THE INJECTION or suction of fluid into the turbulent boundary layer through a porous surface is an effective means of promoting the thermal protection of walls [l, 21. The transpired fluid convects thermal energy away from the wall, improving the ability of surfaces to withstand a high-temperature environment. Despite this classical application, transpired turbulent boundary layers have also been extensively studied in connection with a variety of other applications. For example, injection or suction of fluid can be effectively used to reduce the drag offlows around bodies [3-S]. Also, variations of the transpiration problem are found when the injected fluid is a chemical reagent or when either evaporation or sublimation occurs. For low speed flows, a good account of the transpiration phenomenon is given in the literature by a number of analyses of the problem [3-141. Studies of the velocity boundary layer have established solid expressions for the law of the wall and for the law of the wake [3, 51. These expressions have been derived for a range of flow conditions, yielding a bilogarithmic expression for the skin friction. This skin friction equation is much less sensitive than the momentum-integral equation to small changes in the flow parameters and so gives much more reliable results [8]. Unfortunately, for the thermal turbulent boundary layer with transpiration, no equivalent equation has been derived for predictions of the friction temperature. Most of the work has been conducted at Stanford University [l, 2, 6, 71 to the study of universal correlations for the temperature profile and for the prediction of Stanton number. These studies, together with the analyses of refs. [9-l I], show that close to the wall the temperature profile has, as predicted by simple theoretical mixing-length based approaches, a logarithmic behaviour. In this case, the similarity parameters and the constants in the expressions must be such that their dependence on the Prandtl number