Scholarly article on topic 'Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths'

Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths Academic research paper on "Materials engineering"

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Abstract of research paper on Materials engineering, author of scientific article — F. Mebarek-Oudina

Abstract A numerical investigation of natural convection heat transfer stability in cylindrical annular with discrete isoflux heat source of different lengths is carried out. The adiabatic unheated portions and the discrete heat source are mounted at the inner wall. The top and bottom walls are adiabatic, while the outer wall is maintained at a lower temperature. The governing equations are numerically solved using a finite volume method. SIMPLER algorithm is used for the pressure–velocity coupling in the momentum equation. The numerical results for various governing parameters of the problem are discussed in terms of streamlines, isotherms and Nusselt number in the annulus. The results show that the increase of heat source length ratio decreases the critical Rayleigh number. We can control the flow stability and heat transfer rate in varying of the length of heat source.

Academic research paper on topic "Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths"

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Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths

F. Mebarek-Oudina

Department of Physics, Faculty of Sciences, University 20 août 1955 - Skikda, B.P 26 Route El-Hadaiek, Skikda 21000, Algeria

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Article history:

Received 21 September 2016 Revised 14 March 2017 Accepted 3 August 2017 Available online xxxx

Keywords:

Numerical modeling Hydrodynamic stability Finite volume method Natural convection Cylindrical annulus Heat source

ABSTRACT

A numerical investigation of natural convection heat transfer stability in cylindrical annular with discrete isoflux heat source of different lengths is carried out. The adiabatic unheated portions and the discrete heat source are mounted at the inner wall. The top and bottom walls are adiabatic, while the outer wall is maintained at a lower temperature. The governing equations are numerically solved using a finite volume method. SIMPLER algorithm is used for the pressure-velocity coupling in the momentum equation. The numerical results for various governing parameters of the problem are discussed in terms of streamlines, isotherms and Nusselt number in the annulus. The results show that the increase of heat source length ratio decreases the critical Rayleigh number. We can control the flow stability and heat transfer rate in varying of the length of heat source.

© 2017 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4XI/).

1. Introduction

In the last decades, the natural convection heat transfer and fluid flow in different configurations are the subject of various studies.

We can found the important application particularly for the transport process of the flow in an annular geometry with heat source (s) in several engineering applications (electronic industries, nuclear industries...)

The natural convection is the preferable method of cooling the electronic components, in other to eliminate the power consumption, operating noise, reliability concerns..etc.

Also, it may offer the basic understanding of the cooling process necessary during the shutdown periods of nuclear reactors.

Where appear clearly, the possibility of natural convection for cooling the heat source.

The microelectronic devices area needs a deep strategy for the understanding of the thermal behavior of the fluid flow in an enclosure with different arrangements of heat sources, and the natural convection cooling provides a way to facilitate and enhance heat transfer for their devices. In this area, a various experimental and theoretical studies are performed for these flows. [8-20]

The study of the cooling processes has a very large interest for the electronics industry where the excessive generation of heat

E-mail addresses: oudina2003@yahoo.fr, f.oudina@hotmail.com

can be the cause of damage and loss of device or electronic system used.

The control of the temperature of these components is the principal goal of this cooling process. From where, the heating effects can generate a partial or total failure which involve significant variations of the electric performances, or a rupture of welding connecting the component to the substrate.

Saravanan and Sivaraj [1] investigated the effect of surface radiation on convection in a square cavity driven by a discrete heater.

Analytical study of the natural convection in a vertical channel containing of the heat sources is made by Gunes [2]. With two and three dimensions, analytical expressions describing variations of the variables fields in stationary regime are presented. In all calculation fields and for small Grashof numbers, these expressions are in good agreement with numerical solutions. The analytical expression of the flow rate of volume through the channel and the variation of the number of Nusselt are also obtained.

Bazylak et al. [3] made an estimated numerical analysis of heat transfer due with a whole of sources laid out on the bottom wall of a horizontal enclosure. They found that the optimum rates of heat transfer and the onset of thermal instability depend on the length and on the spacing of the sources and the aspect ratio of the enclosure. For small lengths of the source the structure of the Rayleigh -Benard cell is transformed into small broad cells, indicating the presence of a significant heat transfer to the continuation of which, a bifurcation characterized by the existence of instabilities in the physical system is obtained.

http://dx.doi.org/10.1016/j.jestch.2017.08.003

2215-0986/® 2017 Karabuk University. Publishing services by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

D lengths [m] Greek symbols

g gravitational acceleration, m/s2 a thermal diffusivity of the fluid, m2.s

L enclosure height, m b thermal expansion coefficient of the fluid, K 1

l dimensional length of the heater, m c aspect ratio = L/D [-]

Nu local Nusselt number [-] H temperature, K

Nu overage Nusselt number [-] e dimensionless size of the heater = l/L [-]

P dimensionless pressure [-] k radii ratio [-]

Pi distance between the heater and bottom wall, m K thermal conductivity, m2/s

P2 distance between the heater and top wall, m P density of the fluid, kg.m~3

Pr Prandtl number [-] r electric conductivity,

Q heat flux, W/m2 V kinematic viscosity of the fluid, m2.s_1

Ra Rayleigh number [-] w dimensionless stream function [-]

r,z radial and axial coordinates, respectively

ri, ro inner and outer radii, m Indices

T dimensionless temperature [-] cr critical

Tmax maximum temperature of the heater [-] c conditions at the cold wall

t dimensionless time [-] h conditions at the heater wall

At dimensionless time increment [-] r,z radial and axial directions, respectively

u,v dimensionless radial and axial velocities, respectively [-]

The phenomenon of convective flow in the enclosure with discrete heat source(s) at the vertical wall has a great interest in several engineering applications.

The convective flow and heat transfer in a vertical annulus due to a single isoflux heat source, arranged at different locations of the inner wall, are investigated by Sankar et al. [21]. In same configuration with two finite sized heaters, Sankar and al. [22] are studied numerically the natural convection in laminar steady state case. They found the effect of the size of discrete heater on the flow and heat transfer.

The stabilization of the flow with the application of magnetic field for deceleration of the fluid is found in many references as [4-7].

The study of hydrodynamic stability is necessary to detect exactly the temperature which can make damage and loss of the device or the electronic system used.

To our knowledge the effect of heater length on heat transfer in a cylindrical annulus has never been studied, except the study of Sankar and al. [21] but in laminar case. Our objective is to determine the critical Rayleigh numbers, Racr associated with heat source lengths ratio, showing the effect of heat source length ratio on the hydrodynamic and thermal stability.

This paper is organized as follows. First, the problem formulation is described in Section 2. Section 3 discusses the numerical method and techniques used to solve the governing equations. Section 4 presents the results and discussion. Finally, conclusions in form of the important remarks are given.

2. Mathematical model

A schematic of the flow presented in Fig. 1 is a two-dimensional cylindrical annulus formed by two vertical concentric cylinders in the inner and outer radii ri and ro, respectively. D and L are the width and height of the annulus, respectively. The cylindrical coordinates (r, z) with the corresponding velocity components (u, v) are also specified in Fig. 1.

The top and bottom boundaries of the annulus are thermally insulated and outer wall is isothermally cooled at a lower temperature 0c providing the heat sinks. One finite-sized heater of length i is placed at the inner wall and the heat flux generated by this

////////.

adiabatic ¡wall

•//////I

I />1 I

////!/

'A///////

adiabatic wall

Fig. 1. Problem configuration with cylindrical coordinate system, where S1, S2 and S3 are the monitoring points of hydrodynamic and thermal instabilities detection.

heater is Q, whereas the unheated portion of the inner wall is thermally insulated.

The position of this isoflux heat source of the length i is fixed at the medium of the inner wall of the annulus.

The dimensionless length ratio e = L of the heater kept different values e = 0.8, 0.6, 0.4, 0.2. Whereas, the distance between the heater and bottom wall pj equal the distance between heater and top wall p2.

The fluid of kinematic viscosity t in the annulus is assumed to be laminar, Newtonian with negligible viscous dissipation and constant of thermophysical properties. The Boussinesq approximation is employed with gravity, g, acts in negative z-direction.

The annular gap, (ro - r,), is used as the length scale, the thermal diffusion time across the gap, (ro - r,)2/a, is the time scale, and the temperature scale is Q(ro - r,)/K, where k is the thermal

conductivity and a is the thermal diffusivity of the fluid. The non-dimensional temperature relative to the temperature of the outer cylinder is T = (0 - 0c)j/Q(ro - ri).

The flow is governed by the Rayleigh number, Ra = ^fQD4, and the Prandtl number, Pr = a, as well as these geometric parameters: the aspect ratioc = D, the radii ratio, k = p-, the heater location and length.where, a = K/pCp is thermal diffusivity of the liquid, k is the thermal conductivity and Cp its specific heat to constant pressure.

The non-dimensional governing equations are:

d(ru) + d(rv) = 0

du du du dP 1 d f du\ d2u

--h u--h V- =---h--r- H--

dt ^ dr ^ dz dr ^ \r dr\ dr ! ^ dz2

dv dvv dv_Ra _@p Лdf dv) d2v

dt+ u dr+ v dz - Pr '1 ~~dz + lr dry dr + dz2

dT dT dT _ 11 d f dT\ d2T dt+ u dr+ v dz " Pr( r dr[r ~dr) + ldz2

The boundary conditions are no-slip and as all boundaries are stationary the velocity equals zero on all boundaries. For the temperature, it is held constant T = 0 on the outer cylinder, r = ro. The top and bottom, z = 0/D, have zero heat flux so dT/dz = 0. On the inner cylinder r = ri, dT/dr = -1 at the heater, and dT/dr = 0 elsewhere.

The initial and boundary conditions in dimensionless form are:

at t = О, u = v = T = 0

for t > 0,

atr = 1 : u = v = 0, dr = 0(unheated portion), 'dT = -1 (heater) inner wall

at r = 2 : u = v = 0, T = 0 outer wall (6b)

atz = 0 : u = v = 0, = 0 Bottom adiabatic wall (6c) dz '

at z = L : u = v = 0, = 0 Top adiabatic wall (6d)

D dz '

3. Numerical solution

The heat transfer process governed by two dimensional natural convection Eqs. (1)-(4), with the associated boundary conditions, is obtained using a finite-volume method.

The components of the velocity (u and v) are stored at the staggered locations, and the scalar quantities (P and T) are stored in the center of these volumes [5]. The numerical procedure called SIMPLER [24] is used to handle the pressure-velocity coupling. The second-order accurate central difference scheme is used to dis-cretize the convection and diffusion terms.

The Thomas algorithm (TDMA) is employed to solve the dis-cretized difference equations after its arrangement in tridiagonal matrix.

Convergence at a given time step is declared when the maximum relative change between two consecutive iteration levels fell below than 10~4, for u, v and T. The calculations are carried out on a PC with CPU 3 GHz.

Table 1

Grid independence test for c = 1, k = 2 and Ra = 107

Grid size Nu Tmax

42 x 42 22.39233 5.673707 x 10-2

82 x 82 21.772 5.726366 x 10-2

122 x122 21.70649 5.716366 x 10-2

162 x162 21.700320 5.709647 x 10-2

202 x 202 21.700961 5.704113 x 10-2

0.06 -

- 42x42

- 82x82

- 122x122

162x162 - 202x202

0.04 -

0.02 -

122x122

162x162

202x202

Fig. 2. Profiles of dimensionless temperature T with z (a), and dimensionless axial velocitycomponent v with r (b), in the middle of the cavity for various grids.

In heat transfer problems, the overall rate of heat transfer across the heater is the important parameter.At a given point on the heater surface, the average Nusselt number at the heater is given by Nu = 1 J1 Nu ■ dz, where 1 and 2 are, respectively, the bottom and top positions of the heater.

Ra = 107

Fig. 3. Streamlines and isotherms for e = 0.25, y = 1, Ra = 104 and Ra = 107. a) Our numerical results; b) Sankar et al. 2012.

Our results Sankar et al. 2012

LU_I_I.........I_I.........I_I.........I_I.......

10' Ra

Fig. 4. Comparison of average Nusselt number with the results of Sankar et al. 2012 for y = 2 and e = 0.25.

Fig. 5. Temporal evolution of the dimensionless velocity for y = 1, Ra = 104 and 108, and e = 0.2, 0.4, 0.6.

Grid effect

Several non uniform grids close to the annulus walls, where large velocity and temperature gradients exist, thus requiring a larger number of nodes in order to resolve the specific characteristics of the flow, also in order to reduce numerical errors.

Therefore, the increments Dr and Dz are irregular, they are chosen according to geometric progressions of ratio 1.07, which permitted grid refinement near the walls.

In order to examine the grid size effect on the numerical solution, a number of sizes are investigated for grid independence study: 42 x 42, 82 x 82, 122 x 122, 162 x 162 and 202 x 202 nodes.

For this purpose, the average Nusselt numbers and maximum temperatures at the heater are monitored for a grid system of 42 x 42, 82 x 82, 122 x 122, 162 x 162 and 202 x 202, which is presented in Table 1.

From Table 1, it can be observed that the maximum difference between 162 x 162 and 202 x 202 grid systems is within 0.01%.

Consequently, a 162 x 162 grid system for y =1 is used to carry out further calculations. To choose the optimum grid sizes, similar tests of the grid independency are conducted for other aspect ratios. These results are not presented here because of the same conclusion obtained.

By examining the curves illustrated in (Fig.2a-b), a change of less than 1% in computed values is observed between 122 x 122 and 162 x 162 or 162 x 162 and 202 x 202. For this, the grid corresponding to 162 x 162 nodes is adopted for all numerical simulations, in order to optimize the CPU time and the cost of computations.

This house code is developed for the present model and was validated successfully in the next section.

4. Results and discussion

Numerical investigations are carried out in a vertical annulus to study the effects of size of an isoflux discrete heater on the flow and heat transfer.

The outer wall is maintained at lower temperature while the top and bottom walls are thermally insulated.

In the steady and oscillatory state, the flow pattern, temperature distribution, and heat transfer characteristics are obtained for a wide range of physical and geometrical parameters like: the Rayleigh number >103, the Prandtl number is fixed at Pr = 0.71, which corresponds to air, the aspect ratio (y) and radius ratio (k) are, respectively, taken as y = 1 and k = 2 and the length of heater ratio, e = 0.2, 0.4, 0.6 and 0.8 are chosen in the present computations. In the annular enclosure the streamlines and isotherms are presented.

4.J. Validation

The variation of average Nusselt number for various values of Ra at the heater with length e = 0.25 and aspect ratios y = 2 is presented in Fig. 4 in order to validate our numerical results.

The average Nusselt number at the discrete heater tends to increase with the increase of Rayleigh number.

From the Figs. 3 and 4, comparisons show good agreement between our results and that of Sankar et al. [21].

The good concord between the present results and literature proves the credibility of the present code.

4.2. Steady state flow

For two Rayleigh number Ra = 104 and 108, the flow is laminar in all hydrodynamic cases. In Fig. 5, the temporal evolution of the radial velocity of the fluid is presented for Ra = 104, 108 and various heater lengths e. The radial velocity increases with the increase of the values of Rayleigh numbers and heater lengths e.

For two different Rayleigh numbers, the hydrodynamic flow and thermal fields inside the discretely heated annular enclosure are illustrated through streamlines and isotherms in Figs. 6 and 7.

At Ra = 104, the parallel isotherm pattern near the discrete heater indicate that the conduction mode dominates heat transfer from the discrete heater and convection is developed in a layer adjacent to the heater.

We can see that the streamlines form a single cellular pattern circulates in the conventional clockwise and occupies the entire

г = 0.8

г = 0.6

г = 0.4

£ = 0.2

Ra = 104

Fig. 6. Streamlines and isotherms for c = 1, Ra = 104 and various e.

£ = 0.8

£=0.6

£ = 0.4

£ = 0.2

Ra = 108

Fig. 7. Streamlines and isotherms for y = 1, Ra = 108 and various e.

- ,R<i=104

— \ \ e = 0.2 /

- \ e = 0.4

- = 0.8 , i e = 0.6 , i

0.2 0.4 0.6 z («) 0.8

Ra= 10s

- ,e = 0.2

- 1 le = 0.4

\ e=o.a \

- eVo.8 \ \ \

0.2 0.4 0.6 0.8

Fig. 8. Effect of the heater length ratio on the local Nusselt number for various e and C = 1. (a) Ra = 104, (b) Ra = 108.

Fig. 9. Effect of the heater length on temperature profiles along the discretely heated wall for two different values of Ra and c = 1.

e=0.2,y=l,Ra = Racr

0.349 0.35

0.351 0.352 t

Fig. 10. Comparison of the results between two time steps for the probe S3, c = 1 and heaterlength e = 0.2 in oscillatory state flow, Racr = 9 x 1010.

annular enclosure, but the flow sustained a slight distortion with the length and position of heater.

Due to convection, flow pattern rising along the heater wall, travels towards the outer wall, then the flow descends down along the cold wall before with the center of rotation at the middle of the annulus.

By increasing the Rayleigh number to 108, the convection becomes the dominant mode of heat transfer in the annulus and a significant change in the streamline pattern are observed as the heater length change. With the presence of this strong convection, the parallel isotherms for the low value of Ra are deformed and move toward the outer.

The effect of heater length on the rate of fluid circulation is presented in Fig. 6.

The rate of fluid circulation given in terms of the maximum absolute value of the dimensionless stream function is found to be higher for the small heater.

The hydrodynamic and thermal boundary layers are developed near the heater and the cold wall.

For more details of the alter of the flow and thermal fields, the effect of heater size on the dimensionless temperature profiles, along the mid-height of the annulus, is displayed in Fig. 9 for two different values of the Rayleigh number. In general, the dimen-sionless temperature increases at the leading edge of the heater and descends at the end of the heater, leading to crimped temperature profile over heater.

Strong dependence of the temperature variation at heater surface with its length.

The analysis of the dimensionless temperature profiles in Fig. 9 indicates that the maximum temperature on the heater increases as the value of e increases from 0.2 to 0.8. On the contrary, as the Rayleigh number is increased to (Ra = 108). A significant variation of the profile of temperature over the heater with the change of the value of e.

1650 -

1600 -

1550 -

1500 -

1450 -

0.349 0.3495 0.35 0.3505 0.351 t

Fig. 11. Time evolution of the dimensionless radial velocity u for c = 1, e = 0.2 and Racr = 9x1010 at the probe S3. Dimensionless streamlines W and isotherms T traced at various dimensionless times (ta, tb, tc) (increments as in Fig. 11).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 12. Effect of the heater length ratio on the local Nusselt number for Racr = 2.5 -x 1010, 3.75 x 1010, 5 x 1010, 9 x 1010 corresponding to e = 0.8, 0.6, 0.4, 0.2 respectively and y = 1.

In various practical applications, the importance of the average Nusselt number for the measure of the local and average heat transfer rate from the heat sources is clear.

For different heater length and fixed Rayleigh number, the local Nusselt number variation along the discretely heated surface is shown in Fig. 8a-b.

At Ra = 104, y =1 and for the small heater length the local heat transfer is higher and dissipates more heat compared to a larger sized heater.

The main conclusion is that the size of heater plays an important role in augmenting the heat transfer performance amongst the heater.

The variation of Nu along the heater reveals that, the local Nusselt number is higher at the leading edge of each heater and

Fig. 13. Stability diagram (Racr - e) for c =1 and k = 2.

decreases up to the middle of the heater, and increases thereafter. This is due to perturbations caused by the interaction between the hot air leaving trailing edge and the cold air near the top adiabatic wall. The minimum value of local Nusselt number is found at the heated surface and varies with the physical and geometrical parameters.

It is observed that the minimum value is around the middle of the heater surface for Ra = 104 and e = 0.2, and it shifts toward the right as the value of the Rayleigh number or e increases.

4.3. Oscillatory state flow

In this part, we are interested in the critical point for which the flow becomes oscillatory. For that, we have made a series of experiments simulations, by increasing progressively the value of Ray-leigh number Ra, and we examine the behaviour in time of the

numerical solutions at the point selected. For each value of the dimensionless size of the heater e, we detect the critical Rayleigh number, Racr.

In the literature, several references such as [4,5] explain how the critical Rayleigh number is found.

Thus, the periodic flows is characterized by the critical Rayleigh numbers, transition from steady to time-dependent flow (Fig. 10).

A general observation concerning the stability characteristics of all flow patterns shown in Fig. 11. It is known from hydrodynamic stability theory (e.g. [23]) that when light fluid is below heavy fluid, the density distribution could lead to an instability and oscillations.

The time evolution of the flow may be steady or unsteady depending on the stability characteristics of the flow, value of Ray-leigh number and the source length.

A detail of the periodic time evolution of the radial velocity at the probe S3 of the enclosure is shown in Fig. 11 for e = 0.2. The streamline plots correspond to characteristic locations in the period of the velocity time series.

In order to eliminate a numerical solution, we use two dimen-sionless time step Dt and Dt/2. The physical solutions are defined for the same oscillations, or for the very low variations between the oscillations with Dt and Dt/2.

Fig. 10 presents example of test Dt used in this study for e = 0.2. The results of both dimensional time steps Dt, Dt/2 are in quite good agreement.

The time evolution of the dimensional components (u, v and T) is oscillatory and periodic, indicating that the flow undergoes a bifurcation (see the example of radial velocity, u in Fig. 11). We found that, the oscillation amplitudes of the dimensionless temperature T are smallest compared to the dimensionless axial and radial velocities. These results are in good agreement with those obtained by Mebarek-Oudina & Bessai'h [4-6].

The value of the local Nusselt number decreases with the increase of the value of heater length ratio corresponds of the onset of instabilities. Therefore, the high value of the local Nusselt number is showed for low heater lengths ratio corresponds to high critical Rayleigh numbers. (Fig. 12).

Stability diagram (Racr - e) is presented in Fig. 13 for aspect ratio y = 1 and radii ratio k = 2. This diagram shows the dependence of the critical Rayleigh numbers with the dimensionless heater length. The critical Rayleigh number decreases with the increasing of heater length.

5. Conclusion

A numerical investigation of natural convection of air in a vertical annulus is presented for y = 1, k = 2 and various heater lengths. A heater element placed at the inner cylinder provides a constant heat flux, and the rest of the inner cylinder is insulated. The outer wall of the annular is kept at a lower temperature. The dimension-less heater length, e, is used for the variation of the size of heater. In steady and oscillatory states flow the study is predicted.

The important conclusions drawn from this investigation are:

1. We found that the heat transfer rates are higher for a smaller heater length.

2. The decrease of the rate of heat transfer at the heater with an increase in heater length.

3. The increase of maximum temperature with the dimensionless heater length.

4. The position of a small heater near the middle portion of the inner wall of the annulus will achieve a high heat transfer. At this location, the appearance of hot spot is minimal for both the heater lengths.

5. The increase of critical Rayleigh numbers reinforces the natural convection flows which results in the reduction of heat source temperature.

6. The decrease of critical Rayleigh numbers with an increase in heater length.

Acknowledgments

The author acknowledges the support of this work through the Algerian Ministry of Higher Education.

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