The analytical solution to Graetz problem with uniform wall heat flux is extended by including the viscous dissipation effect in the analysis. The analytical solution obtained reduces to that of Siegel, Sparrow and Hallman neglecting viscous dissipation as a limiting case. The sample developillg temperature proIiles, wall and bulk temperature distributions and the local Nusselt Ilumber variations are presented to illustrate the viscous dissipation effects. It is foulld that the role of viscous dissipation on thermal entrance region heat transfer is completely different for heatillg and cooling at wall. In the case of coolillg at wall, a critical value of Brinkman number, Bre = --11/24, exists beyond which (--11/24 < Br < 0) the fluid bulk telnperature will always be less than the uniform entrance temperature indicating the predominance of cooling effect over the viscous heatillg effect. On the other hand, with Br < Bre the bulk temperature Tb will approach the wall temperature Tw at some downstream position and from there onward the bulk temperature Tb becomes ]ess than the wall temperature Tw with Tw > Tb > To illdicatillg overall heatillg effect for the fluid. The numerical results for the case of coolillg at wall Br < 0 are believed to be of some interest in the design of the proposed artctic oil pipe]ine.
Bomenclature Br
Brinkman number, I~U2m/kTc = Pr Ec Cn, Dn, Kn coefficients in the series expansion of 0, equation ( 13) Cp specific heat at constant pressure Ec Eckert number, U~m/CpTe h local heat transfer coefficiellt k thermal conductivity Nu local Nusselt Ilumber, (2Ro) h/k P axial pressure --2 8 9 --