Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

ISSN 2278 – 0149 www.ijmerr.com Special Issue, Vol. 1, No. 1, January 2014 National Conference on “Recent Advances in Mechanical Engineering” RAME – 2014 Research Paper © 2014 IJMERR. All Rights Reserved

ANALYSIS OF SINGLE PHASE SLIP FLOW HEAT TRANSFER IN A MICRODUCT

M K S Senger1* and Satyendra Singh1

This paper presents a work to derive closed form expressions for the temperature field and Nusselt number under slip flow conditions, analytically, for which the Knudsen number lies within the range 0.001

expressions have been obtained for the temperature distribution and local, NuLand fully developed Nusselt number, Nu in terms of dimensionless parameters; Prandtl number Pr, Knudsen number Kn, Brinkman number Br. The resulting expressions are presented for both the local and bulk temperature profiles. Though, the results are obtained for the microscale problems, they can be reduced to the macroscale counterparts by assuming Kn = 0. The resulting Nusselt number are compared and validated with the theoretical data available in the open literature and found to be in good agreement.

Keywords: Knudsen number, Microduct, Nusselt number, Velocity slip

INTRODUCTION slip, temperature jump, and other newly Analysis of heat and fluid flow at microscale is developed issues (Gad-el-Hak, 2001). For of great importance not only for playing a key example, the no-slip condition is valid for Kn = role in the biological systems (Kakac and 0 and the continuum flow assumption works Yener, 1980), but also for its application in well for Kn< 0.001 while for 0.001

1 Department of Mechanical Engineering, BCTKIT, Dwarahat, Almora, Uttarakhand.

234 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014 added into the governing equations to include Equation (2) is the temperature jump for the non-continuum effects, such that macro flow cylindrical coordinate system, where is the conservation equations are still applicable. As specific heat ratio, is the Prandtl number of explained through kinetic theory of gases, the fluid and is the thermal accommodation (Gad-el-Hak, 2001) introduces slip velocity and factor, which represents the fraction of the temperature jump as follows, molecules reflected diffusively by the wall and It is known that Navier–Stokes equations accommodated their energy to the wall should be combined with the slip flow condition temperature. Its value also depends on the gas so that the results can match the experimental and solid, as well as surface roughness, gas measurements. This slip velocity can be found temperature, gas pressure, and the as temperature difference between solid surface and the gas. has also been determined  F  2  u u   experimentally, and varies between 0 and 1.0. s  F  x ...(1)   wall It can take any arbitrary value, unlike As the fluid temperature at the wall can be momentum accommodation factor (Eckert different from that of the wall, the temperature and Drake (Jr.), 1972). jump reads as Applying slip-velocity and temperature jump, Temperature jump conditions (Maxwell’s Tunc and Bayazitogl (2001) investigated the condition): thermal entrance region of a microduct for two different boundary conditions. F  2  2 T T T  t Tso and Mahulikar (2000) have investigated s  w F P ( 1) r ...(2) t r wall the role of the Brinkman number in analyzing Equation (1) is the slip velocity for the heat and fluid flow through microchannels cylindrical coordinate system, where is the while, based on their experimental data, Koo momentum accommodation factor, which and Kleinstreuer (2004) developed one- represents the fraction of the molecules, dimensional analysis of forced convection in undergoing diffuse reflection. For idealized microducts. An interesting application of smooth surfaces, is equal to zero, which means viscous dissipation in measuring fluid friction secular reflection. For diffuse reflection, is coefficient for flow in a microchannel has been equal to one, which means that the tangential addressed in (Aydin and Avci, 2006). momentum is lost at the wall. The value of Recently, in two separate papers, Aydin and depends on the gas, solid, surface finish, and Avci (2006 and 2007) reported closed form surface contamination, and has been solutions to the fully developed forced determined experimentally to vary between 0.5 convection in micropipes and microchannels and 1.0. For most of the gas-solid couples and later applied their results to present the used in engineering applications, this Second Law (of Thermodynamics) aspects of parameter is close to unity (Eckert and Drake the problems (Avci and Aydin, xxxx). However, (Jr.), 1972). Therefore for this study, in a quick check of their results reveals that when Equation (1) is also taken as unity.

235 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

Kn = 0 their results fail a check versus the SPECIAL CASE available data in the literature. Moreover, the For steady and fully developed incompressible results in (Aydin and Avci, 2006 and 2007) are laminar flow with constant thermo not in agreement with those of (Jeong and physical properties through a parallel-plate Jeong, 2006a and 2006b) with non-zero values microchannel, the continuity equation is of K and B . For example, examining the n r automatically satisfied and the Navier-Stokes results of (Jeong and Jeong, 2006a and equations reduce to: 2006b) one observes that with a fixed value of 2 Kn, the fully developed Nusselt number is 1 dp d u    0 ...(4) independent of for isothermal walls and this is  dx dy2 in line with previous investigations for Kn = 0 (Nield et al., 2003; and Nield and Hooman, 1 dp   0 ...(5) 2006) while Nu in (Aydin and Avci, 2006 and  dy 2007) changes with B . One knows that in the r The pressure must be constant across any fully developed region the longitudinal cross-section perpendicular to the flow, thus temperature gradient vanishes and the momentum equation is now reduced to generated energy, as a result of viscous heating, should be conducted to the walls, as d2 u 1 dp explained in (Kuznetsov et al., 2003; and  2  dx ...(6) Hooman et al., 2007). Moreover, the dy temperature difference in the denominator of , is a function of the longitudinal coordinate and ANALYSIS the fully developed thermal energy equation, Fully developed forced convection of a fluid Equation (22) of (Aydin and Avci, 2006 and with constant property in a micro-duct is 2007), is no more an ordinary differential assumed. Figure 1 shows the coordinate equation as the dimensionless temperature systems for the problem under consideration. defined by those authors is a function of both transverse (radial) and streamwise Figure 1: Definition Sketch coordinates; see Nield (2006). Furthermore, with isoflux walls, following the application of the First Law of Thermodynamics, the longitudinal bulk temperature gradient in the fully developed region should (Shah and London, 1978; Hooman et al., 2007; Hooman et al., 2006; and Hooman and Merrikh, 2006).

dT m  q p  dA mc Hydrodynamic Aspects    p ...(3) dx* For velocity profile of Microduct where  is viscous dissipation. Momentum equation

236 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

2  2 2K (F  2) 2  d u 1 d p 1 d p y 2 H  ...(7) u    n H   d y2  d x  d x  2 F 2    Integrating twice: 2  2 4K (2  F) H d p y n du 1 d p u  1    (y c ) ...(8) 2 d x  H 2 F  d y  d x 1  

2 1 d p H 2  H d p u    c H  c  u  1 0  0 y  H  d x 2 1 2 ...(9) 0 2 d x   y  0;K  0 Boundary Conditions (At ) n From Figure 1 Boundary conditions are as, 4K (2  F) u  2   1 y  n  du (u ) F ...(12) At centre line y  o; 0  c  0 o   d y 1 u = Dimensional velocity; At surface u = Non-dimensional velocity (2 F) u y H; u    ...(10) (u orU ) y  H F y y  H Mean velocity m :

From Equations (3) and (4) 2 4K (2  F) u    n  m 3 F  (2  F) 1 d p 1 d p H 2      H     c  F  d x  d x  2 2    4K (2  F)  1 y2  n u   From depth of micro channel: u   3 F  K (2  F) U  n  K  2K  2 12   n n  F 

( K Defined from centre line or reference With F = F = 1, similar to [4], the velocity n t axis) profile is, for a Micro duct

2  2  (2  F) 1 d p 1 d p H  3u 1 4K  y  (2KH ) H    c  u   n  n   2   ...(13) F  d x  d x  2  2  1 6K     n 

(2  F) H 2 where K = 0.5 /L with L being equal to H c  2K H 2  ...(11) n c c 2 n F 2 for micro-duct. With Kn = 0 Equation (7) lead

237 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014 to known (no-slip) Poiseuille flow through a  2L q  parallel plate channel or a circular tube. c Nu    ...(17) L  KT( T ) For F = 1  w m 

hL  2  L  H 1 4K  y Note: * for duct c ; N  u  3 n  u K  2 12K  n   q  h( T T ) N  2N w m ; u uL Note: * – First order slip coefficient and Isothermal Boundaries momentum accommodation coefficient for one knows that far away from the duct inlet, velocity profile F (MAC) and thermal the temperature shows no change with the accommodation coefficient Ft (TMAC) T F = F = 1 for most of the engineering  0 t longitudinal coordinates, i.e., * that application. x leads to the following form for the thermal FIRST LAW ANALYSIS FOR energy equation now from Equation (1) we get: MICRO-DUCT 2 For this case thermal energy equation in the 2T  u  c u 0  k     absence of axial conduction reads p *2  *  y  y  T 2T c u  k   p ...(14) 2 x* y*2 2T   u      ...(18) *2 k  *  y  y  2  du  Further, as      *  2  2 2   dy   T   9U y  Or  ...(19) y*2 K (1 6K )2  Equation (8) will be as,  n 

2 Reference temperature:- T 2T  u  c u  k     p * *2  *  ...(15) x y  y  3U 2 T  r ...(20) 4K (1 6K )2 As we know Bulk mean temperature Tm as n

uTdA Applying Temperature jump conditions: T  ...(16) m UA F  2  2 T And the Nusselt no is defined a T T  t s  w F P ( 1) x t r wall

238 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

(Maxwell’s condition) Now from Equation (23)

16K  Nu T  T  n T L s w P (1  ) r ...(21) r 8T  2H  r  80K   1 T T T 32 168K  n (7 42K )   r n P (1  ) n (35 210K ) Note: *  r  n x wall y y  R

  T  T  T (1 y4) s R    (35 210K )  N  n u  70K    16K   n ...(24) 4 n 4 21K  (1  6K ) T  T T 1 y    n P (1  ) n  w r P (1  ) ...(22)  r   r  Now Nu is defined in terms of the channel Now for Bulk mean wall temperature: L width while with Dh = 2H; H uTdA 1 N  2N T     uTdy u uL m UA UH 0     3U (70 420K ) T   n  m N    2H (1 6K ) u 70K  ...(25) n 4 21K  n (1  6K )   16K    n P (1  ) n  2 n  r   (1 4K ) T T 1 y  dy n  w r  P (1  )    r  K  0 For n Bulk mean wall temperature: 70 N  17.5 u 4  80K   Isoflux Boundaries T T  T 32 168K n (7 42K ) m w r n P (1  ) n For this case fully developed definition  r  1 T T   m (35 210K ) required that leads to n x* x* ...(23)  2  Now Nusselt No is: 1 *     3Uy  * q" 1  dy  q"  2   2L 4KT   01 6KH   c r dT   n   ...(26) Nu    m    L  KT( T ) * UHc  w m  dx p

239 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

  q H 2 dT dT 2 (  )   m q  3U  2  s w K P ( 1) dr  1  r * UHc 2 ...(27) dx p  q" 1 6K  2H   n  q H 2 q (  )  2KH  s w K P ( 1) n K Combining Equation (21) with the thermal r energy equation, Equation (9), the thermal  energy equation reads Cause K  n 2H   dT 6B dT m q   q  K  1 r  * UHc 2 dr dx p  1 6K    n     4K  (  )  n s w P (1  ) ...(30) U 2 r B  ; D  (2H ) r q( D ) h h Now for Bulk mean wall temperature: u dy 2   dT d2 T  du  m UH c u  m  K     p ...(28) dx* dy*2  dy*    1    m w 840(1 6K ) Or n   18B   1 2  2   3 r  d  1 1 4K  y 18B   2    n 3  r  2(1 6K )  (1 6K )      n  n  dy2 2  (1 6K )  (1 6K )2    n  n  64 336K  336(1 4K )(1 5K ) 18B  n n n   r y2 (1 6K )2 n 18B 3360K    r (64 336K )  n (1 6K ) (1 6K ) n P (1  ) n ...(29) n r  Applying Temperature jump conditions: Finally, Nusselt number can be found as: 2 T T T     s  w 2 P ( 1) r wall N    r u (  ) or  w m  (Maxwell’s condition)

240 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

 2  Figure 2: Variation of Dimensionless N    Velocity with y for Different K u (  ) n  m w  K  0 For n

 70  N    u (17 54B )  r 

RESULTS AND DISCUSSION In the present analysis the momentum and energy equation are coupled to obtain the Nusselt number as a function of Knudsen number and Brinkman number. The flow is considered to be laminar, steady, increase in velocity convection along the incompressible, hydrodynamically and surface of the microduct increases plays an thermally fully developed. First the velocity is important role to increase the heat transfer. The derived from the momentum equation by slip velocity increases at higher velocity which employing the surface and symmetric suggests that they should be retained at large boundary conditions. The resulting velocity is values of Kn. It is observed from the graph that then used into the energy equation to obtain the difference in velocities between the the temperature distribution and the Nusselt centerline and wall decreases at large number. It is observed that there are various Knudsen numbers and therefore the velocity factors, such as: viscous dissipation, profile becomes flat in such cases. Because rarefaction, and velocity slip, temperature the Knudsen number is a function of jump, which affect the heat transfer and fluid streamwise location, the slip velocity is also flow characteristics. The results are obtained the function of streamwise location; specifying for the isothermal and isoflux conditions and the correct slip velocity is an important source compared with the other theoretical results of difficulty in numerical simulations and available in the literature. The results are analytical formulations of such flows. Hence, summarized as below: velocity plays an important role in the heat Effect of Slip Velocity transfer in micro-flows Figure 2 illustrates the variation of the non- Variation of Nusselt Number with dimensional velocity with the increasing Knudsen Number at Different transverse co-ordinate. The variation of the Prandtl Number for Isothermal velocity for the different values of Knudsen Micro-duct num ber with increasing y is shown here. It is Figure 3 presents the variation of Nusselt number with Knudsen number at different clear from the graph as Kn increases the rarefaction effect increases. Also, due to the Brinkman numbers for the isothermal

241 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

Figure 3: Variation of Nu with Kn Figure 4: Variation of Nu with Kn for Isothermal Microduct for the Isofluxmicroduct

microduct under slip flow conditions. It is observed that Nu decreases with the increase for first order model for a given value of Br. In in K . It may be noted that for a given set of n this case, Nu decreases with Kn because of operating parameters, Nusselt number varies temperature jump at the wall of the micro duct. inversely with the temperature difference between wall temperature and the mean fluid However, the results obtained by the first temperature. Temperature jump indicates the order model are found to be sufficiently lower transport of heat energy in the fluid close to compared to the continuum value as depicted the microduct wall. Knudsen number varies in Figure 3. A significant difference in the Nusselt number is obtained compared to the linearly with mean free path. A higher Kn denotes a higher mean free path which indicates a continuum model. higher temperature jump leads to a decrease Variation of Brinkman Number in Nusselt number. with Nusselt Number for different Knudsen number In addition, the Nu obtained by the first order model compared with the continuum value. The Figure 5 depicts the variation of Nusselt first order model predicts a lower value of number with Brinkman number at different Nusselt number compared to the continuum Knudsen number. Brinkman number indicates the ratio of viscous energy dissipation to heat value for various values of Kn. For the continuum conduction from the wall to fluid or fluid to the flow (Kn = 0) the Nusselt number is obtained as, Nu = 17.5 for the isothermal case. wall. Positive values of Brinkman number corresponds to the wall heating case, i.e. the Variation of Nusselt Number with heat transfer takes place from the tube wall to Knudsen Number for Different Prandtl the fluid, while negative values of Brinkman Number for the Isofluxmicro Duct number corresponds a wall cooling case. In Variation of Nu with K for various values of n both the cases, wall heating and wall cooling, Prandtl number for the Isoflux micro duct is the Nusselt number decreases with Knudsen depicted in Figure 4. The results are presented number. However, the rate of decrease in Nu

242 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

Knudsen number and Brinkman number, Figure 5: Variation of Br with Nu analytically. The theory is compared and for different Kn validated with the available analytical results. The flow is assumed to be hydro-dynamically and thermally fully developed. The iso-flux and isothermal boundary conditions are employed at the surface of the micro duct. Viscous dissipation, velocity slips and temperature jump effects are included in the analysis. The following conclusions are drawn from the analysis: • On the basis of the Knudsen number, the Prandtl number, and the Brinkman number, heat transfer in micro channels with Kn is higher in case of wall cooling can be significantly different from

(negative value of Br). This may be explained conventionally sized channels. by the fact that in the wall heating case, the • Velocity slip and temperature jump affect heat transfer takes place from wall to the fluid the heat transfer in reverse ways: a large leading to an increase in mean temperature slip on the wall will increase the convection of the fluid. Therefore, the difference in the along the surface due to an increased mean temperature of the fluid and temperature bulk velocity. Conversely, a large at the wall decreases and the drive for the heat temperature jump will decrease the heat transfer decreases leading to a decrease in transfer byreducing the temperature heat transfer rate and the Nusselt number. gradient at the wall. Therefore, neglecting On the contrary, in case of wall cooling the temperature jump will result in the case, the viscous dissipation increases overestimation of the heat transfer temperature in the fluid and the heat transfer coefficient. takes place from fluid to the wall and the • A Nusselt number reduction is observed difference in the mean fluid temperature and as the flow deviates from the continuum the wall temperature is higher leading to an flow. increase in heat transfer rate and Nusselt number. It is apparent from the analysis that The Prandtl number is important, since it Brinkman number plays an important role in directly impacts the magnitude of the heat transfer. temperature jump. From the temperature jump equation, as Pr increases, the difference CONCLUSION between wall and fluid temperature at the wall In the present analysis closed form decreases.Therefore, greater Nu values for expressions are obtained as a function of large Pr are witnessed.

243 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

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APPENDIX NOMENCLATURE

Across-sectional area

U 2 Br Brinkman number, Br  q Dh

Dh hydraulic diameter

F tangential momentum accommodation coefficient

Ft thermal accommodation coefficient

H half channel width

K thermal conductivity

Kn Knudsen number

n (coordinate) normal to the wall

245 Int. J. Mech. Eng. & Rob. Res. 2014 M K S Senger and Satyendra Singh, 2014

APPENDIX (CONT.)

Nu Nusselt number

c UL Pe Peclet number Pe  p c k Pr Prandtl number

r * radial coordinate

r dimensionless radial coordinate  r* / R T temperature Tr reference temperature u velocity U mean velocity

 x*, y*  streamwise and transverse coordinate

 x, y  x* , y*  / H

Greek Symbols  specific heat ratio  Molecular mean free path  dynamic viscosity  dimensionless temperature  viscous dissipation Subscripts c centerline m mean s fluid properties at the wall w wall

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