Thermal slip flow

Even with an adequate description of molecular velocities near the particle surface, it is not possible to completely establish all variables influencing thermal force. This is because there also exists a so-called thermal slip flow or creep flow at the particle surface. Reynolds (see Niven, 1965) and others have pointed out that as a consequence of kinetic theory, a gas must slide along the surface of a solid from the colder to the hotter portions. However, if there is a flow of gas at the surface of the particle up the temperature gradient, then the force causing this flow must be countered by an opposite force acting on the particle, so that the particle itself moves in an opposite direction down the temperature gradient. This is indeed the case, known as thermal creep. Since the velocity appears to go from zero to some finite value right at the particle surface, this phenomenon is often described as a velocity jump. A temperature jump also exists at the particle surface. 

Bar-Cohen, A., State of the Art and Trends in the Thermal Packaging of The Electronic Equipment, ASME Journal of Electronic Packaging, 1992, 114, 257-270. Barron, R.F, Wang, X. Ameel, T.A. and Warrington, R.O., The Graetz Problem Extended to Slip-Flow, Int. J. Heat Mass Transfer, 1997, 40(8), 1817-1823. 

The laminar gaseous flow heat convection problem in the slip flow region was solved both analytically and numerically for various geometries [3-6], The compressibility effects were included in [7],[8-11] and the results were compared with the experimental results of [12], Thermal creep effects were studied by [13], Exact solutions for flows in circular, rectangular, and parallel plate microchannels were given in [14-17],... 

Their results for the non-slip flow ease agreed with [26], who also used the integral transform teehnique to solve for the Nusselt number for flow through a maerosized reetangular ehannel. They did not inelude viscous dissipation in the work, but they did inelude variable thermal aeeommodation eoefficients. Similar to [15], they concluded that the Knudsen number, Prandlt number, aspeet ratio, velocity slip and temperature jump can all cause the Nusselt number to deviate from the eonventional value. 

The earliest studies related to thermophysieal property variation in tube flow conducted by Deissler [51] and Oskay and Kakac [52], who studied the variation of viscosity with temperature in a tube in macroscale flow. The concept seems to be well-understood for the macroscale heat transfer problem, but how it affects microscale heat transfer is an ongoing research area. Experimental and numerical studies point out to the non-negligible effects of the variation of especially viscosity with temperature. For example, Nusselt numbers may differ up to 30% as a result of thermophysieal property variation in microchannels [53]. Variable property effects have been analyzed with the traditional no-slip/no-temperature jump boundary conditions in microchannels for three-dimensional thermally-developing flow [22] and two-dimensional simultaneously developing flow [23, 26], where the effect of viscous dissipation was neglected. Another study includes the viscous dissipation effect and suggests a correlation for the Nusselt number and the variation of properties [24]. In contrast to the abovementioned studies, the slip velocity boundary condition was considered only recently, where variable viscosity and viscous dissipation effects on pressure drop and the friction factor were analyzed in microchannels [25]. 

In this lecture, a variety of results for convective heat transfer in microtubes and microchannels in the slip flow regime under different conditions have been presented. Both constant wall temperature and constant wall heat flux cases have been analyzed in microtubes, including the effects of rarefaction, axial conduction, and viscous dissipation. In rough microchannels the abovementioned effects have also been investigated for the constant wall temperature boundary condition. Then, temperature-variable dynamic viscosity and thermal conductivity of the fluid were considered, and the results were compared with constant property results for microchannels, with the effects of rarefaction and viscous dissipation. 

Rarefied flows in the slip flow regime, rarefaction increases the hydrodynamic and thermal entrance lengths owing to slip at the walls (see Pressure-Driven Single-Phase Gas Flows ). 

The steady-state heat convection between two parallel plates and in circular, rectangular, and annular channels with viscous heat generation for both thermally developing and fully developed conditions is solved. Both constant wall temperature and constant heat flux boundary conditions are crmsidered. The velocity and the temperature distributions are derived from the momentum and energy equations, and the proper slip-flow boundary conditions are considered. 

Heat Transfer in Circular Microchannels Many researchers have analytically and numerically studied the heat transfer through circular microchannels. For a hydrodynamically and thermally developed flow through a microtube with uniform wall heat flux in the slip-flow regime at radius, ro, and no viscous heating, an expression for the Nu can be written as [7, 30]... 

Hadjicrnistantinou and Simek [35] investigated the case for fuUy developed flow with uniform wall temperature. They found that Eqs. 7 and 17 with both F = 1 and = 1 were adequate in determining the physics in this slip-flow problem. They compared their results with the direct simulation Monte Carlo method. They concluded that slip-flow models neglecting viscous dissipation, expansion cooling, and thermal creep were adequate in describing the heat transfer. However, when considering viscous dissipation, the Nu expression becomes [34]... 

The description by [2] of thermophoresis in gases is based on entropy. In complex fluids the entropy of the interaction between molecules or particles and solvent leads to thermophoretic motion. Schimpf and Semenov [8] consider the slip flow caused by a local pressure gradient set up by the temperature-dependent properties of the particle and solvent. They find the same thermal expansion proportionality factor as Brenner [6] who does not consider these interactions. 

Equation (2.86) displays a marked dependence of on the thermal conductivity ratio. A, which may range over several orders of magnitude. As experimental data became available for aerosol particle-gas systems with large A, it was apparent that (2.86) was seriously in error for the slip-flow regime 0 < Kn. < 0.2, the regime where (2.86) might be expected to be valid. 

Entropy generation of forced convection heat transfer of liquid fluid over the horizontal surface with embedded open parallel microchannels at constant heat flux boundary conditions may be formulated by an integral of the local entropy generation. Embedded open parallel microchannels within the surface can sufficiently reduce both friction and thermal irreversibilities of liquid fluid through slip-flow conditions (Kandlikar et al., 2006 Yarin et al., 2009). 

M. Renksizbulut, H. Niazmand, G. Tercan, Slip-flow and heat transfer in rectangular microchannels with constant wall temperature. International Journal of Thermal Science, 2006, 45, 870-881. 

The thermal creep coefficient G is also obtained from the Navier-Stokes equation, but applying the thermal slip boundary condition (9). It is verified that this coefficient does not depend on the type of the cross-section in the slip flow regime, hut in any case considered here it takes the form... 

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