherms and streamlines for Gr =103, Re=750

and ɛ=0.7 are shown in Figure. 4. At first glance, it can be seen that the variation of

inclination angle affects sorely the flow and temperature patterns. The corresponding

isotherms are denser at the vicinity of the heated plate, whereas a thermal plume appeared and

raised from the top surface of the fin towards the insulated top wall, suggesting a better

convective heat exchange


φ = 0° φ = 20° φ = 45° φ = 65° φ = 90°

φ = 110° φ = 135° φ = 160°

Figure 4.Isotherms and streamlines for Gr=103, ɛ=0.7, Re=750 with various inclination angle.

with the surrounding fluid and with the cold walls. A path is created among the isotherms by

the entering fluid that travels from the inlet then mixes with air within the cavity and finally

exits through the outlet. A temperature gradient is also remarked at the interface of the

incoming flow and the primary convection cell (surrounding the heated plate), which leads to

heightened heat transfer with the entering cold air. When the inclination of the plate is

increased (φ > 20°), the isotherms located under the heated plate are less tightened started too

be distorted while the upper thermal plume became larger, indicating the predominance of

convective heart transfer. For inclinations φ > 90°, the thermal plume localized on the plate

dissipated to reappear for an angle of φ > 160°. Streamlines are presented in the second line

of Figure. 4. The cold air entered into the cavity through the inlet and expanded in the inside

as result of pressure rise. Inter alia it can be noticed a close spacing between the streamlines in

the vicinity of the incoming flow from the inlet to the outlet indicating high velocities. It can

also be seen the development of two major cells, extracting heat from the two sides of the

thin, while two insignificant cells appeared at the downright an up left corners of the cavity.

As the plate angle increases, the recirculating cell above the heated plate enlarged and shifted

to the upper right section of the cavity, whereas the below cell shrank gradually to dissipate

when the plate became in vertical position (φ = 90°). For φ = 110°, the two previous cells

merged to create on major cell surrounding the plate, and then split again for superior

inclinations. The same behavior as before was observed for the left upper small cell, which

grew proportionally to the increase of the tilt angle until φ = 90°, then began shrinking again.


φ = 0° φ = 20° φ = 45° φ = 65° φ = 90°

φ = 110° φ = 135° φ = 160°

Figure 5.Isotherms and streamlines for Gr=106, ɛ=0.7, Re=750 with various inclination angle.

For higher values of Grashof Number (Figure. 5), the isotherms appearance change

thoroughly indicating an increase in buoyancy mechanism effect. It is accompanied by a

notable distortion in temperature contours and an enlargement of the thermal plume above the

heated plate. Furthermore, a better thermal stratification at both sides of the fin was observed

leading to an enhanced convective heat transfer. It can also be noticed that the thermal

stratification in the lower part of the fin became more visible as the inclination angle

increased until φ = 110° , then gradually decreased for φ ≥ 135°. The Streamlines presented in

the second line of Figure.5 transformed to multicellular flow as Grashof number increased,

which resulted from the augmentation of the buoyancy mechanism. The couple of cell located

above the heated plate gradually weakened while the lower located cell gradually spread for

the augmentation of the inclination angle until φ = 90°, then progressively regained its

original size for superior angles. Furthermore, it can be noticed that forφ > 90°, the incoming

flow divide the cavity in two principal rotating cells.


a b c

Figure 6.Nusselt variation of heated plate NuH and left cold wall Nuc For Gr=103,ɛ=0.7 and

Ri=0.1 (a),Ri=1 (b), Ri=10 (c).

a b c

Figure 7.Nusselt variation of heated plate NuH and left cold wall NuC For Gr=105,ɛ=0.7 and

Ri=0.1 (a),Ri=1 (b), Ri=10 (c).

IV.1 Heat transfer rating

The Streamlines and isotherms presented previously (Figure. 3 and 4), do not provide

information about the effect of the studied parameters on the quality of heat transfer within

the cavity. Therefore, the variation of convective, radiative and total Nusselt number for

several configurations is illustrated in Figure 6-7. For low Grashof number i.e Gr=103, It is

noted that the plate inclination has no significant effect on heat transfer rate for the considered

flow parameters, since the convective, radiative and total Nusselt number remain nearly

constant at the heated plate and the left cold wall for each Richardson number respectively.

For Higher Grashof number i.e Gr=105, the behavior is different from the precedent case. It

can be seen from Figure. 7 that the convective and total Nusselt number varied with heater

inclination and with heat transfer mode (Richardson Number). For Ri=0.1 and for a heater tilt

angle of 45°, the convective and total Nusselt numbers at the heated plate are higher in

comparison with other orientation, while the radiative Nusselt number stayed nearly constant.

For Ri=1 and Ri=10, it is seen from Figure. 7 b-c that orientations of 90° and 110° favored

heat transfer since the convective and total Nusselt number for the heated plate and for the left

cold wall are higher than those corresponding to other inclinations. On the other hand, the

values of the average convective Nusselt number were always bigger than the average

radiative Nusselt number. This fact indicates convection heat transfer is predominant

compared to radiative heat transfer, but the contribution of the latter is not negligible (40% of

total heat transfer).


V. CONCLUSION

The problem of combined mixed convection with radiation in a square vented cavity with an

inside inclined thin heater has been studied numerically. The effects of inclination angle,

Grashof Number and Richardson number on flow and heat transfer has been investigated.

Based on the obtained results, the conclusions of this work are the following:

The Temperature field and the flow pattern are very influenced by thin orientation

angle.

For low Grashof numbers, the variation of the inclination angle has no compelling effect

on enhancing heat transfer and that for any Richardson value considered, however the

behavior changed for higher value of Grashof number as there is an optimal inclination,

which enhance heat transfer.

Convective heat transfer is more prevalent than radiative heat transfer in the present

study. Furthermore, fin inclination has no significant influence on Radiative Nusselt

number for all the considered Grashof values.

There is not an inclination angle that improves heat transfer for all flow and thermal

configuration, but for every case there is an orientation that gives best thermal

performance.

VI. REFERENCES

1. Abdul Hakeem .A.K, Saravanan. S and Kandaswamy. P, 2011, Natural convection in a

square cavity due to thermally active plates for different boundary conditions,Computers

and Mathematics with Applications, N°62, Pages 491–496.

2. Saravanan. S, Abdul Hakeem A. K, Kandaswamy. Pand Lee.J, 2008, Buoyancy

convection in a cavity with mutually orthogonal heated plates,Computers and

Mathematics with Applications, N°55, Pages 2903–2912.

3. Kumar De. A and Dalal A, 2006, A numerical study of natural convection around a

square, horizontal, heated cylinder placed in an enclosure,International Journal of Heat

and Mass Transfer, N° 49, Pages 4608–4623.

4. waheebM., 2011, Numerical simulation of free convection of nanofluid in a square cavity

with an inside heater,International Journal of Thermal Sciences, N° 50, Pages 2161–

2175.

5. Waheb J. H, Jalil S. M and Ibrahim .M, 2015, The Natural Convective Heat Transfer in

Rectangular Enclosure Containing Two Inclined Partitions, American association for

science and technology Journal, N° 2, Pages 36–43.

6. Ben-Nakhi .A and Chamkha A. J , 2007, Conjugate natural convection around a finned

pipe in a square enclosure with internal heat generation,International Journal of Heat and

Mass Transfer, N° 50, pages 2260–2271.

7. Saravanan .S and Sivaraj .C, 2010 , Coupled thermal radiation and natural convection heat

transfer in a cavity with a heated plate inside,International Journal of Heat and Fluid

Flow, N° 40, pages 54–64.

8. Rahman .M, Alim M. A, and Saha .S, 2008, Heat-Conducting Horizontal Square

Cylinder,Suranaree Journal of Science and Technology, N°17, Pages 139–153.

9. Saeidi S. M, and. Khodadadi J. M , 2006 , Forced convection in a square cavity with inlet

and outlet ports,International Journal of Heat and Mass Transfer, N° 49, pages 1896–

1906.


10. A. H, Shahi .M, and Talebi .F, 2010,Effect of inlet and outlet location on the mixed

convective cooling inside the ventilated cavity subjected to an external

nanofluid,International Communication in Heat and Mass Transfer, N° 37, Pages 1158–

1173.

11. Bahlaoui .A, Raji .A, Hasnaoui .M, 2009, Mixed convection cooling combined with

surface radiation in a partitioned rectangular cavity,Energy Conversion Management, N°

50, Pages 626–635.

12. Premachandran .B and Balaji .C,2006,Conjugate mixed convection with surface

radiation from a horizontal channel with protruding heat sources,International Journal of

Heat Mass Transfer, N° 49, Pages 3568–3582.

13. H. C. Hottel, A.F. Saroffim, (1967) Radiative Heat Transfer, McGraw Hill, New York.

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