Chimney effect in a "T" form cavity with heated isothermal blocks: The blocks height effect

Chimney effect in a ‘‘T’’ form cavity with heated isothermal blocks: The blocks height effect

M. El Alami

^a

, M. Najam

^a,*

, E. Semma

^b

, A. Oubarra

^a

, F. Penot

^c

aGroupe Energetique, Departement de Physique, Facultedes Sciences, UniversiteHassan II Ain Chock, Km 8 route d’El Jadida, Maarif, BP 5366 Casablanca, Maroc

bFST, UniversiteHassan I^er, Settat, Maroc

cLET-ENSMA, UMR CNRS 6608, BP 40109, 86961 Futuroscope Cedex, France Received 14 November 2003; accepted 9 January 2004

Available online 12 March 2004

Abstract

In this paper, a numerical study of natural convection from a two dimensional ‘‘T’’ form cavity with rectangular heated blocks is conducted. The blocks are identical, and the domain presents a symmetry with respect to a vertical axis passing through the middle of the opening. The governing equations are solved using a control volume method, and the SIMPLER algorithm for the velocity–pressure coupling is em- ployed. Special emphasis is given to detail the effect of Rayleigh number and block height on the heat transfer and the flow rate generated by the chimney effect. The results are given for the parameters of control as, 10⁴6Ra6310⁶, Pr¼0:71, opening diameter (C¼l⁰=H⁰¼0:15), blocks gap (D¼d⁰=H⁰¼ 0:5) and blocks height (1=86B¼h⁰=H⁰61=2). These results show that the heat transfer variation withRa is in the same manner as those met in the case of the vertical smooth or ribbed channels.

2004 Elsevier Ltd. All rights reserved.

Keywords:Heat transfer; Chimney effect; ‘‘T’’ form cavity; Isothermal blocks

1. Introduction

Two dimensional channels formed by parallel plates are a frequently encountered configuration in convection air cooling of electronic equipment, ranging from transformers to main frame computers and from transistors to power supplies. Packaging constraints and electronic consid- erations, as well as devices or system operating modes, lead to a wide variety of heat dissipation

www.elsevier.com/locate/enconman

*Corresponding author. Fax: +212-22-23-06-74.

E-mail address:mnejam@yahoo.fr(M. Najam).

0196-8904/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.01.012

profiles along the channel walls. Many kinds of thermal wall conditions are proposed to yield approximate conditions in the prediction of the thermal performance of such configurations [1].

Along the same lines, a numerical study of natural convection was conducted by Penot et al. [2].

The authors considered a vertical channel, which simulates a chimney, placed in a closed and differentially heated cavity. The chimney walls were capped isotherms. Their results show that the chimney effect can sometimes be absent. A numerical investigation of free convection in a vertical

Nomenclature

B dimensionless block height (h

⁰=H⁰

) C dimensionless opening diameter (l

⁰=H⁰

) d

⁰

space between adjacent blocks

D dimensionless space between adjacent blocks (d

⁰=H⁰

) h

⁰

block height

H

⁰

channel height l

⁰

opening diameter

L

⁰

length of calculation domain (cavity) M mass flow rate (Eq. (5))

n normal coordinate

Nu local Nusselt number (Eq. (6)) N u mean Nusselt number (Eq. (7)) Pr Prandtl number (Pr

¼m=a)

Ra Rayleigh number (Ra

¼

gbDTH

⁰³

/(am)) T

⁰

temperature of fluid

T

_H⁰

imposed temperature on blocks T

_C⁰

temperature of cold surface

T dimensionless temperature of fluid (T

¼ ðT⁰

T

_C⁰Þ=ðT_H⁰

T

_C⁰Þ)

U

⁰

, V

⁰

velocities in x

⁰

- and y

⁰

-direction

U , V dimensionless velocities in x- and y-direction [

¼ ðU⁰;

V

⁰Þ

H

⁰=a]

x

⁰

, y

⁰

Cartesian coordinates

x, y dimensionless Cartesian coordinates [

¼ ðx⁰;

y

⁰Þ=H⁰

]

a

thermal diffusivity

b

volumetric coefficient of thermal expansion

k

thermal conductivity of fluid

m

kinematic viscosity of fluid

q

fluid density

W

dimensionless stream function

Subscripts

C critical value, cold wall H heated wall

max maximum

isothermal channel was conducted by Desrayaud and Fichera [3]. Two rectangular blocks are symmetrically mounted on the channel surfaces, and two cases were investigated. The blocks were capped isotherms or insulated. It was found that the heat transfer in the case of isothermal ribs is more important than the heat transfer generated by the channel with adiabatic ribs. An experi- mental and numerical investigation about the effect of the position of wall mounted rectangular blocks on the heat transfer, taking into account the angular displacement of the block, was conducted by Bilen et al. [4]. The experiments were conducted in a rectangular horizontal channel.

The results showed that the most efficient parameters were Reynolds number and angular dis- position. Murakami and Mikic [5] presented an optimization study using a method of determining optimum values of the channel diameter, flow rate and number of channels for minimum pressure drop. Various strategies were developed for enhancing the heat transfer, including placement of an obstacle in the flow path of the coolant to destabilize the flow [6], using openings between blocks in recirculating movements spaces [7,8] or variable space length between two adjacent blocks [9].

These strategies, certainly, enhanced the heat transfer between the blocks, but in the case of a horizontal channel submitted to a convective horizontal jet, the vertical planes of the blocks do not appear well ventilated. Hence, Najam et al. [10,11] direct the jet perpendicularly to the hor- izontal channel with rectangular blocks on its lower plane. Because of the problem periodicity, rigid adiabatic planes were introduced to suppress an eventual exchange of heat and mass between two contiguous modules. The calculation domain is a ‘‘T’’ form cavity. Their results showed that the heat transfer from the cavity is maximum when the natural convection regime dominates (cases of lower Re). These results are also mentioned by Hung [12]. The chimney effect in such configurations cannot be neglected at low values of Reynolds number. In this paper, our objective is to examine the natural convection effect in the ‘‘T’’ form cavity studied by Najam et al. [13].

Thus, we consider the flow rate at the inlet as a fundamental unknown of the problem, and we will determine it as a function of the control parameters.

2. Physical problem and governing equations

A schematic representation of the configuration is depicted in Fig. 1(a). Because of the problem periodicity, the studied domain is reduced to a ‘‘T’’ form cavity (Fig. 1(b)). The blocks are heated with constant temperature. The upper wall is cold, while the other sides of the cavity are insu- lated.

The flow is considered steady, laminar and incompressible, and the Boussinesq approximation has been applied. The thermophysical quantities are assumed to be constant except for the density in the buoyancy force. With these assumptions, the dimensionless governing equations in terms of pressure P , velocities U and V and temperature T can be written as:

oU ox þoV

oy ¼

0

ð1Þ

oU

ot þ

U

oU

ox þ

V

oU

oy ¼ oP

ox þ

Pr

o²

U

ox²

þo²

U

oy²

ð2Þ

oV

ot þ

U

oV

ox þ

V

oV

oy ¼ oP

oy þ

Pr

o²

V

ox²

þo²

V

oy²

þ

Pr Ra T

ð3Þ

oT

ot þ

U

oT

ox þ

V

oT oy ¼o²

T

ox² þo²

T

oy² ð4Þ

Referring to Fig. 1, the dimensionless variables are:

x

¼

x

⁰

H

⁰;

y

¼

y

⁰

H

⁰;

U

¼

U

⁰

H

⁰

a ;

V

¼

V

⁰

H

⁰

a ;

T

¼

T

⁰

T

_C⁰

T

_H⁰

T

_C⁰;

P

¼ðP⁰þqgy⁰Þ qa²

Ra

¼

gbDTH

⁰³

am ;

Pr

¼m

a

with

ðDT ¼

T

_H⁰

T

_C⁰Þ

The boundary conditions associated with the system of Eqs. (1)–(4) are given below:

T

¼

1; U

¼

V

¼

0 (on the blocks surfaces), T

¼

U

¼

0,

oV

oy ¼

0, P

¼

M

²

2 (at the inlet opening).

The mass flow rate

M

is calculated at the inlet opening as:

M

¼ Z 0:575

0:425

V dx

y¼0

ð5Þ

At the evacuation opening, U , V , T are extrapolated by adopting similar processes as shown in Refs. [14,15] (their second spatial derivative terms in the vertical direction are equal to zero).

L’=H’ l’

Cavity H’

Micro cavity h’

openings

d’

(a)

T=0

Air aspired (b)

x’, U’

y’, V’

T=1

Adiabatic walls g

l’

Fig. 1. (a) Model of two dimensional channel with openings; (b) Calculation domain.

For reasonably high values of Ra (

P

5

·

10

³

), any region outside the cavity can be neglected, the upstream thermal diffusion through the inlet being negligible. Thus, because the fluid is not pre-heated before entering, the computational domain can be restricted to the cavity.

The local Nusselt number over the block surfaces (active walls) is calculated as:

Nu

¼ oT on

n

with n

¼ ðx;

yÞ

ð6Þ

The mean Nusselt number over the active walls is:

N u

¼ Z B

0

oT ox

x¼0:25

dy

þ Z 0:25

0

oT oy

y¼B

dx

þ Z B

0

oT ox

x¼0:75

dy

þ Z 1

0:75

oT oy

y¼B

dx

ð7Þ

3. Numerical method

For this problem, the governing equations were solved numerically using a control volume method [16]. An upwind scheme was used for the discretization of all convective terms in the vorticity and energy equations (respectively, Eqs. (2)–(4)). The final discretized forms of Eqs. (1)–

(4) were solved by using the SIMPLER algorithm. As a result of a grid independence study, a grid size of 81

·

81 was found to model accurately the flow fields described in the corresponding results.

The time steps considered ranged between 10

⁵

and 10

⁴

. The accuracy of the numerical model was verified by comparing the results from the present study with those obtained by De Val Devis [17]

for natural convection in a differential heated cavity and then with the results obtained by Des- rayaud and Fichera [3] in a vertical channel with two ribs, symmetrically placed on the channel walls. Good agreement was obtained in the

Wmax

and Nu terms. When a steady state is reached, all the energy furnished by the hot walls to the fluid must leave the cavity through the cold surface (with the opening). This energy balance was verified within less than 3% in all cases considered here.

4. Results and discussion

The objective of this investigation is to analyze the chimney effect in a ‘‘T’’ shaped cavity. The flow rate at the inlet is an unknown of the problem.

Heat transfer rates across the cold and hot walls, and the flow and thermal fields are examined for the range of Rayleigh number as: 5 10

³6

Ra

6

3 10

⁶

) and other parameters of the problem (D

¼

0:5, C

¼

0:15, A

¼

1 and Pr

¼

0:71). Streamlines and isotherms are presented for B

¼

0:5, 0.25 and 0.125 and various Ra.

4.1. Flow and thermal fields 4.1.1. Case of

B

¼1=2

In this case and when the Rayleigh number is less than 10

³

, the heat transfer is essentially by

conduction. The streamlines and isotherms presented in Fig. 2(a) for Ra

¼

10

⁴

show that the

solution is symmetric with respect to the vertical axis passing through the middle of the openings.

Air aspired by the chimney effect at the inlet opening is canalized around this axis. The intensity of the jet increases with increasing Ra. Consequently, two recirculating cells appear inside the jet as shown in Fig. 2(b) for Ra

¼

10

⁵

. The corresponding thermal distribution shows that the evacuated heat transfer is more important than in the last case: the isotherms are too tight near the vertical planes of the blocks, and so, the thermal gradient is important in these zones. The flow structure is the same as that of the boundary layer on a vertical plane placed in a semi-infinite domain. We note that the isotherms become practically horizontal in the cell zones. The air is stratified in this area of the domain. By increasing Ra more and more, the cell sizes increase and constrain the fresh aspired air to go along the active walls. Consequently, the vertical and horizontal planes of the blocks are more ventilated (Fig. 2(c)) Ra

¼

5

·

10

⁵

. The flow is similar to a boundary layer on a vertical plane. In this case, small recirculating cells appear at the inlet. The flow coming into the cavity vertically through the inlet opening becomes more and more powerful with increasing Ra.

So, to return to the boundary layer, near the vertical walls where the pressure drop is important, it creates a recirculation flow just after entering the cavity. At high Rayleigh number, the jet be- comes sensitive to the Coanda effect. Hence, the symmetry of the problem solutions is destroyed (Fig. 2(d)) for Ra

¼

10

⁶

, and novel solutions that are very sensitive to the initial conditions and to the roundness of the computer emerge. The jet has a tendency to get closer to the vertical plane in the left. We can note that there is an important thermal gradient near this plane, as shown by the isotherms that are very tight. At Ra

¼

3

·

10

⁶

, oscillations appear, and the problem solutions are unsteady.

4.1.2. Cases of

B

¼1=4 and 1/8

In these cases, the block height B is reduced, and the flow domain is increased. For B

¼

1=4, the flow structure and thermal field are presented in Fig. 3(a)–(d), respectively, for Ra

¼

10

⁴

, 10

⁵

and 5

·

10

⁵

. For Ra

¼

10

⁴

, Fig. 3(a), the stream lines are open lines which represent the air jet, aspired by the chimney effect, and recirculating cells inside the jet. The problem solution is symmetric. The

(a) Ra=10⁴

(b) Ra=10⁵

Ra=10⁶ (c)

Ra=5x10⁵

(d)

Fig. 2. Streamlines and isotherms case ofB¼0:50; (a)Ra¼10⁴, (b)Ra¼10⁵, (c)Ra¼5·10⁵, (d)Ra¼10⁶.

isotherm lines show that the heat transfer is weak near the horizontal planes of the blocks, however, it is important near the vertical ones. The cell sizes increase with increasing Ra and constrain the jet of fresh air (open lines) to deviate near the adiabatic walls as shown in Fig. 3(b).

We can note that there is good heat transfer in the vertical direction from the horizontal heated planes. Along the vertical planes, the heat transfer is more important, as shown by the isotherms, and the flow is a boundary layer on a vertical plane kind. The zone around the middle of the cavity is more stratified than the previous case. The symmetry of the solution is conserved. For Ra

¼

6

·

10

⁵

, this symmetry is destroyed, but for this value of Ra, the solution is always steady.

The isotherms show that all the active walls of the blocks are well ventilated. For Ra up to 7

·

10

⁵

, the problem solution is non-stationary and numerical oscillations are reached. For B

¼

1=8, The major remark is that the recirculating cells appear earlier than in the previous cases as shown in Fig. 4(a) for Ra

¼

5

·

10

³

. It is the same for the oscillations that appear at Ra

¼

5

·

10

⁵

. The cell sizes are important in this case, Fig. 4(b) for Ra

¼

5

·

10

⁴

. Consequently, the fresh air aspired by natural convection is constrained to go near the active walls (horizontal and vertical planes of the block), and the heat transfer is very important through the block surfaces, as shown by the isotherms in Fig. 4(b) and (c). The solution given in Fig. 4(c) for Ra

¼

4

·

10

⁵

is the limit of the steady state solutions.

(a) Ra=10⁴

(b) Ra=10⁵

(c) Ra=5x10⁵

Fig. 3. Streamlines and isotherms case ofB¼0:25; (a)Ra¼10⁴, (b)Ra¼10⁵, (c)Ra¼5·10⁵.

Concerning the flow structure, we can note that the recirculating cells appear as earlier the block height is smaller. This is due to the increase of the ‘‘T’’ form cavity free zone when reducing B.

4.2. Heat transfer

The relationships between mean Nusselt number Nu and Rayleigh number Ra for various values of B are shown in Fig. 5 in which the heat quantity leaving the active planes of the blocks are presented. Generally, we can note that Nu increases with Ra. The Ra increase leads to the appearance of boundary layer flows along the vertical planes of the blocks, and so, there is an important temperature gradient in these zones. The Nu variations with Ra may be correlated as follow:

Nu

¼

0:56 Ra

^0:25

for B

¼

0:50 with 10

³6

Ra

6

10

⁶

Nu

¼

0:34 Ra

^0:25

for B

¼

0:25 with 10

³6

Ra

6

6 10

⁵

Nu

¼

0:29 Ra

^0:25

for B

¼

0:125 with 10

³6

Ra

6

4 10

⁵

(a) Ra=5x10³

(b) Ra=5x10⁴

(c) Ra=4x10⁵

Fig. 4. Streamlines and isotherms case ofB¼0:125; (a)Ra¼5·10³, (b)Ra¼5·10⁴, (c)Ra¼4·10⁵.

The maximum deviation of these correlations is less than 3%. We note that this kind of correlation is presented in the cases of steady state natural convection in smooth and ribbed vertical channels [18,19]. On the other hand, Nu decreases when reducing the block height B. The B reduction leads to reductions of the vertical active surfaces of the blocks. ItÕs important to note that the Nu correlations exponents are equal to 0.25. These correlations are obtained in the cases of vertical isothermal plates [20].

4.3. Mass flow rate

An important outcome of the computations is the rate of induced mass flow through the cavity.

In Fig. 6, we present the flow rate variation with Ra for various values of B. We note that this flow rate is calculated at the inlet and the outlet openings with a maximum deviation less than 0.1%. As for the heat transfer, the mass flow rate increases with Ra and B. The most important remark is that the gap between the different curves greatly decreases when Ra increases. These curves intersect for Ra

¼

3

·

10

⁵

and 6

·

10

⁵

, respectively, in the case of B

¼

0:125 and 0.25 in the first hand and in the case of B

¼

0:125 and 0.50 in the other hand. Up to Ra

¼

6

·

10

⁵

, the numerical oscillations appear in the cases of B

¼

0:125 and 0.25. The following correlations link B and M and Ra:

M

¼

0:48 Ra

^0:36;

for B

¼

0:50 with 10

³6

Ra

6

10

⁶

M

¼

0:17 Ra

^0:43;

for B

¼

0:25 with 10

³6

Ra

6

6 10

⁵

M

¼

0:03 Ra

^0:57;

for B

¼

0:125 with 10

³6

Ra

6

4 10

⁵

The maximum deviation is less than 1% in the cases of B

¼

0:50 and 0.25. For B

¼

0:125 the deviation is equal to 4.5%.

Nu=0.56Ra^0.25, B=0.50

Nu=0.34Ra^0.25, B=0.50 Nu=0.29Ra^0.25, B=0.125 u

N

1E+4 1E+5 1E+6 1E+7

1 10 100

Ra

Fig. 5. Mean Nusselt number variation withRafor various values ofB.

5. Conclusion

The chimney effect and natural convection heat transfer are numerically studied in a ‘‘T’’

shaped cavity.

•

Qualitatively, from the flow structure (streamlines and isotherms), the appearance of convective cells constrains the aspired air to go along the active planes of the blocks when Ra increases.

Consequently, the heat transfer is amplified.

•

In the range of the studied parameters, the results show that the block height has an important effect on the mass flow rate and average Nusselt number. M and Nu increase with B. These parameters are well correlated with the Rayleigh number based on cavity height.

•

For relatively high values of Ra (Ra

P

3

·

10

⁵

), the block height effect on the mass flow rate becomes negligible.

•

The Nusselt correlations with Ra, obtained with these numerical simulations are of the same kind as those of natural convection in vertical smooth or ribbed channels.

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