mixed convection heat transfer in truncated cone …

Kufa Journal of Engineering

Vol. 9, No. 4, October 2018, P.P. 286-307 Received 25 October 2017, accepted 31 January 2018

MIXED CONVECTION HEAT TRANSFER IN TRUNCATED

CONE ENCLOSURE AS SOLAR CONTAINER WITH

INTERNAL CENTERED TRIANGLE OBSTACLE

Kadhum A. Jehhef1 and Mohamed A. Siba2

1 Ph.D., Department of Mechanics, Institute of Technology, Middle Technical

University, Iraq. Email: [email protected]

2 Ph.D., Department of Mechanics, Institute of Technology, Middle Technical University,

Iraq. Email: [email protected]

http://dx.doi.org/10.30572/2018/kje/090420

ABSTRACT

In this study, the mixed convection heat transfer of the air inside a truncated cone enclosure

with aspect ratio of (0.75, 1.75, 2, 2.45 and 2.65) with centered triangle obstacle height varying

by (0, 2.5, 5, 7.5, 10, 12.5 and 20 cm) and the heated wall inclination angles of (20o, 30o, 40o,

50o, 60o), the Richardson number in the range of (7 to 11) was investigated numerically. The

results are addressed to automotive a suggested solar container with titled solar collector. The

heat transfer from the heat source (inclined solar collector) of the enclosure walls is investigated

for mixed convection as interaction of the forced convection flow between the inlet and outlet

port in the bottom wall. The parameters of heat source, Reynolds number, obstacle height,

enclosure aspect ratio, and left and right walls titled angles are considered in this work. The

numerical simulation of the problem is carried out using commercial CFD code. The results are

given in terms of the streamlines, isothermal, and the enclosure Nusselt number that

characterizes the heat transfer from the heat source and from the interior fluid to the enclosure

walls, respectively. The results show that the interaction of the main flow and the flow at the

heated walls and the buoyancy force at the heated walls increased by using a triangular obstacle

and by increasing the obstacle heights, it increased by 20% when using obstacle at position of

(h=5 cm). Also, it is found that the Nu increased with increasing Re and the wall heat flux. The

Nu increased with increasing Ri in the case of using (h=0 and 5 cm) but it decreased slightly in

the other cases and showed that the minimum value of Nu present at a heated wall inclination

of θ=50o but the maximum value at θ=30o.

KEYWORDS: Mixed convection, truncated cone, triangle obstacle, enclosures.

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.30572/2018/kje/090420

Kufa Journal of Engineering, Vol. 9, No. 4, October 2018 287

1. INTRODUCTION

The natural convection heat transfer problem in simple geometries enclosures is extensively

studied in the previous literatures in the recent years. The most cases are the Rayleigh Bernard

cell, vertically heated walls enclosures, and solar container Sridhar and Reddy, (2007). A

review of natural convection in cavity with various shapes has been presented in several studies

such as (but not limited to) Wang and Li, (1999); He et al., (2005); and Dixit and Babu, (2006).

A various shapes as cavities with inclination angles and wavy walls studied by Aounallah et al.,

(2007), and various trapezoidal cavities presented by Varol et al., (2009). The study of natural

convection was presented for triangular enclosures mostly performed by Fuad et al., (2007).

Jan, et al., (2007) published a theoretical paper on convective heat transfer in an enclosure with

titled wall to study the automotive headlights containing Light Emitting Diodes (LED). They

showed that the source convective heat transfer of the wall increases with increasing Reynolds

number and also increased with increasing the aspect ratio but it increasing with decreasing the

titled angles of the enclosure wall. Merk and Prins, (1953); Merk and Prins, (1954) studied the

iso-thermal axi-symmetric forms of the vertical cones and they showed that the vertical cone

has such a solution in steady state. Furthermore, Hossain and Paul , (2001) studied the effects

of the transpiration velocity on the laminar free convection flow from a vertical non-isothermal

cone due to increase the temperature gradient, the velocity and the surface temperature will

decrease. Ramanaiah and Kumaran, (1992) have performed an investigated study on…the

natural convection about a permeable cone and a cylinder subjected to boundary condition of

radiation. Alamgir, (1989) studied the natural convection with laminar flow from vertical cones

by using the integral method. Pop and Takhar, (1991) studied the compressibility effects in

laminar natural convection from a vertical cone. Papanicolaou and Jaluria, (1995) investigated

the effect of varying the position of the heat source inside an enclosure with a constant shape

and using a finite thickness of the enclosure sidewalls. Bapuji, et al., (2008) presented a

numerical analysis on the natural convection with unsteady laminar flow from an

incompressible viscous fluid flows past a vertical cone with uniform surface heat flux. Finally,

Elsayed, et al., (2016) studied the effect of heat generation or absorption and thermal radiation

on free convection flow and heat transfer over a truncated cone. Comparisons with previously

published works, the present study focused on the effect of presence of triangular obstacle

inside a truncated cone enclosure with tilted vertical walls on the mixed convection heat transfer

coefficients.

288 Kadhum A. Jehhef and Mohamed A. Siba

2. NUMERICAL SOLUTION

2.1. Geometry Specification

To study the effect of buoyancy-induced flow regime inside a truncated cone enclosure with

heated inclined vertical walls, a two-dimensional physical model and required boundary

conditions are displayed in Fig. 1a.

a) Enclosure dimensions

B) Numerical boundary conditions

Fig. 1. Description of problem geometry and boundary conditions.

2.2. Governing Equations

The presentation of the buoyancy-induced regime inside an enclosures performed by the same

basic equations of the general fluid motion, namely the conservation of mass, momentum, and

energy. The free convection heat transfer scenarios require at least one fluid parameter, density,

to vary within the domain of interest. The equation of the buoyancy-induced flows inside a

vertical truncated cone for incompressible flow presented by Ganguli et al., (2009):

dρ

dt+ ∇. (ρv⃗ ) = 0 1

d

dt(ρ. v⃗ ) + ∇. (ρv⃗ v⃗ ) = −∇P + ∇. τ⃗ + ρg⃗ 2


d

dt(ρ. T⃗⃗ ) + ∇. (ρcpv⃗ T⃗⃗ ) = ∇. [k∇. (T⃗⃗ )] 3

The stress tensor in the momentum expression, 𝜏⃗ , can be calculated by:

�⃗� = 𝛍 [(𝛁�⃗� + 𝛁�⃗� 𝐓) −𝟐

𝟑𝛁�⃗� 𝐈] 4

By making these simplifications and expansions, Eqs. 1 and 2 can be rewritten for a steady state

as ANSYS:

∇. (ρv⃗ ) = 0 5

∇. (ρv⃗ v⃗ ) = −∇P + μ(∇2. v⃗ ) + ρg⃗ 6

∇. (ρcpv⃗ T⃗⃗ ) = ∇. [k∇. (T⃗⃗ )] 7

The fluid density can be simplified at all points within the employed domain, by employing the

Oberbeck-Boussinesq approximation to linearize the temperature dependency of density. The

buoyancy force is redefined as Ganguli et al., (2009):

ρg⃗ = g⃗ [ρref − ρrefβ(T − Tf)] = ρrefg⃗ [1 − β(T − Tf)] 8

Where 𝛽 represent a fluid compressibility found via:

β = −1

V

∂V

∂P 9

The air can be considered as an ideal gas at nominal atmospheric pressures and temperatures,

the air compressibility found by:

β =1

Tf 10

A necessary simplification was made to the mass and momentum equations present in Eqs. 4

and 5 allows for the development of the final forms for incompressible flow:

∇(v⃗ ) = 0 11

v⃗ . (∇. v⃗ ) = −1

ρref∇P + vref(∇

2. v⃗ ) + g⃗ [1 − β(T − Tf)] 12

2.3. Boundary Conditions

Obstacle Surface: the fluid velocity at the surface of the solid must be zero in all directions:

vx = vy = 0 13

For adiabatic obstacle boundary condition:

∂T

∂n= 0 14


Right and left walls: the heat flux is assumed to be constant along the Right and left walls:

∂T

∂n=

q′′

kref 15

where 𝑛 refers to the direction normal to the surface and 𝑞′′ is the heat flux.

Top and bottom walls: the confining wall is assumed to be adiabatic, signifying a zero

temperature gradient normal to the partition surface.

∂T

∂n= 0 16

Air Inlet: the air inlet with constant velocity and temperature as following:

vy= Vin= 0.2, 0.3, 0.4 and 0.5, T=Tin=298 K 17

Air Outlet: The outlet vent, which specify a zero pressure differential normal to the open

surface:

∂P

∂n= 0 18

At the outlet section, the flow and temperature fields are assumed to be fully developed. The

outflow boundary condition has been implemented in the outlet section. This boundary

condition implies zero normal gradients for all flow variables except pressure. Thus, by using

sweeping outlet temperature boundary conditions as:

∂T

∂y= 0 19

2.4. Grid Independence Test

A grid independence test was performed to evaluate the effects of grid sizes on the results. Four

sets of mesh were generated using tetrahedral three dimensional meshes with 15552, 17565,

19225 and 20170 nodes. It was observed that the 19225 and 20170 nodes produce almost

approximated identical results with a percentage error of 0.135%. Hence, a domain with 20170

nodes was chosen to increase the computational accuracy. The summary of the grid

independence test results is shown in Table 1.

Table 1. Grid independence test results.

Nodes Outlet temperature (K) Percentage error (%)

15552 307.2546 1.165

17565 308.4568 0.833

19225 310.0887 0.186

20170 310.6256 0.135


2.5. Computational Grids and Numerical Method

To study the problem of the heat transfer by mixed convection parameters of the air inside a

truncated cone enclosure with heated left and right inclined walls and a triangular obstacle in

the middle of the enclosure. Also, the finite-volume computational nodes were generated for

the present two dimensional enclosure geometry. In an effort to simplify the construction of

each of this geometry and mesh files, the Auto-CAD and Gambit v. 2.3.16 software are used.

The air space modeled using a quadrilateral mesh by 20170 nodes and Fig. 2 displays an

example of this computational domain.

Fig. 2. The two dimensional truncated cone enclosure with aspect ratio.

The 2D computational domain exported to the FLUENT, in this study, to couple the inlet and

outlet pressure boundary conditions the SIMPLE algorithm was performed. The simulation

iteration was terminated when the mass, momentum, and energy residuals drop below 10-9 for

each simulation iteration. By using under-relaxation factors of 0.70 and 0.30 for the pressure

and momentum equations respectively, but the density and energy relaxation factors were left

at the default values of unity.

3. RESULTS AND DISCUSSION

The present numerical work has been presented in mixed convection system where the

Richardson number is in the range of Ri =7 to 11 for two different inlet Reynolds number

Re=600 to 1600. The mixed convection problem presented in the truncated cone enclosure, top

and bottom walls Were assumed as adiabatic. In this section, the laminar steady flow results

have been achieved. Firstly, the streamline and isotherm contours were presented for different

obstacle height varying for six different positions within the geometry (h= 0 or without obstacle,

2.5, 5, 7.5, 10, 12.5 and 20 cm (see Fig. (1b)) from the inlet port.


The present calculations were performed at constant Pr=0.712 for air [18], and constant position

and length of the heated wall of the left and right surfaces of the cone. In addition, its length is

about Lp=45 cm to simulate a solar container model of truncated cone enclosure geometry. The

heated wall with different inclination angles of θ= 20o, 30o, 40o, 50o and 60o, in order to obtain

various aspect ratio in the range of A= 0.75, 1.75, 2, 2.45 and 2.65. At each A, the Reynolds

number, Richardson number, Rayleigh number was varied in order to obtain the flow

conditions.

3.1. Effect of obstacle Position

In all cases of position of the obstacle chosen in this study the cases of h=0, 5, 10, and 20

presented in Fig. 3 for the same angle of inclination of the legs of the cone. However, the results

showed that the presence of the obstacle in the way of the inlet air flow will increase the

temperatures contour values near the heated walls. Furthermore, it makes a thermal stratified

zone on the upper region of the triangular obstacle due to redirection the air flow toward the

heated left and right walls instead of the upper wall in order to extract the heat to the bottom

port. But, when changing the obstacle position toward the top wall leads to increasing the air

flow distance that the researchers observed that the temperatures near the heated wall decreased,

and appeared the circulation region near the heated zone. Also, the results showed that the

thermal stratified zone upper the obstacle start decreasing until it disappeared gradually in the

case of h=20 cm.

The effect of changing the position of the triangular obstacle on the air flow streamlines, shown

in Fig. 4. It displayed the streamlines inside the truncated cone cavity with obstacle height 0 to

20 cm. The results showed that increasing the obstacle height will increase the air circulation

near the heated walls. The buoyancy force direction and momentum force identical along the

source heated surfaces at h=20 cm and h=5 cm. Long vortices appeared in the region below the

mainstream. The mainstream viscous forces drive rotating eddy against the buoyancy force at

the heated surfaces. These two positions delimit the possible minimum and maximum heat

transfer. Basically, the interaction of the main stream flow and heated surfaces flow increased

by using a triangular obstacle and by increasing its height. Where the main flow attached to the

heated wall but the eddies that generated by the viscous driven cause a heated walls which is

driven the heated buoyancy force. This effect can be observed clearly in the case of plotting the

y-direction velocity vector in Fig. 5 plotted the dynamic field for different buoyancy ratio values

and can show the vortices intensity increased by increasing the obstacle height. The fluid falls


downward from the upper region and the flow divided into two small cells turning from the

center to the cold sides.

3.2. Effect of enclosure angle

The effect of increasing the enclosure inclination angles on the isothermal lines is plotted in

Fig. 6. The results show that increasing inclination angles lead to decrease the thermal layers

near the heated walls due to transfer the walls from inclined to vertical position and decreasing

the buoyancy ratio values near the heated walls. Also, the increasing angles will decrease the

temperature upper the obstacle. For a various enclosure shape that caused by various titled

angles, the streamline were plotted, as shown in Fig. 7. The results obtained for a constant

Ra=2.36×107, Richardson number Ri=11, Reynolds number Re=1600, and obstacle position

h=7.5 cm. The results show that there are no significant changes in the streamlines between the

obstacle height and the heated walls. If the cavity size is increased strongly there is a small

effect on the streamlines in the area above the obstacle. In the zone between heated walls and

the outlet port which show that the flow affected by the enclosure inclination angle.

3.3. Effect of Rayleigh number and Reynolds number

Fig. 8 and Fig. 9 presented the results of the isothermal and streamline respectively at h=5.0

cm, inclination angle fixed at θ=60o and Richardson number at Ri = 11. And varied the

Rayleigh number from Ra = 1.57 ×106 to Ra = 2.09×108, and varied the inlet Reynolds number

from 600 to 1300 the results show the increasing Rayleigh number will increase the temperature

distribution between the obstacle and the heated surface. The Figs. also show that increasing

the Reynolds number lead to increase the streamlines above the obstacle region. And show that

the secondary flow (flow induced by the heated surface) increased when using the low Reynolds

number such as Re=600 and 1000 as compared with the main flow (flow induced by the inlet

velocity), but when increasing the Reynolds number up to 1000 show that the secondary flow

will disappear gradually at Reynolds number 1600.

Fig. 10 and Fig. 11 showed the velocity profile in term of obstacle height and enclosure angles

of inclination respectively. The figures show that the increasing the obstacle height from 0 to

20 cm lead to increase the velocity profile between the obstacle and the heated surfaces. But,

when increasing the inclination angles it was shown that the velocity in the region between the

obstacle and the heated surface is increased too.


3.4. Local and overall Nusselt number

The local Nusselt number Nu along the right side of the enclosure is presented in Fig. 12. It

shows also a dependence of the source power of the heated wall for high Reynolds number

Re=1600, however, it differs from the behavior of the Nuoverall. Cavity with obstacle of

h=7.5cm performed maximum heat transfer. The main flow is directed to the cavity surfaces by

the heated walls itself where this develops the convection at the enclosure walls. The results

show that the local Nusselt number (Nu) increased with increasing the heat flux and decreased

with percent by 76 % as transferred from the top to the bottom location of the heated wall at

Ra=2.36×107. However, Fig. 13 shows the dependence of the local Nusselt number on the

Reynolds numbers. It is clearly shown that the Nusselt number increased with increasing the

Reynolds number from 1000 to 1600 but the maximum Nusselt number found in Reynolds

number from 600 about Nu=35 in the region between the obstacle and the heated right wall.

For high Re, the main flow inside the cavity takes apart from the secondary flow. The two flows

(main flow and secondary flow) do not interact for positions like h=0 to 20 cm. However, the

convective fluid flow from the inlet to the outlet port stays at low temperature lead to very high

cavity Nu in Fig. 14. Moreover, the cavity Nu increases with increasing the height of the

obstacle toward the upper region of the enclosure but decreasing in the lower region. Fig. 15

presented the effect of the cone legs titled angle on the local Nu. The obstacle position remains

constant at h=7.5 cm. The results show that the increasing in the inclination angle will lead to

increase the maximum value of the local Nusselt number from 25 to 36 when increased the

inclination angle from 20o to 60o. The overall Nu and the cone cavity Nu are used to present

the heat transfer from the heated right wall as presented in Fig. 16. The results show that the

overall Nusselt number increases upon increasing the Reynolds number and with increasing the

obstacle position θ=20o, Ri=11, Ra=2.36×107. The overall Nusselt number increased by 20%

when using obstacle at a position of h=5 cm. The inlet Re has the sufficient effect on all studied

parameters. There is a considered enhancement of the convective heat transfer that can be

achieved with varying the obstacle height. But in case of using Richardson number as presented

in Fig. 17, it was shown that the overall Nusselt number increased with increasing the

Richardson number only case of h =0 and 5 cm but for cases of above 5 cm show that Nusselt

number decreased with increasing the Richardson number.


a) Without obsticale

b) h=5 cm

c) h=10.0 cm

d) h=20.0 cm

Fig. 3. Isothermal lines for air flow at θ=20o, Re = 1600, Ri = 11, Ra=2.36×107.

312.48

312.48

312.48

312.48

312.

48

312.

48

337.44

337.44

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4433

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362.40

362.

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362.40387.37

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412.

33

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310.20

310.20310.20

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X

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-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.01

0.02

0.03

0.04

0.05


a) h=0.0 cm

b) h=5.0

c) h=10 cm

d) h=20 cm

Fig. 4. Streamlines at air flow at θ=20o, Re = 1600, Ri= 11, Ra=2.36×107.

-0.3

2

-0.3

2

-0.2

1

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0.03

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X

Y

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.01

0.02

0.03

0.04

0.05


a) h=0.0

b) h=5.0 cm

c) h=10 cm

d) h=20 cm

Fig. 5. Velocity vector for air flow at θ=20o, Re = 1600, Ri= 11, Ra=2.36×107.

X

Y

0 0.025 0.05 0.0750

0.01

0.02

0.03

0.04

0.05

X

Y

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

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0.02

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Y

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

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X

Y

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.01

0.02

0.03

0.04

0.05


a) θ=30o

b) θ=40o

c) θ=50o

d) θ=60o

Fig. 6. Isothermal lines at air flow at h=7.5cm, Re = 1600, Ri = 11, Ra=2.36×107.

309.7

6

309.7

6

309.7

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309.7

6 329.2

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309.76 309.76

339.04 348.80

358.56

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0.02

0.03

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27

310.127

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36

350.636

350.636

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370.8

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370.891 370.891

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391.145

41

1.4

00

310.127

330.382

350.636

360.763370.891

X

Y

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

309.784

309

.78

4

309.784

30

9.7

84

329.

351

329.3

51

329.351

329.3

51

348.9

19

348.919

348.919

348.9

19

348.919

35

8.7

02

358.702

358.702

358.7

02

378.270

378.

270

378.2

70

39

7.8

38

397.8

38

436.973

329.351 329.351339.135348.919

358.702

368.4

86

368.486

309.784

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

309.896

309.8

96

309.

896

309.

896

329.687

329.6

87

329.687

329.6

87

349.4

79

349

.479

349.479

349.

479

369.2

7

369.27

369

.27

389.062

309.896319.791

309.896319.791

339.583

359.374359.374

349.479

369.27

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


a) θ=30o

b) θ=40o

c) θ=50o

d) θ=60o

Fig. 7. Streamlines for air flow at h=7.5 cm, Re = 1600, Ri= 11, Ra=2.36×107.

-0.2

8

-0.1

8

-0.18

-0.18-0.08

-0.0

8

-0.0

8

-0.0

8

-0.0

8

0.03

0.0

3

0.03

0.0

3

0.03

0.0

3

0.0

3

0.0

3

0.03

0.1

3

0.1

3

0.1

3

0.1

3

0.2

40.2

40

.24

0.24

0.34

0.3

4

X

Y

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.01

0.02

0.03

0.04

0.05

-0.2

92

-0.1

86

-0.1

86

-0.1

86

-0.1

86

-0.1

86

-0.081

-0.0

81

-0.081

-0.0

81

-0.081

-0.0

81

-0.0

81

0.02

5

0.025

0.0

25

0.0

25

0.0

25

0.0

25

0.025

0.025

0.0

25

0.1

31

0.13

1

0.131

0.1

31

0.1

31

0.2

36

0.2

36

0.2

36

0.3

42

0.342

0.3

42

0.4

47

X

Y

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-0.298

-0.1

98

-0.198 -0.198

-0.198

-0.0

98

-0.098

-0.0

98

-0.098

-0.0

98

0.002

0.0

02

0.0

02 0

.00

20.002

0.002

0.002

0.0

02

0.002

0.0

02

0.101

0.1

01

0.1

01

0.1

01

0.1

01

0.10

10.2

01

0.201

0.2

01

0.2

01

0.3

01

0.3

01

0.4

00

0.4

00 0

.40

0

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-0.2

83

-0.283

-0.1

78

-0.1

78

-0.1

78

-0.1

78

-0.1

78

-0.0

74

-0.0

74

-0.074

-0.0

74

-0.0

74

0.0

30

0.0

30

0.030

0.0

30

0.03

00.030

0.0

30

0.0

30

0.1

35

0.1

35

0.135

0.1

35

0.1

35

0.2

39

0.2

39

0.2390.2

39

0.3

44

0.3

44

0.3

44

0.4

48

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


a) Ra = 1.57E+06

b) Ra = 1.83E+07

c) Ra = 2.09E+08

Fig. 8. Isothermal lines for air flow at h=5.0 cm, θ=60o, Re = 1600, Ri = 11.

306.709

306.7

09

306.

709

30

6.7

09

320.126

320.1

26

320.1

26

320.

126

320.1

26

333.543

333.5

43

333.543

333.543

34

6.9

60

346.960

34

6.9

60

360.377

360.3

77

373.795

306.709

313.417

320.126

326.834 333.543

340.252

346.960

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

307.750

307.7

50

307.

750

307.7

50

323.250

323.2

50

323.2

50

323.2

50

338.7

50

338.

750

338.750

338.750

338.

750

354.250

35

4.2

50

354.250

354.

2503

69.7

50

385.250 400.750

307.750 307.750

323.250 323.250

346.500

354.2

50

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

308.8

03

308.803

308.8

03

326.409

326.4

09

326.4

09

326.

409

344.015

344.0

15

344.015

344.

015

361.

622

361.622

36

1.6

22

361.

622

379.2

28

379.228

308.803 308.803

326.409326.409 335.212

344.015

352.81936

1.62

2

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


a) Re= 600

b) Re= 1000

c) Re= 1300

Fig. 9. Streamlines lines for air flow at h=5.0 cm, θ=60o, Ri = 11.

-0.074

-0.074

-0.074-0.0

32

-0.0

32

-0.0

32

-0.032

-0.032

-0.0

32

0.010

0.0

10

0.0

10

0.0

10

0.0

10

0.0100

.010

0.0

10

0.05

20

.05

2

0.052

0.0

52

0.0

52

0.0

52

0.0

52

0.0

52

0.0

950

.095

0.1

37

0.1

79

-0.011

-0.0

11

-0.032

-0.074

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-0.108

-0.108

-0.1

08

-0.1

08-0

.04

6 -0.0

46

-0.0

46-0

.046

0.0

17

0.017

0.017

0.0

17

0.0

17

0.0

17

0.017

0.0

17

0.0

17

0.080

0.080

0.0

80

0.0

80

0.1

43

0.1

43

0.2

06

0.2

06 0.269

-0.046 -0.046

-0.014

-0.014

-0.014

0.017 0.017

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-0.2

28

-0.2

28

-0.1

44

-0.1

44

-0.1

44

-0.060

-0.060

-0.060

-0.0

60

-0.0

60

0.0

23

0.023

0.023

0.0

23

0.0

23

0.0

23

0.0

23

0.1

07

0.107

0.1

07

0.107

0.107

0.1

91

0.1910

.19

1

0.2

750

.2750.2

75

0.3

58

-0.0

18 -0.018

-0.018-0.018

X

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


Fig. 10. velocity centerline of the enclosure for different of the obstacle position at

θ=60o,h=5.0cm, Re=1600, Ra=2.36×107.

Fig. 11. velocity centerline of the enclosure for different of the obstacle position a h=5.0cm,

Re=1600, Ra=2.36×107.

x-direction

y-

ve

locity,(m

/s2

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

0.3

0.4

0.5h=0 cm

h=5 cm

h=10 cm

h= 20 cm


Fig. 12. Enclosure Local Nusselt number independence of the left heater length for different of

the heat source at θ=20o,h=7.5cm, Re=1600, Ra=2.36×107.

Fig. 13. Enclosure Local Nusselt number independence of the left heater length for different

Reynolds number for, θ=20o, h=7.5 cm, Ri=11, Ra=2.36×107.

Heater Length, L (m)

Lo

ca

lN

usslte

Nu

mb

er,

Nu

0 0.01 0.02 0.03 0.045

10

15

20

25

30

35

40

q''= 900 W/m2

q''= 800 W/m2

q''= 700 W/m2

q''= 600 W/m2


Lo

ca

lN

usslte

Nu

mb

er,

Nu

0 0.01 0.02 0.03 0.045

10

15

20

25

30

35

40

Re=1600

Re=1300

Re=1000

Re=600



obstacle position for, Re=1600, θ=20o, Ri=11, Ra=2.36×107.


inclination angle for, h=7.5 cm, Ri=11, Ra=2.36×107.


Lo

ca

lN

usslte

Nu

mb

er,

Nu

0 0.01 0.02 0.03 0.045

10

15

20

25

30

35

40

h=0.0 cm

h=2.5 cm

h=5.0 cm

h=7.5 cm

h=10.0 cm

h=12.5 cm

h=20.0 cm


Fig. 16. Effect of obstacle height on overall Nusselt number for Richardson number, θ=20o,

Ri=11, Ra=2.36×107.

Fig. 17. Effect of obstacle height on overall Nusselt number for Richardson number, Re=1600,

θ=20o, Ri=11, Ra=2.36×107.


4. CONCLUSIONS

This study emphasized on the mixed convection heat transfer of a suggested solar container

with subjected heat flux on the inclined right and left walls with different angles of inclination.

The forced flow of one inlet with different values of the Reynolds number and two outlet ports

is thermally coupled. The results of streamline and isothermal give a good indication up on the

air thermal and flow distribution inside the enclosure for all conditions employed. The mixed

convection heat transfer is presented by the local and overall Nu. The effect of Reynolds

number, the Richardson number, the heat flux in addition to the cavity specification such as the

triangular obstacle height, cone legs title angles and aspect ratio are studied in this work. The

present numerical results show that the Nu increases with increasing the inlet Re, with

increasing the triangular obstacle positions. On the other hand, by increasing the cone titled

angles in the case that the cavity area is maximized yields a rise of the overall Nu where natural

and forced convection compete.

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mixed convection heat transfer in truncated cone …

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