Temperature jump boundary condition

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

Deissler, R.G., An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases, Int. J. Heat Mass Transfer, Vol. 7, pp. 681-694, (1964). 

The temperature jump boundary conditions and inlet condition are also used in the same form for this case. The non-homogeneous boundary conditions can be written as follows... 

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]. 

It can be seen from Fig. 1 that gas flows in micron size channels are typically relevant to the slip flow regime, at any rate for usual pressure and temperature conditions. For lower sizes, i.e., for Knudsen numbers higher than 10 , the slip flow regime could remain valid, provided that classical velocity slip and temperature jump boundary conditions are modified (taking into account higher-order terms as explained below) and/or that Navier—Stokes equations are extended to more general sets of conservation equatiOTis, such as the quasi-gasodynamic (QGD), the quasi-hydrodynamic (QHD), or the Burnett equatiOTis [3]. 

For moderate pressure levels, and Knudsen numbers up to 0.5 or so, pressure-driven gas microflows can be modeled with the compressible Navier-Stokes equations, slip velocity, and temperature jump boundary conditions, considering die gas as an ideal gas. 

Temperature Jump Boundary Conditions Another characteristic of rarefied gas flow is that there is a finite difference between the fluid temperature at the wall and the wall temperature. Temperature jump is first proposed to be... 

Maxwell s derivation of velocity slip and temperature jump boundary condition is based on kinetic theory of gases. A similar boundary condition can be derived by an approximate analysis of the motion of gas in an isothermal condition, which has been presented in this section. 

Figure 9.4 Temperature jump boundary condition near a boundary...

Similar to velocity (Figure 9.3), we need to have appropriate boundary conditions for temperature. Figure 9.4 shows the temperature jump boundary condition near a boundary. 

Hence, the temperature jump boundary condition, equation (9.14), can be written as... 

The above equation is the dimensionless representation of temperature jump boundary condition. Other boundary conditions are... 

The surface temperature can be determined from temperature jump boundary condition as... 

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