The Transient Liquid Crystal Technique: Influence of Surface Curvature and Finite Wall Thickness

The transient liquid crystal technique is nowadays widely used for measuring the heat transfer characteristics in gas turbine applications. Usually, the assumption is made that the wall of the test model can be treated as a flat and semi-infinite solid. This assumption is correct as long as the penetration depth of the heat compared to the thickness of the wall and to the radius of curvature is small. However, those two assumptions are not always respected for measurements near the leading edge of a blade. This paper presents a rigorous treatment of the curvature and finite wall thickness effects. The unsteady heat transfer for a hollow cylinder has been investigated analytically and a data reduction method taking into account curvature and finite wall thickness effects has been developed. Experimental tests made on hollow cylinder models have been evaluated using the new reduction method as well as the traditional semi-infinite flat plate approach and a third method that approximately accounts for curvature effects. It has been found that curvature and finite thickness of the wall have in some cases a significant influence on the obtained heat transfer coefficient. The parameters influencing the accuracy of the semi-infinite flat plate model and the approximate curvature correction are determined and the domains of validity are represented. * Present address: Institut für Luftfahrtantriebe (ILA) University of Stuttgart D-70569 Stuttgart Germany NOMENCLATURE Symbols Bi Biot number b radius ratio c specific heat capacity d wall thickness D diameter of cylinder h heat transfer coefficient hfp heat transfer coefficient from flat plate analysis Eq. (1) hR heat transfer coefficient from simplified model Eq. (21) k thermal conductivity Lx turbulent length scale Ma Mach number Nu Nusselt number r radius r ~ dimensionless radial coordinate R radius of curvature Re Reynolds number s coordinate at surface from stagnation point T temperature t time Tu turbulence level Greek α thermal diffusivity λ eigenvalues ρ density Θ dimensionless temperature σ curvature parameter (1-cylinder, 2-sphere) τ dimensionless time 1 Copyright © 2004 by ASME Subscripts c coolant d based on wall thickness fp evaluated with semi-infinite flat plate assumption g hot gas i initial R based on radius of curvature tot total w wall 1 inner wall 2 outer wall INTRODUCTION The demand for gas turbines with improved efficiency and higher power output has led to a continuous increase of the turbine inlet temperature. Consequently, turbine components such as blades and platforms are exposed to ever-higher thermal loads. To guarantee safe operating conditions and acceptable blade life the designers have to optimize the cooling systems and to predict with sufficient precision the blade temperatures. To fulfill those tasks, detailed and accurate knowledge of the heat transfer characteristics of the blades are required. Transient techniques have been successfully and widely used to measure the heat transfer coefficients on both internal and external surfaces of turbine blades (e.g. Clifford et al. (1983), Guo et al. (1995) and Drost and Bölcs (1998)). Ireland and Jones (1985) used for the first time liquid crystals together with the transient technique. They measured the heat transfer coefficient in blade cooling passages with high spatial resolution, and showed the importance of the method for gas turbine applications. The transient experiment is usually generated by a step change in the gas temperature. The evolution of the surface temperature during the transient test is measured and compared with the predictions of a one-dimensional heat conduction model for a prescribed heat transfer coefficient. This heat conduction model generally considers the wall of the test object as a flat, semi-infinite solid. This assumption is correct as long as the penetration depth of the heat compared to the thickness of the wall and to the radius of curvature is small. Schulz and Jones (1973) evaluated that the semi-infinite model holds if the duration of the experiment is smaller than tmax= d/16α. More recently, Vogel and Weigand (2001) compared the semi-infinite model with a more complex finite model and showed that this criterion could be extended to tmax < d/4α. Buttsworth and Jones (1997) proposed a simple curvature correction based on a semi-infinite assumption and valid for tmax << R/α. However, on thin and highly curved walls like in the leading edge region of a film-cooled turbine blade, those limitations on tmax might be violated leading to incorrect heat transfer measurements. In this paper, an exact heat conduction model taking into account the finite thickness of the wall and the curvature is presented. The difference between, the semi-infinite flat plate approach, Buttsworth and Jones’s curvature correction and the exact solution are shown for typical experimental situations. To verify the theoretical considerations, transient liquid crystal experiments were carried out on hollow cylinders with different wall thickness. Heat transfer coefficients are obtained by evaluating the measurement data with the different heat conduction models. The results are compared with recent correlations for convective heat transfer on a cylinder in crossflow (Dullenkopf and Mayle 1994, Van Fossen at al. 1995 and Oo and Ching 2001). The rigorous treatment of wall thickness and curvature effects presented in this paper should help to further improve accurate transient liquid crystal measurements. EXPERIMENTAL SETUP AND MEASURING TECHNIQUE Experimental Setup A free jet facility supplied by a continuously running compressor was used for this study (Fig. 1). The diameter of the free jet nozzle was 150mm. A square-meshed turbulence grid with a mesh size of 15mm and a bar width of 3mm was installed. The turbulence intensity at the location of the cylinder leading edge was about Tu=7%. The transient experiments were performed at two Mach numbers, Ma=0.11 and Ma=0.14 with a gas total temperature of 45°C. A non-dimensional integral length scale of Lx/D=0.30 at Ma=0.14 was measured by Reiss (2000) using a hot-wire probe. Cameras and light sources Turbulence Generator Cylinder Model Ø 30