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Energy Procedia 36 (2013) 293 - 302

TerraGreen13 International Conference on "Advancements in Renewable Energy and Clean Environment", Beirut, Lebanon ,15-17 February, 2013

THE NATURAL CONVECTION IN ANNULAR SPACE LOCATED BETWEEN TWO HORIZONTAL ECCENTRIC CYLINDERS: THE GRASHOF NUMBER EFFECT

Chahinez Ghemouga* , Mahfoud Djezzara, Abdelkarim Bourasb

a Department of physics , University Constantine 1, Am El bey Road, Constantine, 25000, Algeria

bM'sila University Algeria

Abstract

The natural convection in an annular space, located between two horizontal eccentric cylinders is studied. This latter is oriented at an arbitrary angle a. The annular space is filled by a Newtonian and incompressible fluid. The number of Prandtl is fixed at 0.702 (case of the air), the eccentricity is fixed at 0.4 but the number of Grashof varies. By using the approximation of Boussinesq and the vorticity-stream function formulation, the flow is modeled by the differential equations with the derivative partial. For the conditions of heating, we suppose the two cylindrical walls of the enclosure isotherms, T1 for the internal wall and T2 for the external wall, with T1>T2.

A computer code was developed, the latter uses finite volumes, for the discretization of the equations and in order to show its reliability. We examine the effect of the number of Grashof on the results obtained that it is qualitatively or quantitatively.

© 2013 The Authors. Published by Elsevier Ltd.

Selection and/or peer-review under responsibility of the TerraGreen Academy

Keywords: Natural convection; annular space; eccentric cylinders; Grashof number effect; vorticity-stream function formulation

* Corresponding author. Tel.: +213558278533; fax: +0-000-000-0000 . E-mail address: chahigher@yahoo.fr

1876-6102 © 2013 The Authors. Published by Elsevier Ltd.

Selection and/or peer-review under responsibility of the TerraGreen Academy

doi: 10.1016/j.egypro.2013.07.034

1. Introduction

Natural convection between horizontal eccentric cylinders has been widely studied experimentally and numerically over the past three decades because of the importance of this subject in industries, such as transmission cable cooling systems, latent energy storage systems, nuclear reactor designs, etc. Many experimental and numerical investigations have been done on the thermal convection between two concentric annuli. A comprehensive review of the relevant literatures was given by Kuehn and Goldstein [1, 2].

Nomenclature

a defined constant in the eccentric coordinates (m)

Ci radius ratio

C2 eccentricity of the annular space formed by eccentric cylinders

cP Specific heat at constant pressure (j.kg-1.k-1)

g Gravitational acceleration (m.s-2)

Gr Grashof number defined by Gr = ¿22-AT V2

H, H Dimensional and dimensionless metric coefficients (m)

Nu, Nu Local and average Nusselt numbers

q Heat flux density (W.m-2)

ri, r2 Inner and outer radius respectively (m)

So Source term

T Fluid's temperature (K)

Ti, T2 Hot's and cold's wall temperatures (K)

AT Temperature difference AT=Ti-T2 (K)

Vn,Ve Velocity components n et 0 (m.s-1)

V Velocity vector (m.s-1)

Greek symbols

a Inclination angle of (°)

P Thermal expansion coefficient (K-1)

r^ Diffusion Coefficient

1 Thermal conductivity (W.m"1.K"i)

u Kinematic viscosity (m2. s-1)

P Fluid density (kg. m-3)

n ,e,z Bicylindrical coordinates

¥ Stream function (m2. s"1) ra Vorticity (s"1) ^ General function

Studies on the fluid flow and heat transfer inside an eccentric annulus have been carried out for straight annuli considering either horizontal or vertical cases which include both the fully developed and developing flows. The natural convection flow in eccentric annulus is simulated numerically by Lattice Boltzmann Model (LBM) based on double-population approach which studied by Fattahi et al [3] One of the early works is related to Snyder [4] who studied heat transfer in an eccentric annulus under the slug flow assumption by an analytical approach. He considered fully developed condition with the outer pipe as insulated. Laminar forced convection in eccentric annuli has been studied by Cheng and Hwang [5], Trombetta [6], and Susuki et al. [7] using analytical and numerical tools at different thermal boundaries. Manglik and Fang [8] have studied the effect of eccentricity on the heat transfer using different thermal boundary conditions in laminar fully developed conditions. They have indicated that the Nusselt number decreases as the eccentricity increases.

1. Problem formulation

Let's consider an annular space, filled with an incompressible Newtonian fluid, situated between two eccentric cylinders. Figure 1 represents a cross-section of the system.

The cylinders are assumed to be isothermal with the inner cylinder being held at a higher temperature. The physical properties of the fluid are constant, except the density p whose variations are at the origin of the natural convection. Viscous dissipation is neglected, just as the radiation (emissive properties of the two walls being neglected). We admit that the problem is bidimensionnal, permanent and laminar.

Vertical plane

Tl> T2

Horizontal plane

Fig. 1 a cross-section of the system

• Continuity equation:

div V = 0

• Momentum equation:

a v ^^^ p ^ vp

— + (V.grad)V g +-

dt Po p0

Heat equation

- + ( V . grad)T = -

The coordinates are:

x = a-sh(n) ch(nJ— cos(&)

a.sin(9 ) C^1 cos{0)

The equations (1), (2) and (3) becomes :

dl ) d( )

— ihV 1+— IhV 1= 0

a„ V n) e'

T^T^'H ^ <* e M+GM ).*)]£

+ (,.e)sin(a)~G n)cos(a)]§ +

5 T dq

de p cp h

d q2 de2

The introduction of vorticity defined by:

d2 W + d2 W

{cHn)~ code))

' <* e »"OT^

We pass directly to the writing dimensionless equations, by posing the following dimensionless quantities:

Dh = a, H = — , V+ = V„ Dhh

n - — , Ve = Ve — , œ1 = œ ■

h + V + T - T2

h , v = — et T+ = -—

v v Ti - T2

The equations (5), (6), (7) and (8) becomes :

in ( H vq ■+Jk( H V tJ = 0

— ^ + ^ ^ = H \[F («.ô)cos(a)+G (n,0)sn(a)]^L H I dq

H dq H

[F (,, ô )sin (a) - G {q, ô )coS{a) ]*!+- [ + -L

~22 + ~22 + a œ + a œ

dq2 de2

d T+ dq

■ + H V

. dj+_ _ J_ dt ~ Pr

8q2 8e2

d2v+ + s2 v+

H2 8 q2 8 e2 The boundary conditions are the following ones: • inner cylinder wall Condition (n=ni=constant ):

v ; = V j

N N \ h e-VL = 0 dq

1 " 52 V , S2 V+

H2 8 q2 se 2

t1 = 1

Outer cylinder wall Condition (n=ne=constant):

8 v+ Sv+

V q = V e = if =iT = 0

8 2 v + + 8 2 v+

T+2 = 0

(12) (13)

The temperatures distribution obtained local Nusselt number value relation:

Nu =--

T]=cste

82T+ . 82T

œ —

œ —

2. Numerical Formulation

To solve the equations (11) and (12) with the associated boundary conditions, we consider a numerical solution by the method of finite volumes, exposed by Patankar [9]. For the equation (13), we consider a numerical solution by the method of the centered differences, exposed by Nogotov [10].

Physical Domain ^ computational d°maiti

Fig. 2 physical and computational domain

2.1 Discretization equation transfer of a variable <p The general differential equation is:

i (H Vn * -% > + -I(H Vi * -f ' -

We illustrate sources and diffusion coefficients in table 1

Table 1

Sources and diffusion coefficients

HGÄF (n, 6 \:osa + Gin, 6 )sina] —— dn

+ [F(y, 6")sina - G(tj, 6)oosa\—— }

The final discretization equation is:

aP VP = aN <N + aS VS + aE <E + aW VW + b (16)

The equation coefficients are well defined in Patankar [9]. The power law scheme is used to discretize the convective terms in the governing equations.

3. Results and discussion

We consider three annular spaces formed by eccentric cylinders with different values of Grashof number, inclination angle (a=0°) and relative eccentricity (C2=0.4).

3.1 Grid study

In this study several grids were used arbitrarily, to see their effect on the results. Table 2 shows us the variation of average Nusselt number and the maximum of the stream function value according to the number of nodes for each grid. We choose the grid (101x111).

Table2

Variation of average Nusselt number and the maximum of the stream function value according to the number of nodes

Gr=104 Gr=5.104 Gr=105 Gr=106

^max |Er| Vmax |Er| Vmax |Er| Vmax |Er| %

21x31 1.32 - 5.23 - 8.86 - 34.99 -

31x41 1.34 1.49 5.28 0.94 9.04 1.99 34.41 1.65

41x41 1.34 0.00 5.29 0.19 8.93 1.21 33.99 1.65

51x61 1.33 0.75 5.26 0.57 8.81 1.34 33.58 1.17

61x71 1.31 1.50 5.20 1.14 8.72 1.02 33.29 0.83

71x81 1.29 1.52 5.15 0.96 8.64 0.91 33.06 0.67

81x91 1.28 0.77 5.09 1.16 8.56 0.92 32.85 0.63

91x101 1.27 0.78 5.04 0.98 8.46 1.16 32.64 0.63

101x111 1.25 1.57 4.98 1.19 8.37 1.06 32.45 0.58

111x121 1.24 0.80 4.93 1.00 8.30 0.83 32.29 0.49

3.2 Comparison with other Results of literature

Kuehn et al. (1976) developed a numerical study on natural convection in the annulus between two concentric cylinders and horizontal with a radius ratio was taken equal to 2.6, they calculated a local equivalent thermal conductivity, defined as the report of a temperature gradient in a convective heat exchange on a temperature gradient in an exchange conduction:

Keq --

conduction

They calculated an average value of the conductivity.

We applied our computer code to this case and we compared the average value of our results with theirs, we notice that they are in concord. Table 3 illustrates this comparison well.

Table3

Comparaison of the average thermical conductivity of Kuehn with our results

Numerical pr 0.7 0.7 0.7 0.7

Study Ra 102 103 6 x 103 104

Kuehn 1.00 1.08 1.73 2.01

Inner our calculs 1.00 1.06 1.73 2.06

wall |E (%)| 0 1.4 0.3 2.8

Outer Kuehn 1.00 1.08 1.73 2.01

wall our calculs 1.00 1.06 1.73 2.07

|E (%)| 0 1.7 0.05 3.5

3.3 The thermal condition: isotherm inner cylinder

• Influence of the Grashof number

Figures 4 and 5 represent the isotherms and the streamlines for different values of the Grashof number when the inclination angle a=0°, the relative eccentricity C2=0.4.

These figures show that the structure of the flow is bicellular. The flow turns in the trigonometrically direction in the left side and in the opposite direction in the right one.

When the Grashof number is equal to 104, the heat transfer is essentially conductive, so the isotherms of figure 3 are almost parallel to the walls. For Gr=5.104 the isotherms change appreciably to follow the direction of the flow, and the values of the streamlines mentioned on the same figure increase also appreciably, which translates a transformation of the conductive transfer to the convective transfer. Nevertheless there is a movement of the fluid: the particles, which warm up on the hot wall, tend to rise along this one, then to go down again along the cold wall. Thus the flow is organized in two principal cells which turn very slowly in opposite directions.

However, for Gr = 106 the isothermal lines are modified and eventually take the form of a mushroom. The temperature distribution is decreasing in the hot wall towards the cold wall. The direction of the deformation of the isotherms is consistent with the direction of rotation of the streamlines. In laminar flow, we can say that under the action of the movement of particles flying from the hot wall at the symmetry axis, the isothermal lines are away from the wall there. The values of the Stream function increase which means that convection increases.

Fig. 3 Isotherms and streamlines for C2=0.4, a=0° and Gr=104

• Variation of local Nusselt number on the inner and outer walls

We determine the local Nusselt numbers for which changes along the walls are closely related to distributions of isotherms and isocourants, so that, qualitatively, these variations and distributions can often be deduced from each other.

The figure 6 represent the curves of variation of the local Nusselt numbers NUi and NUe along the inner and outer walls (for a = 90 °). We note, that the Nue is maximum in the summit region (0 = 110 ° and 0 = 250 ° C 2 = 0.4), and minimum in the lower part of the annular space (0 = 0 ° and 0 = 360 ° C 2 = 0.4).

If the point of the wall is located directly between two vortices, the local Nusselt number is minimum if the fluid moves away from the wall and, maximum, if the fluid is supplied to the wall.

Fig 6. The Variation of local Nusselt number on the inner and outer walls 5. Conclusion

We have developed a program of numerical calculation, based on the finite volume method, which determines the thermal and dynamic fields in the fluid and the dimensionless number of local Nusselt number on the active walls of the enclosure, depending on the quantities characterizing the state of the system. The influence of the Grashof number on the flow was Particularlyexamined.

The results of numerical simulations have shown that conduction is the regime of heat transfer dominant for Grashof numbers lower than 5.104. For Grashof numbers greater than 5.104, the role of convection becomes dominant and the values of stream function increase.

References

[1] T. H. Kuehn and R. J. Goldstein, An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders, J. FluidMech74 (1976) 695-719.

[2] T. H. Kuehn and R. J. Goldstein, An experimental study of natural convection in the concentric and eccentric horizontal cylindrical annuli, ASME J. Heat Transfer100, (1978) 635-640.

[3] E. Fattahi, M. Farhadi and K. Sedighi, Lattice Boltzmann simulation of natural convection heat transfer in eccentric annulus, International Journal of Thermal Sciences. 49 (2010) 2353-2362.

[4] W.T. Snyder, An analysis of slug flow heat transfer in an eccentric annulus, AIChE J. 9 (4) (1963) 503-506.

[5] K.C. Cheng, G.J. Hwang, Laminar forced convection in eccentric annuli, AIChE J.14 (1968) 510-512.

[6] L. Trombetta, Laminar forced convection in eccentric annuli, Int. J. Heat Mass Transf. 14 (1971) 1161-1173.

[7] K. Susuki, J.S. Szmyd, H. Ohtsuka, Laminar forced convection heat transfer in eccentric annuli, Heat Transf. Jpn. Res. 20 (1991) 169-183.

[8] R.M. Manglik, P.P. Fang, Effect of eccentricity and thermal boundary conditions on laminar fully developed flow in annular ducts, Int. J. Heat Fluid Flow 16 (1995) 298-306.

[9] S.V. Patankar, Numerical Heat Transfer and fluid flow, McGraw-Hill book company, New-York, 1980

[10] E.F. Nogotov, Applications of Numerical Heat Transfer, McGraw-Hill book company, New-York, 1978