Extended Graetz problem

The problem given by eq.(59) subject to the conditions (60) is referred to as extended Graetz problem in honor of the pioneering work [5]. To solve this problem we need the eigenvalues m and the eigenfunctions y [R] of the eigenproblem ... 

The solution of the extended Graetz problem, eqns. (59, 60), is a special case from the solution given by Mikhailov and Ozisik in the book [20] ... 

H.-E. Jeong and J.-T. Jeong, Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel, International Journal of Heat and Mass Transfer 49, 2151-2157 (2006). 

Thus, application of the lumping procedure to the original formulation leads to the extended Graetz problem described by the equation and boundary conditions below ... 

Other extended Graetz problems in which the effect of viscous dissipation, inlet velocity, and temperature profiles are considered are reviewed in detail by Shah and London [1]. 

M. A. Ebadian, and H. Y. Zhang, An Exact Solution of Extended Graetz Problem with Axial Heat Conduction, Int. J. Heat Mass Transfer, (32) 1709-1717,1989. 

Jetmg H, Jeong J (2006) Extended Graetz problem including axial conduction and viscous dissipation in microtube. J Mech Sci Technol 20(1) 158—166... 

The cases where axial diffusion cannot be neglected lead to the formulation of the extended Graetz problem for which analytical solutions have been given. If axial diffusion is indeed negligible (and this should be true for long channels), the only two mechanisms at play are axial convection and transverse diffusion. From this interaction, two different regimes appear which describe the behavior for axial distances of the same order of the channel length z = 0 L) ... 

E. Papoutsakis, D. Y. Ramkrishna, and H. C. Lim, The extended Graetz problem with Dirichlet wall boundary conditions, Appl Sci Res 36 13-40 (1980). 

Jeong HE and Jeong JT 2006a Extended Graetz problem including stream wise conduction and viscous dissipation in micro channel. Int. J. Heat Mass Transfer, 49, pp. 2151-2157. 

Sellars, J. R., M. Tribus, and J. S. Klein Heat Transfer to Laminar Flows in a Round Tube or Flat Conduit The Graetz Problem Extended, Trans. ASME, vol. 78, p. 441, 1956. 

J.R. Sellars, M. Tribus and J.S. Klein, Heat transfer to laminar flow to a round tube or flat conduit— The Graetz problem extended, Trans. Am. Soc. Meek Eng. 75 441 (1956). L.P.B.M. Janssen and M.M.C.G. Warmoeskerken, Transport Phenomena Data Companion, Edward Arnold, London, 1987. 

Bar-Cohen, A., State of the Art and Trends in the Thermal Packaging of The Electronic Equipment, ASME Journal of Electronic Packaging, 1992, 114, 257-270. Barron, R.F, Wang, X. Ameel, T.A. and Warrington, R.O., The Graetz Problem Extended to Slip-Flow, Int. J. Heat Mass Transfer, 1997, 40(8), 1817-1823. 

Randall, F.B., Wang, X. and Ameel, T.A, The Graetz Problem Extended to slip flow, Int. J. Heat Mass Transfer, 1997, 40, 1817-1823. 

Heat transfer by forced convection inside micro tube, generally referred as the Graetz problem, has been extended by Barron et al. [11] and Larrode and al. [12] to include the velocity slip described by Maxwell in 1890 [13] and the temperature jump [14] on tube surface, which are important in micro scale at ordinary pressure and in rarefied gases at low-pressure. 

For the uniform temperature boundary eondition in a cylindrical charmel, the fully developed Nusselt number decreases as Kn increases. For the no-slip condition Nu =3.6751, while it drops down to 2.3667 for Kn = 0.12, which is a decrease of 35.6 %. This decrease is due to the fact that the temperature jump reduces heat transfer. As Kn increases, the temperature jump also increases. Therefore, the denominator of Eq. (5.14) takes larger values. Similar results were found by [18]. They report approximately a 32 % decrease. However, [20] extended the Graetz problem to slip flow, where they find an increase in the Nusselt number for the same conditions without considering the temperature jump. We can see the same trend in the other two cases of constant wall heat flux for cylindrical and rectangular geometries. 

In section 5.1.4, the Graetz problem was solved using the semianalytical technique. The solution obtained is numerical in x and analytical in z. The solution is obtained as a function of the Peclet number. The solution obtained compares well with the analytical solution reported in the literature. Our technique avoids calculation of special functions and at the same time provides solutions explicit in the Peclet number. In section 5.1.5, the semianalytical technique developed earlier was extended to the case when the initial condition is a function of x. 

Sellers, J. R., Tribus, M., and Klin, J. S., Heat transfer to laminar flow in a round tube or flat conduit-the Graetz problem extended, Trans. ASME, Vol. 78, No. 2, pp. 441-448, 1956. 

R.F. Barron, X.M. Wang, R.O. Warrington, and T.A. Ameel, The Graetz problem extended to slip flow. International Journal of Heat and Mass Transfer AQ, 1817-1823 (1997). 

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