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Laminar Mixed Convection in the Entrance Region of Semicircular Duct with Constant Heat Flux

Y.M.F.El. Hasadi, I.M.Rustum and A. Abdala University of Garyounis, Benghazi, Libya

A laminar mixed convection in the entrance region for horizontal semicircular ducts with the flat wall at the bottom has been investigated. The governing momentum and energy equations were solved numerically using a marching technique with finite control volume approach following the SIMPLER algorithm. Results were obtained for the thermal boundary conditions of uniform heat input axially and with uniform wall temperature circumferentially (H1 boundary condition) with Pr=0.7 and a wide range of Grashof numbers. These results include velocity, temperature distributions, at different axial locations, covering all aspects of flow, axial distribution of local Nusselt numbers and local average wall friction factor. It was found that Nusselt numbers were close to the forced convection values near the entrance and then decreases to a minimum as the distance from the entrance increases and then rises due to the effect of free convection before reaching constant value (fully developed). As Grashof number increases Nusselt number and average wall friction factor increases in both developing and fully developed regions and the location of the onset of the secondary flow moves upstream.

Nomenclature

Dh = hydraulic diameter

+ππ

)2(2 or

f = friction factor defined by Eq(11) g = gravitational acceleration Gr = Grashof number defined by Eq(8) Nux = local Nusselt number defined by Eq(10)

1P = dimensionless cross sectional average pressure defined by Eq(8) P2 = dimensionless cross sectional excess pressure defined by Eq(8) q = rate of heat input per unit length r,ro = radial coordinate, radius of the circular wall x = axial coordinate T,Tw = dimensionless fluid temperature and wall temperature V,W,U = dimensionless fluid velocities in r,θ,x directions Re = Reynolds number defined by Eq(8) Pr = Prandtl number defined by Eq(8) Ub = dimensionless mean axial velocity Tb = dimensionless mean temperature Subscripts fd = fully developed

9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-3795

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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I. Introduction Laminar combined forced and free convection flows in ducts received much attention in the recent years because

of their wide range of applications, such as compact heat exchangers and chemical processes. This paper is concerned with the problem of laminar mixed convection in the entrance region of horizontal semicircular ducts, with uniform heating which is a particular case of multi passage tubes.

Due to the large amount of literature on fully developed laminar mixed convection for different cross sectinos. Consideration will be given to the geometry of semicircular ducts only. Nandukumar, et al.1 studied numerically the problem of fully developed laminar mixed convection flow in horizontal semicircular ducts for the H1 thermal boundary condition with the flat wall at the bottom. Lei and Trupp 2 solved the same problem considered in Ref 1 with flat wall on top. They reported approximately the same results of Nusselt number as for the flat wall at the bottom (Ref 1). Chinporncharoepong etal. 3 Studied the effect of orientation by rotating the horizontal semicircular duct from 0° (the flat wall on top) to 180° (the flat wall at the bottom) with incremental angle of 45°. Busedra and Soliman 4 investigated the effect of duct inclination on laminar mixed convection in inclined semicircular ducts under buoyancy assisted and opposed conditions. They oriented the flat wall of the duct in vertical position using two thermal boundary conditions H1 and H2. All the previous studies used the water as a working fluid.

The literature on combined free and forced convection in the entrance region is few in comparison with the fully developed case. Most of the results for laminar mixed convection in the entrance region are available for vertical rectangular ducts 5-7, for vertical circular tubes 8-9. Other results for horizontal ducts are also available for rectangular ducts 10 concentric annulus 11, and circular tubes 12. For the case of laminar mixed convection in the thermal entrance region of horizontal semicircular duct was experimentally investigated by Lei and Trupp 13. The heat input was uniformly generated along the duct test section with the flat wall on top they used the water as a working fluid, and obtained results for the local and fully developed Nusselt number for a wide range of flow parameters. Busedra and Soliman 14 studies experimentally the same problem considered in Ref 4 by inclining the semicircular duct upward and downward with α= ±20° while the flat wall was in vertical position. They noted that, the axial variation of Nusselt number followed the trend observed in Ref 4 for the horizontal and upward inclinations. Theses values of Nusselt number increased with Grashof number and angle of inclination.

Numerical studies for the case of laminar mixed convection in the entrance region of horizontal semicircular ducts are to the knowledge of the authors, nonexistent. The present investigation therefore, concerned with buoyancy effects on laminar mixed heat transfer of simultaneously developing hydrodynamically and thermally for horizontal semicircular ducts with the H1 thermal boundary condition. These effects are to be examined over a wide range of Gr. The calculated parameters are local Nusselt numbers and average wall friction factors, as well as, the development of axial velocity and temperature profiles.

II. Mathematical Model Figure1 shows the geometry of a semicircular duct with radius ro with the flat surface always falling in the top.

The fluid enters the duct with uniform velocity equals to ue and a uniform temperature equals to te. The heat rate per unit length is assumed to be constant at any cross section along the flow axis of the duct and equals to q . The flow is assumed to be laminar and simultaneously developing hydrodynamically and thermally and H1 boundary condition was applied. The problem is analyzed for constant fluid properties with negligible viscous dissipation. The variation of density is taken into account only in the body forces (Boussinesq approximation). All terms containing the second derivative of any quantity with respect to x are neglected. The fluid pressure decomposition which quite wide used is given by Ref 15:

),(2)(1),,( θ+=θ rpxprxp (1)

Where 1p is the cross-sectional average pressure ,which is assumed to vary only in the x- direction .On the

other hand , 2p provides the driving force for the secondary flow within the cross section.

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This decoupling of pressure makes it possible to solve the three dimensional problem by using the marching

technique in which the solution is progressed stepwise in the axial direction with a two dimensional elliptic system (in the r and θ directions) to be solved at each axial step. The governing Navier-Stokes equations and the energy equation in cylindrical coordinates can be written in the following non dimensional form:

Figure 1. Geometry and coordinate system

R

W

V ro

θ

Section A-A

Ue,Te

g

X,U

A

A q

q

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Continuity:

0Pr

2

22)( =

∂∂

π+π

+θ∂

∂+

∂∂

XURWRV

R (2)

Axial Momentum:

UdXPd

XUUU

RW

RUV 2

221

Pr1

Pr

2

22

∇+

π+π

−=

∂∂

π+π

+θ∂

∂+

∂∂ (3)

Radial Momentum:

θ+∂∂

−+

+

θ∂∂

−∇=

∂∂

π+π

+θ∂

∂+

∂∂ sin2

2

2222

Pr

2

22 GrT

RP

RW

R

VW

RV

XVUV

RW

RVV (4)

Angular Momentum:

θ+θ∂

∂−−

−

θ∂∂

+∇=

∂∂

π+π

+θ∂

∂+

∂∂ cos21

2222

Pr

2

22 GrT

PRR

WV

R

WV

RW

XWUW

RW

RWV (5)

Energy:

TXTUT

RW

RTV 2

Pr

2

22Pr ∇=

∂∂

π+π

+θ∂

∂+

∂∂ (6)

Where

2

2

2112

θ∂

∂+

∂∂

∂∂

=∇RR

RRR

(7)

With dimensionless parameters of:

0h2

30pbh

2

20

*2

22b

*1

1e

b

00

h0

r2

2D,k

rqgGr ,kCρ

Pr ,νuDeR

ρνrpP ,

ρupP ,

kq

ttT

uu U,

νwr W,

νvrV ,

PrReDxX ,

rrR

+ππ

=νβ

=ν

==

==−

=

=====

(8)

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The dimensionless initial and boundary conditions are: At X = 0

θ===

=

θ===

and R allfor 02PWV fluid within the0P1P

and R allfor 0T wallsat the0U

fluid withn the1U

(9a)

At X>0

wallsat thewTT

wallsat the 0WVU=

=== (9b)

Starting from uniform flow at the inlet of the duct (defined by the equation (9a)) the solution for U, V, W, 1P ,

2P and T is to be determined as a function of X, R and θ. The solution is to be progressed along X until X= 0.4 for all Gr. This distance is sufficient to insure that the fully developed region has been reached.

Some important engineering relations have been put in dimensionless form:

)(

12)2(

2

bTwTkhhD

xNu−+π

π== (10)

The product fRe where the friction factor f is defined by:

2)2/1( bu

wfρ

τ= (11)

The parameter wτ was calculated by averaging the wall shear stress around the circumference of the duct .Thus average wall friction factor is given by the following relation:

π

=

∂∂

−π=

∂∂

−=

∂∂

+π

π= ∫ ∫ ∫

1

0

1

0 0dθ

1RRUdR

θθRUdR

0θθRU

22)(

4Ref (12)

The numerical method employed to solve these equations is based on the SIMPLER algorithm and using a marching method. It uses a staggered grid with 30, 50, 473 increments in the radial, angular and axial directions. For the discretization of the governing equations (2 through 6), the power-scheme suggested by Patankar16 was used. The procedure used to calculate Tw was described by Parakash and Liu17.

Because there is no theoretical data available for the simultaneously developing laminar forced or mixed

convection or even for laminar fully developed mixed convection for air in the semicircular duct with the flat wall in the bottom position, only the fully developed fRe and Nu for the case of forced convection (Gr=0.0) have been compared with the results obtained from the literature. These values are within 0.1%, 0.12% respectively, from the solution reported by Lei and Trupp18 and Lei and Trupp19.

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III. Results and Discussion The numerical investigations were carried out with Pr = 0.7 and a wide range of Gr. The following sections

represent the development of secondary flow patterns, axial velocity and temperature profiles, and the development of the quantities Nux and fRe along the flow direction.

Development of Secondary Flow: Figure 2 shows the development of secondary flow patterns for the case of Gr = 106. In station (I) the secondary

flow has a maximum cross-stream velocity of 16.8. The flow is symmetric and moves the fluid from retarding areas to the accelerating areas. In station (II) the flow consists of two counter rotating secondary flow cells and the maximum cross-stream velocity increased to the value of 49.3 due to the effect of more intense free convection. While in station (III) the two cells are fully established and the maximum cross–stream velocity is increased to the value of 72.9. Finally in station (IV) the flow reached the fully developed state and the secondary flow pattern remained essentially the same from this station on.

Development of axial velocity: Figure 3 shows the development of axial velocity contours for the case of forced convection (Gr = 0.0). The flow

is symmetric and the maximum velocity is forming at the center of the duct as the flow approaching the fully developed state. The development of axial velocity contours for the case of Gr = 106 is shown in Figure 4. In the absence of buoyancy effects at the early part of the entrance, the isovels are seen to be very much similar to those of pure forced convection. The velocity is constant in most of the flow domain with steep sharp changes very near the duct walls as shown in stations (I) and (II). As the flow proceeds further downstream, as in station (III) the effects of buoyancy begins to appear by shifting the maximum velocity towards the flat wall. In station (IV) the fully developed state has been reached and the flow consists of two maximum velocities near the duct flat wall.

Development of temperature profiles: Figures 6 and 7 show the development of (T-Tw) contours for the case of pure forced convection, Gr = 0.0 and

Gr = 106, respectively. The development of the temperature contours is quite similar to that of the axial velocity and the main effect of Gr is that it reduces the value of (T-Tw) which is due to mixing resulted from free convection. Also, two minimum (T-Tw) are shown to be forming toward the corners of the lower part of the duct.

Effect of Grashof number: The effect of Gr on the development of Nux is presented in Figure 8. The axial development of Nux at Gr<=103 closely follows the curve of pure forced convection indicating weak secondary flow. However for Gr>=105, the Nux is close to the forced convection only at smaller values of X, and as X increases Nux decreases to minimum and then rises up due to free convection effect to the fully developed value. Further, as Gr increases the minimum Nux shifts upstream and the thermal entrance length decreases. This behavior is also predicted experimentally by Maughan and Incropera20 for horizontal and inclined parallel plates, Barozzi, et al.21 for horizontal and inclined circular tubes and Busedra and Soliman14 for horizontal and inclined semicircular ducts and numerically by Mahaney, et al.10 for rectangular ducts. The increase in Gr enhances the Nux in the late stages of the developing and in the fully developed regions. The corresponding values of Nux at stations (III) and (IV) at Gr = 106 are 6.341 and 7.042 respectively which are 112% and 170% higher than those of forced convection (Gr = 0.0) at the same axial stations. Figure 9 shows the effect of Gr on the development of fRe. The axial development of fRe is similar to that of Nux in terms of its increase in the late part of the developing and fully developed regions with Gr. The corresponding values of fRe at stations (III) and (IV) at Gr = 106 are 21.086 and 22.109, respectively, which are 111% and 139% higher than those of forced convection (Gr = 0.0) at the same axial stations. The enhancement of fRe is less generally if compared with that of Nux.

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Max = 16.8Scale: = 73

Max = 49.3Scale: = 73

Max = 72.9Scale: = 73

Max = 63.4Scale: = 73

Umax=1.147 Umax=1.474

Umax=1.601 Umax=2.026

(I) X=1x10-3 (II) X=1x10-2

(III) X=1.67x10-2 (IV) X=1x10-1

Figure 2 Secondary Flow Patterns for Different Axial Stations at Gr=106

(I) X=1x10-3 (II) X=1x10-2

(III) X=1.67x10-2 (IV) X=1x10-1

Figure 3 Axial velocity contours at different axial stations for Gr= 0.0 .

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Umax=1.147Umax=1.472

Umax=1.589 Umax=1.667

(T-Tw)min= -.0152 (T-Tw)min= -.0457

(T-Tw)min= -.0571 (T-Tw)min= -.0954

(I) X=1x10-3 (III) X=1x10-2

(III) X=1.67x10-2 (IV) X=1x10-1

Figure 4 Axial velocity contours at different axial stations for Gr=106. (I) X=1x10-3 (II) X=1x10-2

(III) X=1.67x10-2 (IV) X=1x10-1

Figure 6 (T-Tw) Temperature contours at different axial stations for Gr= 0.0

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(T-Tw)min= -.0152 (T-Tw)min= -.0447

(T-Tw)min= -.0515 (T-Tw)min= -.0489

(I) X=1x10-3 (II) X=1x10-2

(III) X=1.67x10-2 (IV) X=1x10-1

Figure 7 (T-Tw) Temperature contours at different axial stations for Gr= 106

Figure 8 Effect of Gr on the axial development of Nux

1E-4 1E-3 0.01 0.13

6

910

20

30

40

50(IV)(III)(II)(I) Gr=0.0

Gr=103

Gr=105

Gr=106

Nux

X

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Figure 9 Effect of Gr on the axial development of fRe

Onset of the secondary flow: It is defined as the axial distance where the local Nux value departs the forced convection value by 5%. It is a

major indicator to where the free convection effects begin to appear. Results for the location of the onset of secondary flow are tabulated in Table1. From these results it is clear that as the Gr increases the location of the onset of the secondary flow moves upstream.

Table 1 Locations of the onset of the secondary flow for different Gr

Gr Location of the onset of the secondary flow 1x105 4.14x10-2 1x106 1.24x10-2

IV. Conclusions The main objective of this investigation was to study numerically the laminar mixed convection in the entrance

region for simultaneously developing of hydrodynamic and thermal boundary layers for horizontal semicircular ducts with the flat wall at the bottom for H1 thermal boundary condition, Pr=0.7 and a wide range of Gr. From these results the following conclusions can be made:

1) The general trend is that Nux increased in the late part of the developing and fully developed regions with

Gr. The axial variations of Nux follow the trend as noted in the previous works for horizontal ducts. 2) The values of fRe increases with Gr in the late part of the developing and the fully developed regions. 3) As Gr increases the location of the onset of the secondary flow moves upstream thus shortening the

entrance region.

1E-4 1E-3 0.01 0.110

20

30

40

5060708090

100 Gr=0.0

Gr=103

Gr=105

Gr=106

fRe

X

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16Patankar, S.V, " Numerical Heat Transfer and Fluid Flow", Mc-Graw Hill, New York,1980. 17Parakash, Y.D. and Liu,Y.D.," Analysis of Laminar Flow and Heat Transfer in the Entrance Region of an Internally Finned

Circular Tubes", J. Heat Transfer, Vol 107 , 1985, pp84-91. 18Lei, Q.M. and Trupp, A.C., " Maximum Velocity Location and Pressure Drop of Fully Developed Laminar flow in Circular

Sector Ducts", J. Heat Transfer, Vol 111, 1989, pp 1055-1087. 19Lei, Q.M. and Trupp, A.C., " Further Analysis of Laminar Flow Heat Transfer in Circular Sector Ducts", J. Heat Transfer,

Vol 111, 1989, pp 1088-1090. 20Maghan, J.R. and Incropera, F.P., " Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and

Inclined Channel", Int. J. Heat Mass Transfer, Vol 30, 1987, pp 1307-1318. 21Barozzi, G.S., Zanchini, E. and Mariotti, M., " Experimental Investigation of Combined Forced and Free Convection in

Horizontal and Inclined Tubes", MECCANICA, Vol 20, 1985, pp 18-27.