Heat transfer with upstream thermal penetration in flow through porous plate passages A. Haji-Sheikh a, * , W.J. Minkowycz b , Saeed Manafzadeh b a Department of Mechanical and Aerospace Engineering, The University of Texas at Arlington, Arlington, TX 76019-0023, USA b Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, USA article info Article history: Received 1 October 2009 Accepted 14 November 2009 Available online 27 November 2009 Keywords: Heat transfer Parallel plate ducts Thermal entrance Convection Axial conduction abstract The objective of this study is to develop a series solution for determination of the temperature field in parallel-plate ducts with prescribed wall heat flux. Consideration is also given to ducts filled with fluid saturated porous materials. A simple transformation is used to improve the convergence of this series solution. Temperature variations and heat transfer coefficients are determined for infinitely long parallel-plate ducts when the walls undergo a step change in the applied wall heat flux. Axial thermal penetration near the applied wall heat flux location is studied. The solutions with the contribution of axial conduction in these fluid passages are acquired using a modified Graetz-type solution. Finally, this technique is augmented by the contribution of frictional heating. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction For the case of prescribed wall temperature, the extended Graetz-type solutions for flow through parallel plate channels and circular ducts filled with fluid saturated porous materials are given in Minkowycz and Haji-Sheikh [1]. When the Peclet number is small, it is demonstrated in [1] that the axial heat conduction increases the local wall heat flux near the thermal entrance location where there is a step change in the wall temperature. Accordingly, available infor- mation shows substantial differences in the wall heat flux values when compared to results from heat transfer data reported in [2–7] for flow in ducts without the effect of axial conduction. This study modifies the methodology in [1] for flow through parallel- plate ducts in the presence of an abrupt change in the wall heat flux. Earlier, Michelsen and Villadsen [8] presented a study of Graetz- type solutions for flow in pipes. Lahjomri and Oubarra [9] present a detailed analytical study of axial heat conduction in flow through clear circular ducts with prescribed wall temperatures and Lahjomri et al. [10] extended the methodology in [9] to Laminar Hartmann flow. Weigand and Lauffer [11] analytically studied the effect of axial conduction on the heat transfer coefficient in the presence of a step change in the wall temperature, taking into consideration both lam- inar and turbulent flow. Kuznetsev et al. [12] numerically acquired information about the effect of axial conduction in fluid saturated porous circular ducts and Nield et al. [13] present that numerical solution for parallel plate channels. These studies show that axial conduction near the thermal entrance location becomes significant in porous passages filled with metallic foams. Parallel studies for the effect of axial conduction ion micro-channels are found in [14–16], which show that the axial conduction contribution is signif- icant in the entrance regions of micro-channels. The studies of to heat transfer to fluid flowing through clear pipes with variable wall heat flux are well documented by Pap- outsakis and collaborators [17,18]. Akins and Dranoff [19] report experimental studies and compare the experimental results to available analytical/numerical information for a moderately large Peclet number of 100. Heat transfer to fluid flowing through clear rectangular ducts with H1 boundary conditions is analyzed in [20], while [21,22] contain the studies of heat transfer in rectan- gular porous passages with prescribed wall heat flux. Tada and Ichimiya [23,24] numerically acquired heat transfer information for flow in circular tubes filled with fluid saturated porous mate- rials and with constant wall heat flux and include the effect of viscous dissipation in [24]. The effect of axial conduction for flow through clear pipes is investigated in [25], while different thermal entrance conditions produced the results reported in [26,27]. Based on the available information, it is expected to have a signif- icant upstream thermal penetration near a step change in the wall heat flux location. The limiting mathematical solution for slug flow in parallel- plate ducts is given in [28], where the acquired solution is com- pared to that for flow over an infinite flat plate having a step change in the heat flux at the wall. It is shown that the Stanton number for parallel-plate ducts approaches that for flow over a 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.11.011 * Corresponding author. Tel.: +1 817 272 2010; fax: +1 817 272 2952. E-mail address: haji@uta.edu (A. Haji-Sheikh). Applied Thermal Engineering 30 (2010) 639–648 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng