Numerical Solution of Graetz Problem With Axial Conduction

Transient conjugated heat transfer in thermally developing laminar flow in thick walled pipes and minipipes with time periodically varying wall temperature boundary condition

“…(1a)-(1h) and (2a)-(2g) are solved simultaneously by a numerical finite-difference approach. The conductive terms are discretized by central-difference schemes and convective term in the energy differential equation for the fluid side by an exact method given in Bilir [35]. This method of discretization is a two-dimensional cylindrical version of the ''exact or exponential scheme" defined by Patankar [36].…”

“…The finite-difference formulation used in the discretization of the differential equations and the boundary conditions were also used in some previous works, Bilir [14,15,35], Bilir and Ates ß [37], Ates ß et al [38], Darıcı et al [39] and proved to give fast and reliable results in heat transfer problems for thermally developing laminar pipe flows, especially when axial fluid conduction is important, i.e. for low Peclet number flows.…”

Investigation on evaluation criteria of axial wall heat conduction under two classical thermal boundary conditions

“…Their results showed that the axial wall conduction could overwhelm the fluid axial conduction. Other noticeable numerical studies included Nguyen [4] and Bilir [5]. Wijeysundera [6] obtained an analytical solution for laminar forced convection in circular and flat ducts in the presence of axial wall conduction and external convection at the outer surface of the duct wall.…”

Forced convection to laminar flow of liquid egg yolk in circular and annular ducts

“…Some of these results are summarized in Shah and London (1978), Shah and Bhatti (1987) and Kays and Crawford (1993). Additionally, also available in the literature is a great deal of information on the effects of viscous dissipation (Krishnan and Sastri, 1978;Lawal and Mujumdar, 1985) and fluid axial conduction (Vick and Ozisik, 1981;Bilir, 1992;Olek, 1998). In the last fifty years, many extensions of the widely known Graetz problem (or Graetz-Nusselt problem) and the Lévêque solution have been proposed; in addition, the analyses of the accuracy of analytical and numerical results have received attention.…”