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Analysis of Single Phase Slip Flow Heat Transfer in a Microduct

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Analysis of Single Phase Slip Flow Heat Transfer in a Microduct 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<Kn<0.1. The flow is assumed to be laminar, steady, incompressible, fully developed both hydrodynamically and thermally. Thermo-physical properties are assumed to be constant for the analysis and hence no property variation in axial direction. The constant wall temperature and constant heat flux thermal boundary conditions are employed at the surface of the microduct. The viscous dissipation, velocity slip and temperature jump effects are taken into consideration for analyzing the heat transfer characteristics inside the microduct. Closed form 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<Kn< 0.1 the cooling electronic equipment. Modeling heat flow is called slip flow. and fluid flow through such small devices is For the slip flow regime, which is the main different from the macroscale counterparts in interest of this study, slip velocity and being associated with the inclusion of velocity temperature jump boundary conditions are 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.
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