Chemical Engineering Science 59 (2004) 543–555 www.elsevier.com/locate/ces On the Prandtl or Schmidt number dependence of the turbulent heat or mass transfer coecient Bojan M. Mitrovic a , Phuong M. Le a , Dimitrios V. Papavassiliou a; b; * a School of Chemical Engineering and Materials Science, The University of Oklahoma, 100 E. Boyd, SEC T-335, Norman, OK 73019, USA b Sarkeys Energy Center, The University of Oklahoma, 100 E. Boyd, SEC T-335, Norman, OK 73019, USA Received 27 August 2002; received in revised form 22 July 2003; accepted 14 September 2003 Abstract Numerical experiments using a direct numerical simulation (DNS) of turbulent ow between two parallel plates in conjunction with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the ow eld are conducted for Prandtl or Schmidt numbers between 0.01 and 50,000. The LST methodology is used to generate mean temperature proles as a function of the entry distance in the case of a step change in heat or mass ux at the walls of the channel. The heat transfer coecient and the Nusselt number ratio, Nu(x)=Nu(x →∞), downstream from the step change in the wall ux are determined for the range of Pr or Sc uids examined. Relations between the heat or mass transfer coecient at the fully developed part of the channel and Pr or Sc are proposed for low and high Pr or Sc cases. Finally, unied correlations, which provide the heat or mass transfer coecient for all Pr or Sc, in the Reynolds number range examined, are proposed. Also, the exponent of the asymptotic dependence of the eddy diusivity close to the wall is obtained. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Turbulence; Dispersion; Heat transfer; Lagrangian methods; Simulation; Modeling 1. Introduction The dependence of the heat transfer coecient, K + , on the uid Prandtl number, Pr , in wall turbulence has both theoretical signicance and practical interest. There has been a considerable dierence in opinion as to what is the proper relation between K + and Pr . Textbooks (Bird et al., 1960; Hinze, 1987) usually present the heat transfer co- ecient for fully developed ow (i.e., when K + is in- dependent of entry eects), with the Deissler asymptotic prediction, K + ∼Pr -3=4 , or with the Sieder-Tate prediction, K + ∼Pr -2=3 , for Pr →∞. These two relations are deduced from plausible limiting expressions for the eddy diusivity close to a wall. However, based on very accurate measure- ments for turbulent mass transfer, Shaw and Hanratty (1977) suggested that K + ∼Sc -0:704 , where Sc is the Schmidt num- ber. Other laboratory measurements (Incropera et al., 1986; * Corresponding author. School of Chemical Engineering and Materials Science, The University of Oklahoma, 100 E. Boyd, SEC T-335, Norman, OK 73019, USA. Tel.: +1-405-325-0574; fax: +1-405-325-5813. E-mail address: dvpapava@ou.edu (D.V. Papavassiliou). Hubbard and Lightfoot, 1966; Van Shaw, 1963) have also showed dierences from the Deissler and Sieder-Tate pre- dictions. The problem of nding the correct exponent for Pr or Sc, however, has not been conclusively resolved due to the disagreements among the experimental results of dif- ferent investigators, and due to the diculty of obtaining consistent data for a range of Pr or Sc number uids. The contribution of the present work is to provide a state- ment regarding the Pr or Sc dependence of the heat/mass transfer coecient by using results obtained from a La- grangian method (Lagrangian scalar tracking, LST) coupled with a direct numerical simulation (DNS) of turbulent ow in a channel. The Eulerian DNS approach has not been able to give an answer to this issue, since it is limited by the capabilities of high performance computers to simulations for a relatively narrow range of uids (0:025 6 Pr 6 10) (Kim and Moin, 1989; Lyons et al., 1991; Kasagi et al., 1992; Kasagi and Shikazono, 1995; Kawamura et al., 1998; Na et al., 1999; Na and Hanratty, 2000; Tiselj et al., 2003). In the Lagrangian approach, the behavior of a wall source is determined by following the paths of a large number of scalar markers in a DNS of turbulent ow in a channel. The mean scalar eld can be synthesized from 0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.09.039