Slug Flow-Heat Transfer in Parallel Plate Microchannel ... - Core

Available online at www.sciencedirect.com

ScienceDirect Energy Procedia 36 (2013) 268 – 277

TerraGreen 13 International Conference 2013 - Advancements in Renewable Energy and Clean Environment

Slug Flow-Heat Transfer in Parallel Plate Microchannel Including Slip Effects and Axial Conduction Mourad Mecilia*, El Hacene Mezaacheb b

Laboratoire de Recherche sur la Physico-chimie des Surfaces et Interfaces, Université de Skikda, BP 26, Skikda 21000, Algérie b Département des Sciences de la Matière, Faculté des Sciences, Université de Skikda, BP 26, Skikda 21000, Algérie

Abstract In this paper the forced convection with slug flow model in parallel plate microchannel is studied. In the slug flow model, it is assumed that the velocity of the fluid is uniform across any cross-section perpendicular to the axis of the microchannel. The microscale analysis is taking into account by including the jump temperature at the surface. The axial conduction within the fluid is included but the viscous dissipation is neglected. The energy equation is solved by finite integral transform technique. The isoflux and isothermal boundary conditions were applied. The local and fully developed Nusselt numbers have been obtained in terms of the Knudsen number and Peclet number.

© 2013The TheAuthors. Authors. Published by Elsevier Ltd. © 2013 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review underunder responsibility of the TerraGreen Academy Academy. Selection and/or peer-review responsibility of the TerraGreen Keywords: forced convection; slug flow; parallel plate microchannel; jump temperature; integral transform technique.

1. Introduction Convection heat transfer in micro-conduits is encountered in many industrial processes such as biomedical diagnostic technique, biochemical application, microelectromechanical systems (MEMS), cryogenics, thermal control of electronic devices, chemical separation processes, aerospace engineering, vacuum technology, accelerometer flow sensors, micro nozzles, micro valves. The researchers classified the gas flow in microchannel into four flow regimes: continuum flow regime (KnwT wY @Y 1

Where 1

³0 T( X ,Y )dY

Tb ( X )

Kn

O 2L

(26)

2.2. Constant wall heat flux The dimensionless energy equation and boundary conditions are given by: w 2T

wT wX

wY

2

w 2T

1 2

Pe wX

2

T( X

0,Y ) 0

ª wT º « wY » ¬ ¼Y 0

0

ª wT º « wY » ¬ ¼Y

1

(27-30)

1

In this case (constant wall heat flux), the dimensionless temperature is defined as

T

T T0 (qw L / k )

(31)

2.2.1. Determination of fully developed temperature profile The fully developed temperature profile can be expressed by the superposition form

Tf ( X , Y )

A X g (Y )

(32)

Where A is constant that needs to be determined. Substituting eq. (32) into eq. (27), and integrating in Y direction results in:

g Y

AY 2 BY C 2

(33)

Using the boundary condition at the microchannel center, eq. (29), constant B can be determined as zero. Using the boundary condition at the wall, eq. (30), we obtain A=1. Thus:

274

Mourad Mecili and El Hacene Mezaache / Energy Procedia 36 (2013) 268 – 277

Y2 C 2

g Y

T f X ,Y

X

Y2 C 2

(34, 35)

Integrating eq. (27) in X direction, we obtain

T

X

³0

w 2T wY

2

dX

wT Pe wX 1

(36)

2

Integrating eq. (36) in Y direction from 0 to 1 and taking into account the boundary conditions eq. (29) and eq. (30) 1

³ TdY

X

0

1

³

(X

0

w 2 wX Pe 1

Y2 C )dY 2

§ 1 · ¨¨ TdY ¸¸ 0 © ¹

³

X

w Pe wX 1

2

(37)

§ ¨¨ ©

1

³0 ( X (Y

2

· / 2) C )dY ¸¸ ¹

(38)

After the integration, we obtain the constant C : C

1 Pe 2 1 6

(39)

Fully-developed temperature can be obtained by substituting eq. (39) into eq. (35) as

T X ,Y X

1 1 Y2 2 2 Pe 6

(40)

2.2.2. Determination of fully developed Nusselt number Introducing dimensionless quantities, fully-developed Nusselt number can be written as:

Nu f

hf (4 L) k

4 T w f Tb f

(41)

Where Tb∞ is the fully developed dimensionless bulk temperature which is defined as: Tb f X

1

³0 Tf X ,Y dY

(42)

Substituting eq. (40) into eq. (42)

Tb f X X

1 Pe2

(43)

Tw∞ is the fully developed wall temperature and can be determined by the implementation of the temperature jump in dimensionless form

T wf

T sf 2bKn

Where Ts∞ is the fully developed gas temperature at the surface, it is defined as:

(44)

275


Tf Y 1, X X

Tsf

1 2

1 3

Pe Substituting eqs. (43), (44) and (45) into eq. (41), Nu∞ can be determined as:

(45)

12 1 6bKn

(46)

2 FT 2J 1 FT J 1 Pr

(47)

Nu f

With:

b

All calculations are carried out with specific heat ratio γ=1.4, Prandtl number Pr=0.7 and thermal accommodation coefficient FT = 1. The values of Knudsen number Kn is varied between 0.001 and 0.1, which are the applicability limits of the slip-flow regime, and Peclet number Pe is varied between 1 and 100. The present solution is also reduced to that of a macroscale case, by setting Kn=0. Table 1a reported the values of fully developed Nusselt number computed at different Knudsen number values, for isothermal condition, these values (Nu∞) are evaluated from the following equation. Nu f

4H1 sin H1

(48)

1

³0

cosH1Y dY 2b Kn H1 sin H1

Table 1a indicates that fully developed Nu jumps from 9.8696 (for Kn=0, no-slip flow) to 5.41552 (for Kn=0.1, limiting slip flow), indicating 82.25 % decrease. Table 1b reported the values of fully developed Nusselt number computed at different Knudsen number values, for isoflux condition, these values (Nu∞) are evaluated from eq. (46). Table 1b indicates that fully developed Nu jumps from 12 (for Kn=0, no-slip flow) to 6 (for Kn=0.1, limiting slip flow), indicating 100 % decrease. Table 1. Variation of fully developed Nusselt number with Knudsen number Kn

Nu∞ (a: isothermal condition)

Nu∞ (b: isoflux condition)

0.0

9.8696

12

0.02

8.47545

10

0.04

7.42641

8.57143

0.06

6.60846

7.5

0.08

5.95281

6.66667

0.1

5.41552

6.0

In Fig. 1a, the effect of Pe number on local Nusselt number is demonstrated without the rarefaction effect (Kn=0). The data from this figure reveal that when the axial heat conduction within the fluid is included, the conductive heat transfer is assisting to the convective heat transfer, and this assistance is more significant by decreasing Pe number. Fig. 1a indicates also that the local Nusselt numbers (computed for different Pe numbers) take higher values near the microconduit entrance and monotonically decrease longitudinally along the microconduit. It may also be observed that the local Nusselt number and the thermal entrance length increase with decrease in Pe. In the thermally fully


developed region the all Nusselt numbers computed for different Pe numbers converge to the same asymptotic value. In other words the fully developed Nusselt number does not depends on Peclet number, the same results was reported by [7]. Fig. 1b illustrates the effect of rarefaction represented by Kn on local Nusselt number, and with Pe =10. It can be realized that Nusselt numbers are infinite at the entrance (at X = 0) and decays to their asymptotic (fully-developed) values. (a)

Local Nusselt number

35

Kn=0.0

30 25

Pe=1.0 Pe=5.0 Pe=100

20 15 10 0

1

2

3

4

5

6

7

8

Local Nusselt number

X 12,0 (b) 11,5 11,0 10,5 10,0 9,5 9,0 8,5 8,0 7,5 7,0 6,5 6,0 5,5 5,0 0,0 0,5

Pe=10

Kn=0.0

0.05

0.1 1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

X

Fig. 1. Axial variation of local Nusselt number for isothermal conditions with Kn=0.0 (a), and with Pe=10 (b).

Fully developed Nusselt number

276

12

isothermal condition isoflux condition

11 10 9 8 7 6 5 0,00

0,02

0,04

0,06

0,08

0,10

Kn

Fig. 2. Variation of fully developed Nusselt number with Kn, for isothermal and isoflux boundary conditions.


It is seen that Nusselt number always decreases with increase in Kn and this trend is in agreement with results reported in [2,3,7,9], this result may be explained by noting that, as the flow departs from the continuum behavior, a reduction in heat transfer at the surface occurs because the temperature gradient at the surface becomes smaller due to the temperature jump. Therefore, as Kn increases, the temperature jump increases and leads to decrease in the temperature gradient at the wall, consequently, the heat transfer decreases. 3. Conclusion In this study, the conventional slug flow problem in parallel plate microchannel was extended to include both slip-flow characteristics and axial conduction effect. The energy equation is solved analytically by the infinite integral transform technique and, closed form solutions for temperature profile, bulk temperature and Nusselt number are derived in terms of Peclet number and Knudsen number. It is concluded that: (i) In the thermal entrance region, the local Nusselt number increases as Peclet number decreases but decreases as Knudsen number increases. (ii) The thermal entrance length increases as Peclet number decreases. (iii)The fully-developed Nusselt number decreases with increase in Knudsen number but, for a fixed value of Knudsen number, it reaches a constant value for all Peclet numbers. References [1] Larrodè FE, Housiadas C, Drossinos Y. Slip-flow heat transfer in circular tubes. Int J Heat Mass Transfer 2000; 43: 2669–80. [2] Sun W, Kakaç S, Yazicioglu AG. A numerical study of single-phase convective heat transfer in microtubes for slip flow. Int J of Thermal Sciences 2007; 46: 1084–94. [3] Tunc G, Bayazitoglu Y. Heat transfer in microtubes with viscous dissipation. Int J Heat Mass Transfer 2001; 44: 2395–403. [4] Çetin B, Yazicioglu AG, Kakaç S. Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation. Int commun in Heat and Mass Transfer 2008; 35: 535–44. [5] Çetin B, Yazicioglu AG, Kakaç S. Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation–An extended Graetz problem. Int J of Thermal Sciences 2009; 48: 1673–78. [6] Aydin O, Avcı M. Heat and fluid flow characteristics of gases in micropipes. Int J Heat Mass Transfer 2006; 49: 1723–30. [7] Satapathy AK. Slip flow heat transfer in an infinite microtube with axial conduction. Int J of Thermal Sciences 2010; 49: 153–60. [8] Hadjiconstantinou NG, Dissipation in small scale gaseous flows, J of Heat Transfer 2003; 125: 944–47. [9] Joeng N, Joeng JT. Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel. Int J Heat Mass Transfer 2006; 49: 2151–57. [10] Aydin O, Avcı M. Analysis of laminar heat transfer in micro-Poiseuille flow. Int J Thermal Sciences 2007; 46: 30–7. [11] Kuddusi L. Prediction of temperature distribution and Nusselt number in rectangular microchannels at wall slip condition for all versions of constant wall temperature. Int J of Thermal Sciences 2007; 46: 998–1010. [12] Renksizbulut M, Niazmand H, Tercan G. Slip-flow and heat transfer in rectangular microchannels with constant wall temperature. Int J of Thermal Sciences 2006; 45: 870–81. [13] Kuddusi L, Cetegen E. Prediction of temperature distribution and Nusselt number in rectangular microchannels at wall slip condition for all versions of constant heat flux. Int J of Heat and Fluid Flow 2007; 28: 777–86. [14] Mecili M, Mezaache E. Analytical prediction for slip flow-heat transfer in microtube and parallel plate microchannel including viscous dissipation. International Journal of Heat &Technology 2011; 29: 79-86. [15] Mecili M, Mezaache E, Zeghmati B. Solution of the Graetz-Brinkman problem by integral transform technique. Journal of Scientific Research, 2010; 1: 246-249.

277