Interdisciplinary Applied Mathematics

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Regarding heat exchange in microdomains, it is interesting to note that the viscous heating terms are quite significant; see also (Hadjiconstantinou, 2003b), for a discussion on shear work on solid boundaries for Kn < 3. For example, if the reference temperature T0 is taken to be the room temperature and the specified temperature difference of the domain AT is small, the viscous heating effects become nonnegligible for M > 0.05 (see equation (2.22)). An analytical solution of general heat convection problem for steady and thermally fully developed planar microchannel flows under specified uniform heat flux (</) on the boundaries can be obtained by

FIGURE 5.7. Variation of tangential temperature gradient (ppf) along the surface of a pressure-driven channel as a function of Mach number for different levels of heat fluxes (q) (Re = 1.0, AT = 1K, and T0 = 300K).


decomposing the temperature profile into two parts,


T (x,y)



dTs_


dx



x + 0(y),



(5.10)


where jpfx and 6(y) show the axial and cross-flow temperature variations, respectively. The coordinates x and y are also nondimensionalized here with the reference length scale. A global energy balance in the domain with an insulated top surface and a specified constant heat flux on the bottom surface gives the following relation for the tangential temperature gradient along the channel:


dTs_


dx



1


RePrQ



q + ^EcPr



(5.11)


The corresponding cross-flow temperature distribution in the channel is


%) = RePr|| (b^ —    — EcPr^ + Cy + D, (5.12)


where

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