Heat transfer for laminar thermally developing flow in parallel-plates using the asymptotic method
Heat transfer for laminar, thermally developing flow in parallel-plates is investigated using the asymptotic method. There are two asymptotes for local, and mean Nusselt numbers (Nuz and Num) in the parallel-plates thermal entrance problem at both uniform wall temperature (UWT) and uniform heat flux (UHF). The first asymptote corresponds to very small value of dimensionless axial coordinate (z∗). The second asymptote corresponds to very high value of dimensionless axial coordinate (z∗). Using the methods discussed by Churchill and Usagi (1972, “General Expression for the Correlation of Rates of Transfer and Other Phenomena,” AIChE J., 18(6), pp. 1121–1128), the fitting parameter in the proposed model can be determined. Comparisons of the asymptotic model with the analytical and numerical solutions available in the literature are presented.
Laminar thermally developing flow in rectangular channels and parallel plates: uniform heat flux
Numerical simulations were conducted for thermally developing laminar flow in rectangular channels with aspect ratios ranging from 1 to 100, and for parallel plates. The simulations were for laminar, thermally developing flow with H1 boundary conditions: uniform heat flux along the length of the channel and constant temperature around the perimeter. In the limit as the non-dimensional length, x* = x/(DhRePr), goes to zero, the Nusselt number is dependent on x* to the negative exponent m. As the non-dimensional length goes to infinity the Nusselt number approaches fully developed values that are independent of x*. General correlations for the local and mean heat transfer coefficients are presented that use an asymptotic blending function to transition between these limiting cases. The discrepancy between the correlation and the numerical results is less than 2.5 % for all aspect ratios. The correlations presented are applicable to all aspect ratios and all non-dimensional lengths, and decrease the discrepancy relative to existing correlations.
Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model
A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm–Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge–Kutta method. The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter � ) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.