research article numerical investigation of...

Hindawi Publishing CorporationJournal of EnergyVolume 2013 Article ID 327179 7 pageshttpdxdoiorg1011552013327179

Research ArticleNumerical Investigation of Forced Convection in a Channel withSolid Block inside a Square Porous Block

Neda Janzadeh and Mojtaba Aghajani Delavar

Islamic Azad University Science and Research Ayatollah Amoli Branch Amol Iran

Correspondence should be addressed to Mojtaba Aghajani Delavar madelavarnitacir

Received 13 February 2013 Accepted 7 April 2013

Academic Editor S A Kalogirou

Copyright copy 2013 N Janzadeh and M Aghajani Delavar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The fluid flow and heat transfer in a porousmediumhave received considerable attention due to its importance inmany engineeringapplications In this study numerical investigation of fluid flow and heat transfer over a hot solid block inside a square porous blocklocated in a channel was carried outThe lattice Boltzmannmethod with nine velocities D2Q9 was used for numerical simulationsBrinkman-Forchheimer model was successfully used to simulate fluid flow in porous media The effects of parameters such asporosity Reynolds number on flow pattern and heat transfer were studied The different effects of mentioned parameters werediscussed in the paper

1 Introduction

Thefluid flow and transport phenomena in a porousmediumhave been used in many fields of science and engineeringsuch as geothermal energy extraction solar collector solarabsorbers food processing fuel cells petroleum processingcatalytic and chemical particle beds transpiration coolingelectronic cooling drying processes porous bearing nuclearreactors and many others

Thermal Insulation Fluid flow and convective heat transfer inconduits fully and partially filled with porous medium havebeen investigated analytically experimentally and numer-ically by many researchers Poulikakos and Kazmierczak[1] theoretically studied the convection heat transfer in thedeveloped region of two parallel plates and in a circularpipe with porous media attached to the wall Fu et al [2]numerically simulated the convection heat transfer througha single porous block mounted on the heated wall and theystudied the effect of different parameters including the porousblock size and length and porosity of porous block andReynolds number on convection heat transfer enhancementLi et al [3] numerically investigated the effect of staggeredporous blocks mounted on the wall in a laminar flow channelon the velocity field and local heat transfer Huang et al

[4] numerically studied steady-state heat transfer to a fluidpassing through multiple heated block mounted on the wallof a channel with porous covers

Guo et al [5] investigated numerically the heat transfer ofpulsating flow through a pipe with a porousmedium attachedto the pipe wall In their work they used conventional controlvolume method Hamdan et al [6] used an implicit finite-difference method to simulate forced convection in a poroussubstrate inserted in the core of a parallel plate channel Jianget al [7] investigate the forced convection heat transfer inplate channels saturated with air and water which containsintered bronze porous media

The lattice Boltzmann method (LBM) is a powerfulnumerical technique based on kinetic theory for simulationof fluid flows and modeling the physics in fluids [8ndash10]The problem of convective heat in porous medium wasinvestigated numerically by Guo and Zhao [11] In their workthey used the lattice Boltzmann method to simulate thetemperature field Seta et al [12] applied the lattice Boltzmannmethod to simulation of natural convection in porous mediaShokouhmand et al [13] simulated the laminar flow andconvective heat transfer in a porous medium inside the coreof two parallel plates of a conduit using lattice Boltzmannmethod

2 Journal of Energy

Hot solid block inside a square porous block may beapplied in some engineering application such as porousmedia as an active layer in reacting chemical flows fuel cellheat exchangers and electronic devices cooling

2 Numerical Model

The general form of lattice Boltzmann equation with ninevelocities D2Q9 with external force can be written as [9]

119891119896 ( + 119888119896Δ119905 119905 + Δ119905) = 119891119896 ( 119905)

+Δ119905

120591[119891

eq119896

( 119905) minus 119891119896 ( 119905)] + Δ119905119896

119891eq119896

= 120596119896 sdot 120588 [1 +119888119896 sdot

1198882119904

+1

2

( 119888119896 sdot )2

1198884119904

minus1

2

sdot

1198882119904

]

(1)

where 119888119896 is the discrete lattice velocity in direction 119896 119896 is theexternal force Δ119905 denotes the lattice time step 120591 is the latticerelaxation time 119891eq

119896denotes the equilibrium distribution

function 120596119896 is weighting factor and 120588 is the lattice fluiddensity To consider both the flow and the temperaturefields the thermal LBM utilizes two distribution functions119891 and 119892 for flow and temperature fields respectively The 119891distribution function is as same as discussed previously the119892 distribution function is as follows [9]

119892119896 ( + 119888119896Δ119905 119905 + Δ119905) = 119892119896 ( 119905)

+Δ119905

120591119892[119892

eq119896( 119905) minus 119892119896 ( 119905)]

119892eq119896

= 120596119896 sdot 119879 sdot [1 +119888119896 sdot

1198882119904

]

(2)

The flow properties are defined as (i denotes the componentof the Cartesian coordinates)

120588 = sum119896

119891119870 120588119906119894 = sum119896

119891119896119888119896119894 119879 = sum119896

119892119896 (3)

The Brinkman-Forchheimer equation was used for flow inporous regions that is written as [9 12]

120597

120597119905+ ( sdot nabla) (

120576) = minus

1

120588nabla (120576119901) + 120592effnabla

2

+ (minus120576120592

119870 minus

175

radic150120576119870|| + 120576)

(4)

where 120576 is the porosity K is the permeability 120592eff is theeffective viscosity 120592 is the kinematic viscosity and 119866 is theacceleration due to gravity The last term in the right hand inthe parenthesis is the total body force 119865 which was writtenby using the widely used Ergunrsquos relation [16] For porousmedium the corresponding distribution functions are as the

same as (1) But the equilibrium distribution functions andthe best choice for the forcing term are

119891eq119896

= 120596119896 sdot 120588 sdot [1 +119888119896 sdot

1198882119904

+1

2

( 119888119896 sdot )2

1205761198884119904

minus1

2

2

1205761198882119904

]

119865119896 = 120596119896120588(1 minus1

2120591V)[

119888119896 sdot

1198882119904

+( 119888119896 119888119896)

1205761198884119904

minus sdot

1205761198882119904

]

(5)

The forcing term 119865119896 defines the fluid velocity as

= sum119896

119888119896119865119896120588

+Δ119905

2 (6)

According the previous equations is related to so (6) isnonlinear for the velocity A temporal velocity V is used tosolve this nonlinear problem [13] as follows

=V

1198880 + radic11988820+ 1198881 |V|

V = sum119896

119888119896119891119896120588

+Δ119905

2

1198880 =1

2(1 + 120576

Δ119905

2

120592

119870) 1198881 = 120576

Δ119905

2

175

radic1501205763119870

(7)

The effective thermal conductivity 119896eff of the porousmediumshould be recognized for proper investigation of conjugateconvection and conduction heat transfer in porous zonewhich was calculated by [16]

119896eff = 119896119891 [(1 minus radic1 minus 120576) +2radic1 minus 120576

1 minus 120590119861

times((1 minus 120590) 119861

(1 minus 120590119861)2ln( 1

120590119861) minus

119861 + 1

2minus

119861 minus 1

1 minus 120590119861)]

119861 = 125[1 minus 120576

120576]109

120590 =119896119891

119896119904

(8)

3 Boundary Conditions

From the streaming process the distribution functions out ofthe domain are known The unknown distribution functionsare those toward the domain In Figure 1 the unknowndistribution function which needs to be determined areshown as dotted lines Regarding the boundary conditionsof the flow field the solid walls are assumed to be no slipand thus the bounce-back scheme is applied This schemespecifies the outgoing directions of the distribution functionsas the reverse of the incoming directions at the boundarysites For example for flow field in the north boundary thefollowing conditions is used

1198914119899 = 1198912119899 1198917119899 = 1198915119899 1198918119899 = 1198916119899 (9)

In this channel the inlet velocity at west boundary is known119906 = 119906119908 = 119906inlet In LBM method the inward distributionfunctions at the boundaries have to be specified So the

Journal of Energy 3

1

2

3

4

56

7 8

North

South

West East

Computational

domain

1

2

3

4

56

7 8

1

2

3

4

56

7 8

1

2

3

4

56

7 8

Figure 1 Domain boundaries and known (solid lines) and un-known (dotted lines) distribution functions

Adiabatic

Adiabatic

8119867

1198678

1198672

1198674

119867

Figure 2 The computational domain

1198911 1198915 and 1198918 are unknowns for west boundary conditionThe unknown distribution functions are calculated as

120588119908 =1

1 minus 119906119908[1198910 minus 1198912 + 1198914 + 2 (1198913 + 1198916 + 1198917)]

1198911119899 = 1198913119899 +2

3120588119908 1198915119899 = 1198917119899 minus

1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

1198918119899 = 1198916119899 +1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

(10)

In this problem the outlet velocity is unknown The eastboundary condition in Figure 1 represents the outlet condi-tion Then 1198913 1198916 and 1198917 need to be calculated at the eastboundary as

1198913119899 = 21198913119899minus1 minus 1198913119899minus2 1198916119899 = 21198916119899minus1 minus 1198916119899minus2

1198917119899 = 21198917119899minus1 minus 1198917119899minus2(11)

Table 1 Simulation parameters

L 80 cm120576 03-05-07-09Pr 07H 10 cmRe 25-50-75-100119879inletndash119879wall solid block 20∘C ndash40∘C

Table 2 Comparison of averaged Nusselt number between LBMand Kays and Crawford [17]

119902101584010158402119902101584010158401

05 10 15Nu1

Kays and Crawford [17] 1748 823 1119LBM 1725 816 1110

Nu2Kays and Crawford [17] 651 823 700LBM 649 816 691

Furthermore for the temperature field the local temperatureis defined by (3) For isothermal boundaries such as a bottomhot wall the unknown distribution functions are evaluated as

1198922119899 = 119879ℎ (1205962 + 1205964) minus 1198924119899 1198925119899 = 119879ℎ (1205965 + 1205967) minus 1198927119899

1198926119899 = 119879ℎ (1205966 + 1205968) minus 1198928119899

(12)

where 119899 is the lattice on the boundary For adiabatic boundarycondition such as a bottom wall the unknown distributionfunctions are evaluated as

1198922119899 = 1198922119899minus1 1198925119899 = 1198925119899minus1 1198926119899 = 1198926119899minus1 (13)

4 Computational Domain

Computational domain consists of a hot solid block insidea square porous block located in a channel (Figure 2) Thechannel height length and other simulation parameters areillustrated in the Figure 2 and Table 1

5 Validation and Grid Independent Check

In this study flow pattern and thermal field were simulatedin a channel with a hot solid block inside a porous block byusing lattice Boltzmann method Figure 3 compares well thevelocity profile for flow in a clear channel captured by LBMand results presented by Nield [14] In this figure the resultof simulation is compared well with Seta et al [15] Table 2shows the computed Nusselt number by LBM and Kays andCrawford [17] and good agreement is obtained

6 Results and Discussion

In the present work the thermal lattice Boltzmann modelwith nine velocities was used to solve the force convection

4 Journal of Energy

00

02

04

06

08

1

Nield (2004)Present study

0 02 04 06 08 102 04 06 08 10

02

04

06

08

1

Present study (Re = 10)Present study (Re = 1)Seta et al (2006)

119906119906

max

119906119906

max

119910119867119910119867

Figure 3 Comparison of velocity profile for clear channel between Nield [14] and LBM and porous channel flow between LBM and Setaet al [15]

18

44

02 18

38

38

336

12

08

1 15 2 250

05

1

06 2214

42 3628

3632

14

118

3

1 15 2 250

05

1

3235

35

32

21 27

3131

1 15 2 250

05

1

37 35 3233

35

33 36

2631

3124

1 15 2 250

05

1

2

42

02 14 22

34 26

34

42

3

1 15 2 250

05

1

34 30 29

32

28

22

35

29

2621

1 15 2 250

05

1

18

44

02 1 2

4234

12

3426

26

08

1 15 2 250

05

1

28 2635

26

23 322423

21

1 15 2 250

05

1

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

120576=09

120576=07

120576=05

120576=03

Figure 4 Velocity contours (UUinlet (Re=25)) temperature contours for different porosities

Journal of Energy 5

1 15 2 25

1 15 2 25

28

242321

Re=50

Re=25

Re=75

Re=100

1

21

06 13

2 13

0414

0

05

1

35 32 3135

23 28

35

0

05

1

44

04 1418

12

32

443

12

0

05

1

34

33 30

30

22

27

28

0

05

1

15

655

05 2

6

3

45

3

0

05

1

3330

3228

2421

31

25

0

05

1

05

975

4

2

7 75

25

35

1 15 2 25

1 15 2 251 15 2 25

1 15 2 251 15 2 25

1 15 2 25

0

05

1

3332

30

28

0

05

1

119909119867

119909119867

119909119867

119909119867 119909119867

119909119867

119909119867

119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

Figure 5 Velocity vectors and contours (UUinlet (Re=25)) for different Reynolds numbers

heat transfer in a channel containing hot solid block insidea square porous block The effects of porosity and Reynoldsnumber on the flow field and convective heat transfer wereinvestigated

Figure 4 shows the velocity and temperature contoursfor different values of porosity at Re = 50 With increasingthe porosity velocity rises slightly in the top and bottom ofporous block because fluid tends to flow in clear passageswith lower pressure drop At higher values of porosity fluidchanges easier its path to clear passages so velocity increasesin these passages

According to (8) with increasing the porosity the effectivethermal conductivity in the porous block reduces At highervalue of porosity the heat transfer between fluid and solidblock decreases So the fluid temperature will increase

The velocity and temperature contours for differentReynolds number have been drown in Figure 5

It can be seen that as Reynolds number increases the flowvelocity increases With increasing the velocity the convec-tive coefficient (ℎ) increased according to (119876 = ℎ119860(119879wall minus119879fluid)) and the magnitude of heat transfer increases ButAccording to (119876 = 119898119888Δ119879 = 120588119906119860(119879out minus119879in)) with increasingthe velocity the temperature gradient of the fluid reduces Sothe outlet fluid temperature will decrease

The temperature and velocity profiles for variousReynolds numbers are represented in Figure 6 As expectedthe flow velocity increases with increasing the Reynoldsnumber The higher values of Reynolds number cause thetemperature of zones to be decreased due to the decreasein heat transfer process In Figure 7 the average fluid

6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)

Figure 6 velocity and temperature profiles at different Reynolds numbers (porosity = 07 119909119867 = 716)

Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)

Figure 7 Averaged fluid temperature at different porosities

temperature for various Reynolds numbers at differentporosity is drown that shows the effect of Reynolds numberon temperature more than porosity As mentioned beforeat higher porosity obtains a lower average temperature forfluid

7 Conclusion

Fluid flow and heat transfer in a channel with solid blockinside a square porous block were carried out in this paper

The effect of parameters including porosity and Reynoldsnumber on the velocity and thermal field was investigatednumerically by using lattice Boltzmann method Decreasingthe porosity leads to a higher temperature of fluid due toincreasing 119870eff which causes the heat transfer enhancementin porous block With increasing the Reynolds number thefluid velocity increased and the average fluid temperaturewill be reduced

References

[1] D Poulikakos and M Kazmierczak ldquoForced convection in aduct partially filled with a porous materialrdquo Journal of HeatTransfer vol 109 no 3 pp 653ndash662 1987

[2] W S Fu H C Huang and W Y Liou ldquoThermal enhancementin laminar channel flow with a porous blockrdquo InternationalJournal of Heat andMass Transfer vol 39 no 10 pp 2165ndash21751996

[3] H Y Li K C Leong L W Jin and J C Chai ldquoAnalysis offluid flow and heat transfer in a channel with staggered porousblocksrdquo International Journal of Thermal Sciences vol 49 no 6pp 950ndash962 2010

[4] P C Huang C F Yang J J Hwang and M T Chiu ldquoEnhance-ment of forced-convection cooling of multiple heated blocks ina channel using porous coversrdquo International Journal of Heatand Mass Transfer vol 48 no 3-4 pp 647ndash664 2005

[5] Z Guo S Y Kim and H J Sung ldquoPulsating flow and heattransfer in a pipe partially filled with a porous mediumrdquoInternational Journal of Heat and Mass Transfer vol 40 no 17pp 4209ndash4218 1997

[6] M O Hamdan M A Al-Nimr and M K Alkam ldquoEnhancingforced convection by inserting porous substrate in the coreof a parallel-plate channelrdquo International Journal of NumericalMethods forHeat and Fluid Flow vol 10 no 5 pp 502ndash517 2000

[7] P X Jiang M Li T J Lu L Yu and Z P Ren ldquoExperimentalresearch on convection heat transfer in sintered porous plate

Journal of Energy 7

channelsrdquo International Journal of Heat and Mass Transfer vol47 no 10-11 pp 2085ndash2096 2004

[8] S SucciTheLattice BoltzmannEquation for FluidDynamics andBeyond Clarendon Press Oxford UK 2001

[9] A A Mohammad Lattice Boltzmann Method Fundamentalsand Engineering Applications with Computer Codes SpringerLondon UK 2011

[10] P H Kao Y H Chen and R J Yang ldquoSimulations of themacroscopic and mesoscopic natural convection flows withinrectangular cavitiesrdquo International Journal of Heat and MassTransfer vol 51 no 15-16 pp 3776ndash3793 2008

[11] Z Guo and T S Zhao ldquoA lattice Boltzmann model for convec-tion heat transfer in porous mediardquo Numerical Heat TransferPart B vol 47 no 2 pp 157ndash177 2005

[12] T Seta E Takegoshi and K Okui ldquoLattice Boltzmann simula-tion of natural convection in porous mediardquo Mathematics andComputers in Simulation vol 72 no 2-6 pp 195ndash200 2006

[13] H Shokouhmand F Jam and M R Salimpour ldquoSimulation oflaminar flow and convective heat transfer in conduits filled withporous media using lattice Boltzmann Methodrdquo InternationalCommunications in Heat and Mass Transfer vol 36 no 4 pp378ndash384 2009

[14] D A Nield ldquoForced convection in a parallel plate channel withasymmetric heatingrdquo International Journal of Heat and MassTransfer vol 47 no 25 pp 5609ndash5612 2004

[15] T Seta E Takegoshi and K Okui ldquoThermal lattice Boltzmannmodel for incompressible flows through porous mediardquo Journalof Thermal Science and Technology vol 1 no 2 pp 90ndash1002006

[16] S Ergun ldquoFluid flow through packed columnsrdquo ChemicalEngineering Progress vol 48 no 2 pp 89ndash94 1952

[17] W M Kays and M E Crawford Solutions Manual ConvectiveHeat and Mass Transfered McGraw-Hill New York NY USA3rd edition 1993

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

2 Journal of Energy

Hot solid block inside a square porous block may beapplied in some engineering application such as porousmedia as an active layer in reacting chemical flows fuel cellheat exchangers and electronic devices cooling

2 Numerical Model

The general form of lattice Boltzmann equation with ninevelocities D2Q9 with external force can be written as [9]

119891119896 ( + 119888119896Δ119905 119905 + Δ119905) = 119891119896 ( 119905)

+Δ119905

120591[119891

eq119896

( 119905) minus 119891119896 ( 119905)] + Δ119905119896

119891eq119896

= 120596119896 sdot 120588 [1 +119888119896 sdot

1198882119904

+1

2

( 119888119896 sdot )2

1198884119904

minus1

2

sdot

1198882119904

]

(1)

where 119888119896 is the discrete lattice velocity in direction 119896 119896 is theexternal force Δ119905 denotes the lattice time step 120591 is the latticerelaxation time 119891eq

119896denotes the equilibrium distribution

function 120596119896 is weighting factor and 120588 is the lattice fluiddensity To consider both the flow and the temperaturefields the thermal LBM utilizes two distribution functions119891 and 119892 for flow and temperature fields respectively The 119891distribution function is as same as discussed previously the119892 distribution function is as follows [9]

119892119896 ( + 119888119896Δ119905 119905 + Δ119905) = 119892119896 ( 119905)

+Δ119905

120591119892[119892

eq119896( 119905) minus 119892119896 ( 119905)]

119892eq119896

= 120596119896 sdot 119879 sdot [1 +119888119896 sdot

1198882119904

]

(2)

The flow properties are defined as (i denotes the componentof the Cartesian coordinates)

120588 = sum119896

119891119870 120588119906119894 = sum119896

119891119896119888119896119894 119879 = sum119896

119892119896 (3)

The Brinkman-Forchheimer equation was used for flow inporous regions that is written as [9 12]

120597

120597119905+ ( sdot nabla) (

120576) = minus

1

120588nabla (120576119901) + 120592effnabla

2

+ (minus120576120592

119870 minus

175

radic150120576119870|| + 120576)

(4)

where 120576 is the porosity K is the permeability 120592eff is theeffective viscosity 120592 is the kinematic viscosity and 119866 is theacceleration due to gravity The last term in the right hand inthe parenthesis is the total body force 119865 which was writtenby using the widely used Ergunrsquos relation [16] For porousmedium the corresponding distribution functions are as the

same as (1) But the equilibrium distribution functions andthe best choice for the forcing term are

119891eq119896

= 120596119896 sdot 120588 sdot [1 +119888119896 sdot

1198882119904

+1

2

( 119888119896 sdot )2

1205761198884119904

minus1

2

2

1205761198882119904

]

119865119896 = 120596119896120588(1 minus1

2120591V)[

119888119896 sdot

1198882119904

+( 119888119896 119888119896)

1205761198884119904

minus sdot

1205761198882119904

]

(5)

The forcing term 119865119896 defines the fluid velocity as

= sum119896

119888119896119865119896120588

+Δ119905

2 (6)

According the previous equations is related to so (6) isnonlinear for the velocity A temporal velocity V is used tosolve this nonlinear problem [13] as follows

=V

1198880 + radic11988820+ 1198881 |V|

V = sum119896

119888119896119891119896120588

+Δ119905

2

1198880 =1

2(1 + 120576

Δ119905

2

120592

119870) 1198881 = 120576

Δ119905

2

175

radic1501205763119870

(7)

The effective thermal conductivity 119896eff of the porousmediumshould be recognized for proper investigation of conjugateconvection and conduction heat transfer in porous zonewhich was calculated by [16]

119896eff = 119896119891 [(1 minus radic1 minus 120576) +2radic1 minus 120576

1 minus 120590119861

times((1 minus 120590) 119861

(1 minus 120590119861)2ln( 1

120590119861) minus

119861 + 1

2minus

119861 minus 1

1 minus 120590119861)]

119861 = 125[1 minus 120576

120576]109

120590 =119896119891

119896119904

(8)

3 Boundary Conditions

From the streaming process the distribution functions out ofthe domain are known The unknown distribution functionsare those toward the domain In Figure 1 the unknowndistribution function which needs to be determined areshown as dotted lines Regarding the boundary conditionsof the flow field the solid walls are assumed to be no slipand thus the bounce-back scheme is applied This schemespecifies the outgoing directions of the distribution functionsas the reverse of the incoming directions at the boundarysites For example for flow field in the north boundary thefollowing conditions is used

1198914119899 = 1198912119899 1198917119899 = 1198915119899 1198918119899 = 1198916119899 (9)

In this channel the inlet velocity at west boundary is known119906 = 119906119908 = 119906inlet In LBM method the inward distributionfunctions at the boundaries have to be specified So the

Journal of Energy 3

1

2

3

4

56

7 8

North

South

West East

Computational

domain

1

2

3

4

56

7 8

1

2

3

4

56

7 8

1

2

3

4

56

7 8


Adiabatic

Adiabatic

8119867

1198678

1198672

1198674

119867



120588119908 =1

1 minus 119906119908[1198910 minus 1198912 + 1198914 + 2 (1198913 + 1198916 + 1198917)]

1198911119899 = 1198913119899 +2

3120588119908 1198915119899 = 1198917119899 minus

1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

1198918119899 = 1198916119899 +1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

(10)







119902101584010158402119902101584010158401

05 10 15Nu1




1198922119899 = 119879ℎ (1205962 + 1205964) minus 1198924119899 1198925119899 = 119879ℎ (1205965 + 1205967) minus 1198927119899

1198926119899 = 119879ℎ (1205966 + 1205968) minus 1198928119899

(12)









4 Journal of Energy

00

02

04

06

08

1


0 02 04 06 08 102 04 06 08 10

02

04

06

08

1


119906119906

max

119906119906

max

119910119867119910119867


18

44

02 18

38

38

336

12

08

1 15 2 250

05

1

06 2214

42 3628

3632

14

118

3

1 15 2 250

05

1

3235

35

32

21 27

3131

1 15 2 250

05

1

37 35 3233

35

33 36

2631

3124

1 15 2 250

05

1

2

42

02 14 22

34 26

34

42

3

1 15 2 250

05

1

34 30 29

32

28

22

35

29

2621

1 15 2 250

05

1

18

44

02 1 2

4234

12

3426

26

08

1 15 2 250

05

1

28 2635

26

23 322423

21

1 15 2 250

05

1

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

120576=09

120576=07

120576=05

120576=03


Journal of Energy 5

1 15 2 25

1 15 2 25

28

242321

Re=50

Re=25

Re=75

Re=100

1

21

06 13

2 13

0414

0

05

1

35 32 3135

23 28

35

0

05

1

44

04 1418

12

32

443

12

0

05

1

34

33 30

30

22

27

28

0

05

1

15

655

05 2

6

3

45

3

0

05

1

3330

3228

2421

31

25

0

05

1

05

975

4

2

7 75

25

35

1 15 2 25

1 15 2 251 15 2 25

1 15 2 251 15 2 25

1 15 2 25

0

05

1

3332

30

28

0

05

1

119909119867

119909119867

119909119867

119909119867 119909119867

119909119867

119909119867

119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867








6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)


Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)



7 Conclusion



References








Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 3

1

2

3

4

56

7 8

North

South

West East

Computational

domain

1

2

3

4

56

7 8

1

2

3

4

56

7 8

1

2

3

4

56

7 8


Adiabatic

Adiabatic

8119867

1198678

1198672

1198674

119867



120588119908 =1

1 minus 119906119908[1198910 minus 1198912 + 1198914 + 2 (1198913 + 1198916 + 1198917)]

1198911119899 = 1198913119899 +2

3120588119908 1198915119899 = 1198917119899 minus

1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

1198918119899 = 1198916119899 +1

2(1198912119899 minus 1198914119899) +

1

6120588119908119906119908

(10)







119902101584010158402119902101584010158401

05 10 15Nu1




1198922119899 = 119879ℎ (1205962 + 1205964) minus 1198924119899 1198925119899 = 119879ℎ (1205965 + 1205967) minus 1198927119899

1198926119899 = 119879ℎ (1205966 + 1205968) minus 1198928119899

(12)









4 Journal of Energy

00

02

04

06

08

1


0 02 04 06 08 102 04 06 08 10

02

04

06

08

1


119906119906

max

119906119906

max

119910119867119910119867


18

44

02 18

38

38

336

12

08

1 15 2 250

05

1

06 2214

42 3628

3632

14

118

3

1 15 2 250

05

1

3235

35

32

21 27

3131

1 15 2 250

05

1

37 35 3233

35

33 36

2631

3124

1 15 2 250

05

1

2

42

02 14 22

34 26

34

42

3

1 15 2 250

05

1

34 30 29

32

28

22

35

29

2621

1 15 2 250

05

1

18

44

02 1 2

4234

12

3426

26

08

1 15 2 250

05

1

28 2635

26

23 322423

21

1 15 2 250

05

1

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

120576=09

120576=07

120576=05

120576=03


Journal of Energy 5

1 15 2 25

1 15 2 25

28

242321

Re=50

Re=25

Re=75

Re=100

1

21

06 13

2 13

0414

0

05

1

35 32 3135

23 28

35

0

05

1

44

04 1418

12

32

443

12

0

05

1

34

33 30

30

22

27

28

0

05

1

15

655

05 2

6

3

45

3

0

05

1

3330

3228

2421

31

25

0

05

1

05

975

4

2

7 75

25

35

1 15 2 25

1 15 2 251 15 2 25

1 15 2 251 15 2 25

1 15 2 25

0

05

1

3332

30

28

0

05

1

119909119867

119909119867

119909119867

119909119867 119909119867

119909119867

119909119867

119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867








6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)


Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)



7 Conclusion



References








Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















4 Journal of Energy

00

02

04

06

08

1


0 02 04 06 08 102 04 06 08 10

02

04

06

08

1


119906119906

max

119906119906

max

119910119867119910119867


18

44

02 18

38

38

336

12

08

1 15 2 250

05

1

06 2214

42 3628

3632

14

118

3

1 15 2 250

05

1

3235

35

32

21 27

3131

1 15 2 250

05

1

37 35 3233

35

33 36

2631

3124

1 15 2 250

05

1

2

42

02 14 22

34 26

34

42

3

1 15 2 250

05

1

34 30 29

32

28

22

35

29

2621

1 15 2 250

05

1

18

44

02 1 2

4234

12

3426

26

08

1 15 2 250

05

1

28 2635

26

23 322423

21

1 15 2 250

05

1

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119909119867 119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

120576=09

120576=07

120576=05

120576=03


Journal of Energy 5

1 15 2 25

1 15 2 25

28

242321

Re=50

Re=25

Re=75

Re=100

1

21

06 13

2 13

0414

0

05

1

35 32 3135

23 28

35

0

05

1

44

04 1418

12

32

443

12

0

05

1

34

33 30

30

22

27

28

0

05

1

15

655

05 2

6

3

45

3

0

05

1

3330

3228

2421

31

25

0

05

1

05

975

4

2

7 75

25

35

1 15 2 25

1 15 2 251 15 2 25

1 15 2 251 15 2 25

1 15 2 25

0

05

1

3332

30

28

0

05

1

119909119867

119909119867

119909119867

119909119867 119909119867

119909119867

119909119867

119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867








6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)


Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)



7 Conclusion



References








Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 5

1 15 2 25

1 15 2 25

28

242321

Re=50

Re=25

Re=75

Re=100

1

21

06 13

2 13

0414

0

05

1

35 32 3135

23 28

35

0

05

1

44

04 1418

12

32

443

12

0

05

1

34

33 30

30

22

27

28

0

05

1

15

655

05 2

6

3

45

3

0

05

1

3330

3228

2421

31

25

0

05

1

05

975

4

2

7 75

25

35

1 15 2 25

1 15 2 251 15 2 25

1 15 2 251 15 2 25

1 15 2 25

0

05

1

3332

30

28

0

05

1

119909119867

119909119867

119909119867

119909119867 119909119867

119909119867

119909119867

119909119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867

119910119867








6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)


Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)



7 Conclusion



References








Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















6 Journal of Energy

0 02 04 06 08 10

2

4

6

8

10

2550

75100

119906119906

inlet-2

5

119910119867

(a)

0 02 04 06 08 120

119910119867

119879(∘C)

36

34

32

30

28

26

24

22

Re = 25Re = 50

Re = 75Re = 100

(b)


Re40 60 80 100

20

22

24

26

28

30

32

34

0305

0709

119879(∘C)



7 Conclusion



References








Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 7
















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















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