Abstract
The article addresses the extended Graetz–Nusselt problem in finite-length microchannels for prescribed wall heat flux boundary conditions, including the effects of rarefaction, streamwise conduction, and viscous dissipation. The analytical solution proposed, valid for low-intermediate Peclet values, takes into account the presence of the thermal development region. The influence of all transport parameters (PecletPe, KnudsenKn, and BrinkmanBr) and geometrical parameters (entry length and microchannel aspect ratio) is investigated. Performances of different wall heat flux functions have been analyzed in terms of the averaged Nusselt number. In the absence of viscous dissipationBr=0, the best heating protocol is a decreasing wall heat flux function. In the presence of dissipationBr>0, the best heating protocol is a uniform wall heat flux.
Highlights
A correct estimation of heat and mass transfer coefficients is a powerful tool in the design of heat exchangers [1], mass transfer equipment and reactors [2], and microdevices [3, 4] for chemical [5], biomedical [6, 7], and pharmaceutical applications [8, 9].Focusing on laminar forced convection of an incompressible fluid in a duct, the estimation of transport coefficients requires the solution of the classical Graetz–Nusselt problem [2, 10]
Proposed for a sudden step change of the wall temperature at some positions along the duct and no axial diffusion, the Graetz–Nusselt problem is valid for both heat and mass transfer
It has been solved in transient and steady-state [11], for Dirichlet and Neumann boundary conditions [12], for different wall shape and curvature [9, 13, 14], for non-Newtonian fluids [15], and for counterflow streams [16], in the presence of high viscous dissipation [17], axial diffusion [18, 19], and simultaneous heat and mass transfer [20, 21]
Summary
A correct estimation of heat and mass transfer coefficients is a powerful tool in the design of heat exchangers [1], mass transfer equipment and reactors [2], and microdevices [3, 4] for chemical [5], biomedical [6, 7], and pharmaceutical applications [8, 9]. Erefore, a temperature distribution is built up for z < 0 and this may significantly affect the temperature downstream For this reason, Barletta and Magyari [32] addressed the problem of the thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution, including the effect of viscous dissipation but neglecting heat axial conduction. E solution, taking into account the presence of the thermal development region, is valid for low-intermediate Peclet values Pe and for the prescribed heat flux boundary conditions (no wall conjugation effects). In dealing with finite-length channels, in the presence of axial dispersion and wall heat flux, one issue to be addressed is the proper boundary conditions at the inlet and outlet sections.
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