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Simulation of the Forced Convection in the Channel with a Right Miter Bend

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Simulation of the Forced Convection in the Channel with a Right Miter Bend

A. Brima, M. Bentercia and R. Atmani

Department of Mechanical Engineering, University of Biskra, 07 000 Biskra

Abstract – A laminar steady and two dimensional of cold water entering in constant width duct, passing through a 90 degree turn and redeveloping downstream of the separation region is studied. The behavior of the fluid flow and heat transfer along the entire duel are investigated for the range of the Reynolds number between 100 and 540 (range of experimental flow visualisation and measurements). The effects of the recirculation bubble and Reynolds number on the profiles of the velocity components, temperature and heat transfer coefficient are presented. Also the dependence of the bubble length on the Reynold number is considered.

An analytical solution of the flow problem upstream and downstream of the corner region is obtained. The analytical and numerical results are in good agreement with available results for laminar heat transfer along bends.

Résumé - Un écoulement stationnaire, laminaire et bidimensionnel d’eau froide entrant dans une conduite à largeur constante, passant à travers un coude à 90 degrés et reprenant en aval de la région de séparation est étudié. Le comportement de l’écoulement du fluide et celui du transfert de chaleur le long de la conduite entière sont étudiés pour des nombres de Reynolds entre 100 et 540 ( intervalle des mesures et de visualisation expérimentale de l'écoulement ). Les effets de la bulle de recirculation et du nombre de Reynolds sur les profits des composants de la vitesse, sur la température et sur le coefficient de transfert de chaleur sont présentés. La dépendance de la longueur de la bulle, du nombre de Reynolds est aussi considérée,

Une solution analytique du problème d’écoulement en amont et en aval du coude est obtenue. Les résultats analytiques et numériques obtenus sont en bon accord avec les résultats existants.

Key-words: Bends - Laminar steady two dimensional flow - Heat transfer - Reynolds number - Bubble length - Recirculation.

1. INTRODUCTION

One way of cooling many engineering equipment is to recirculate cold fluids through piping systems. Since it can not establish such systems without bends, valves and bifurcation; the analyses of fluid flows and heat convection through these devices is very important.

Unfortunately, the presence of these elements disturbs the flow and introduces many complicated flow phenomena such as : flow separation, recirculation and reattachment. Consequently, the prediction of the flow and heat transfer behaviors regions becomes very difficult.

These problems can also be round in circulation systems, especially in the vicinity of artificial cardiac valves and inside the aortic sinus. However the pulstile nature of the blood flow on one hand and the non newtonian blood behavior on the other hand make the study of the previous flow phenomena more complicated.

The present paper tries to obtain a closed form mathematical solution for the problem and determines the effects of the Reynolds number as well as the recirculation bubble on the velocity and temperature distributions across and along the channel.

2. PROBLEM DEFINITION

A cold and incompressible fluid flows steadily in a two-dimensional channel with a right angled bend (Fig.

1). This geometry eau be seen as a limiting case of the common bend found in engineering equipment. The method used in this work applies for both channel configurations of the previous figure. Except that the miter- bend produces more separation and recirculation behind the inner sharp corner for the same flow Reynolds number. The Reynolds number is taken between 100 and 540 in order to compare the theoretical results with the experimental ones.

The lengths of the channel upstream and downstream of the bend are big enough to have a fully developed flow established at the entrance and exit of the system. Concerning the thermal conditions of the system, the walls temperature is held constant. In this study we aim to obtain the velocity components as well as the temperature distributions in the channel. Furthermore we seek to determine the pressure gradient and the Nusselt number variation along the inner and outer walls and finally we try Io see how the Reynolds number as well as the recirculation bubble affect the previous variables profiles.

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The chosen fluid in this study is the water and its proprieties are : ρ = 103 kg / m3 k = 0.61 W / m . °C µ = 7.0 10-4 kg / m.s Cp = 4.1 kJ / kg . °C

Fig. 1: Fluid flow in both channel configurations (a) Meter-bend, (b) Gradual bend

3. GOVERNING EQUATIONS

The channel is divided into three regions (Fig. 1). In regions 1 and 3, the non dimensional basic equations given in coordinate are :

Y 0 V X

U =

∂ + ∂

∂ (1)





∂ + ∂

∂ + ∂

− ∂

∂ = + ∂

2 2 2 2

Y U X

U Re

1 X

P 2 1 Y V U X

U U (2)

Y P 2 1 Y V V Y U V

− ∂

∂ = + ∂

∂ (3)





∂ + ∂

= ∂

∂ + ∂

2 2 2 2

Y T X

T Pe

1 Y V T X

U T (4)

2V

∇ is neglected in equation 3.

For the region 2, the non-dimensional equation in polar coordinate are : U 0

) V R

R( =

θ

∂ + ∂

∂ (5)









θ

− ∂ θ

∂ + ∂

∂ + ∂ θ

= ∂ θ −

∂ + ∂

∂ V

R 2 U R ) 1 U R R( R 1 R R

1 P R 2

1 R

U V U R U R

V U 2 2

2 e 2

(6)









θ

− ∂ θ

∂ + ∂

∂ + ∂

= ∂ θ −

∂ + ∂

∂ U

R 2 V R ) 1 V R R( R 1 R R

1 R P R 2

1 R U V R U R

V V 2 22 2

e 2

(7)





θ

∂ + ∂

∂ + ∂

= ∂

∂ + ∂ θ

2 2 2 2

2 T

R 1 R T R

1 R

T Pe

1 R V T T R

U (8)

4. METHOD OF SOLUTION

The flow problem, which must be solved before the thermal one, is handled using the integral method. The crucial point in this approach is the suitable choice of the U-distribution across the channel. The specification of the previous profile is inspired from the flow pattern (recirculation) just downstream of the inner corners The following models for U-profile are chosen :

4.1 Miter-Bend 4.1.1 In region 1 and 3

We have :

) Y ( G ) X ( Z ) Y ( F ) Y , X (

U = + (9)

(3)

Where





=

+

=

=

function distorsion

) Y ( Z

) Y Y 3 Y 2 ( ) Y ( G

) Y Y ( ) Y ( F

2 3

2

Using the continuity equation, the V-profile is :





 − −

= 2

Y Y 2 Y X d

Z ) d

Y , X (

V 4 3 2 (10)

Differentiating equation (2) with respect to Y and equation (3) with respect to X and subtracting last equation from the one before we obtain :





∂ + ∂

= ∂

− ∂

− ∂

− ∂

∂ +∂

∂ + ∂

∂ + ∂

∂ + ∂

2 2 2 2 2

2 2

2 2

Y U X

U Y Re

1 Y X

V V Y X

V V X

V V X X

U U Y

U V Y Y

U U Y X

U U Y X

U

U (11)

Expressing each term in equation (2) in terms of the assumed profiles for the velocity components and integrating the equation across the channel, we get :

0 Re Z 3360 X

Z

3

3 − =

∂ (12)

4.1.2 In region 2

We have the assumed following profiles : ) R ( G ) ( Z ) R ( F ) , R (

U θ = + θ (13)

where





+

=

=

R R 3 R 2 ) R ( G

R R ) F ( U

2 3

2

Using again the continuity equation, the V-component becomes :



 

 − +

− θ

=

θ 2

R R 3 R d 2

Z ) d , R (

V 3 2 (14)

Following the same steps described in section 4.1.1, we obtain :

Re 0 Z 1 Re

2 d

Z d d

Z d 560

1 d

Z Zd 210

1 d

Z d 210

1 d

Z d Re 12

1 d

Z d 280

1

2 2 2

2 3

3 − + =

θ θ θ −

θ − θ +

θ − (15)

4.2 Channel with a Gradual bending 4.2.1 In regions 1 and 3

Since the geometry of these regions is the same as that of the miter-bend, the use of the previous flow model, described in section 4.1.1, leads to the same differential equation (12).

4.2.2 In region 2

The application of the following clever variable transformation. 

 

 −

= h

r

R r i allows us to use the same previous profiles, given by equations (13) and (14) and following the later corresponding steps in which the later corresponding steps in which the integration limits are still the same (0 and 1) as before. These mathematical manipulations lead to a differential equation which is mathematically identical to equation (15). The gain from dividing the channel in three regions is the obtaining of an analytical solution for the linear differential equation (12) in regions 1 and 3. Where as, the non linear differential equation (15) can be solved by Newton-Ralphson method. Once the velocity field is obtained, the thermal problem can be solved numerically. Since the flow can be described by the same equations (12) and (15) in both channel configuration (Fig. 1), only the solution of the miter-bend flow is considered on this study. This is because of its better illustration of the flow separation and bubble recirculation phenomena.

5. SPECIFICATION OF THE BOUNDARY CONDITIONS

In order to solve the equations (12) and (15), it is necessary to specify the boundary conditions for each of them.

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1. Region 1 :





β

= β β

= β

=

=

=

) ( tan with V tan

) U ( tan 10 ) X 0 d

Z d

zero to equal is component V

0 ) X( d

Z d

parabolic is

profile U 0

) ( Z

2. Region 2 :





=

=

=

π θ ∆θ

θ

=

= θ

=

= θ

=

=

0 ) 0 Y ( stress shear )

0 ( Z 2) ( Z

component V

of continuity )

( Z ) 0 ( ) Z 0 X X( d

Z

d(X 0) Z( 0) continuityof Ucomponent Z

3.region 3 :





=

=

=

− π =

=

=π θ

=

=

parabolic profile

U 0

) ( Z

0 ) 0 Y ( stress shear 1

2) ( Z ) 0 ( Z

component V

of continuity 2)

( Z ) 0 X X ( d

Z d

The application of the previous boundary conditions to the corresponding equation gives the following solution for the distortion function (Re= 148):

[ ]





<

≤ +

<π θ

<

θ + θ

− θ +

<

=

0 X

e ) X 049 . 0 ( sin 62 . 69 ) X 049 . 0 ( cos

0 2 553

. 0 098 . 2 107 . 1 327 . 0

0 Z e

327 . 0 Z

X 415 . 1 3

2 X

831 . 2

Where the first and last expressions correspond to the analytical solution, whereas the second one is the approximated expression ti the numerical solution. Similar expressions for Re=297 and Re = 539 can be obtained in the same manner. Concerning the thermal problem, the boundary conditions corresponding to the energy equation (in cartesian and polar forms) are :





<

<

=

<

=

<

<

== < <

1 Y 0 0

) Y , X ( d

T d

X 2 1

) 1 , X ( T

X 1

) 0 , X (

T(0,Y) 0 0 Y 1

T

The insertion of the obtained solution for Z in the U and V profiles and the use of these profiles in the momentum and energy equations allow us to get the distributions of the velocity components, pressure gradient, temperature and Nusselt number in the three regions, The results are plotted in figures 2 through 11.

6. RESULTS AND DISCUSSION

Fig. 2 shows how the distortion varies along the channel. It increases as the flow enters in the corner region, then it falls down to reach maximum negative value in the bubble region and increases again to get zero value far downstream. This distribution pattern looks logical because the distortion function indicates how big the departure of the U-Profile is from the parabolic one.

Fig. 2: Distribution of distortion function Fig. 3: Profile of U-Component at region 2 entrance

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In fact, figure 3 and figure 4 depict this departure, where the maximum of the velocity is shitted towards the inner wall at the beginning of the corner region. Then is moves towards the outer wall at the end of region 2.

This is due to the presence of the flow recirculation shown by the U-negative values (Fig. 4). The present profile pattern for this velocity component is also found in [2] and [3].

Fig. 4: Profile of U-Component in bubble region Fig. 5: Profile of V-Component just after the inner corner

Concerning the distribution of V-component, it is almost negligible in region 1, then it increases to reach a maximum value just after the flow separation (at the point X = 0, 1 and Y = 0,5) as it is shown in figure 5. This variation can be seen as a result of the local flow deviation with respect to the channel centreline. The distribution of the pressure gradient along the outer wall is similar to that of Z. This might be due to the reduction of it is expression to (AZ + B) with (A > 0). However, the pressure gradient along the inner wall varies, to some extent, inversely to Z as it is indicated in figure 6. These profiles of the pressure gradients are similar to those shown in [4]. For the comparison of the present computed bubble lengths with experimental ones [1], table. 1 shows a good agreement between both results. The occurrence of the secondary flow is unlikely in this case as it is reported in [5].

Fig. 6: Distribution of pressure gradient along

The inner wall Fig. 7: Distribution of centreline dimensionless Temperature

Concerning the thermal solution, figure 7 shows that the flow heats up more efficiently in the corner region and therefore the biggest increase in centreline temperature occurs in this area. This might be due to the outer wall bending to the right on one hand the flow separation from the inner corner (obstruction of the bubble height) on the other hand which create a sort of instantaneous flow choking between hot boundaries. Another region where tire flow experiences more heating is the recirculation bubble as it is clear in figure 8 and figure 9.

Fig. 8: Dimensionless temperature distribution through the bubble in down stream direction

Fig. 9: Dimensionless temperature distribution across the bubble in normal direction

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Table 1: Bubble length

Re L Experimental results Model results

148 1.111 1.100 297 1.611 1.556 539 2.166 2.118 This heating increase is due to the flow recirculation with small velocities in the vicinity of a hot boundary.

For the Nusselt number, Fig. 10 shows its distribution along the inner wall and it seems that this thermal variables behavior is logical from the physical point of view and the Nusselt number variation is similar to that of [6].

Fig. 10: Nusselt number distribution along the inner wall NOMENCLATURE

A, B = constants Y = dimensionless ordinate (y/h)

H = dimensionless with channel (h/h) α = thermal diffusivity of fluid P = dimensionless pressure 2p / ρ u2 θ = polar coordinate

re = outer curvature radius of the bend ρ = fluid density

U = dimensionless axial Pe = Peclet number

or (tangential velocity component (u /u)) F, G = functions of Y

u = mean entrance flow velocity L = dimensionless bubble length (1/h) R = dimensionless radical coordinate (r/h) β = angle of streamline deviation ri = inner curvature radius of the bend at the entrance of corner region V = dimensionless normal

or (radial velocity component (v/u)) µ = dynamic viscosity

X = dimensionless abscissa (x/h) Re = Reynolds number ( uh / γ ) Z = distortion fonction

REFERENCES

[1] H.J. Hugerup, ‘Manuscript on Flow Visualization and Measurement’, Aero. Eng. Dep., R.P.1 Troy (N.Y) U.S.A.

[2] A.C. Hurd and A.R. Peters, ‘Analysis of Flow Separation in a Confined Two-dimensional Channel’, ASME Journal of Basic Engineering, Vol.92, pp. 908-914, 1970.

[3] P. Orlandi and D. Cunsolo, ‘Two Dimensional Laminar Flow in Elbowc’, ASME Journal of Fluids Engineering, Vol. 101, pp. 276-283, 1979.

[4] G. Heskestad, ‘Two-Dimensional Miter-Bend’, Flow Journal of Basic Engineering, pp. 433-439, 1971.

[5] A.M.K.P. Taylor, J.H. Whitelaw and M. Yinneskis, ‘Curved Ducts with Strong Secondary Motion : Velocity Measurements of Developing Laminar and Turbulent Flow’, Journal of Fluid Engineering, Vol. 104, pp. 350-358, 1982.

[6] R.S Amano, ‘Laminar Heat Transfer in a Channel with Two Right-Angled Bends’, J. Heat Transfer, Vol. 106, pp. 591-596, 1984.

[7] G.J. Hwang and Change-Hsing Chao, ‘Forced Laminar Convection in Acurved Isothermal Square Duc’, J. Heat Transfer, Vol, 113, pp.

48-55, 1991.

[8] A. Brima and M. Bentercia, ‘Forced Laminar Convection along the Channel 90° Turn’, Magister Thesis, 1993.

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