Literature Review. Chapter 2.

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1 Chapter 2. Literature Review Research programs for gas turbine engines have been active for over half a century. D. G. Wilson (1984) notes that gas-turbine design methods have become classics of the scientific method. Technological development of the engines has allowed the turbineinlet temperature to gradually increase over the decades because increasing the temperature is directly related to increasing the thermal efficiency of the system. Figure 2.1 shows this trend, and it has, in fact, reached the order of over 1500 C in the early 1990 s (Najjar, 1996). Understanding fluid mechanics and heat transfer in the turbine channels is crucial in designing turbine airfoils that are capable of withstanding such increasing thermal loads. This chapter reviews other researchers works in the literature that are relevant to the present study. They are divided into sub-categories. The first section is devoted to some experimental works on gas turbine heat transfer. The focus is on near-endwall heat transfer on turbine airfoils and on experimental techniques. Endwalls are located at both ends of a turbine airfoil (Figure 2.2). The second section is devoted to analytical and numerical techniques in heat transfer and fluid mechanics, and the third is devoted to the history of the heat/mass transfer analogy. The latter is included here because one of the two major focuses of the present study is the heat/mass-transfer analogy function. 8

2 Figure 2.1 Increasing turbine-inlet temperature (Wilson 1984) 9

3 2.1 Gas Turbine Heat Transfer Experimental Methods Experiments are powerful tools when it comes to gas turbine heat transfer because of the complex geometry and flows in a gas turbine. The three-dimensional channels in a gas turbine create various types of secondary flows. In this study, near-endwall heat transfer is one of the major interests. Figure 2.2 shows airfoil-endwall junctions and secondary flows in their vicinities. The presence of an endwall creates a horseshoe vortex at the corner made by the airfoil leading edge and the endwall. The vortex is washed downstream and grows, becoming a strong secondary flow. It is called a passage vortex and has significant effects on heat transfer on the suction side of an airfoil. The first half of this section introduces experiments that investigate the effects of the passage vortex. The second half of this section introduces a few experimental techniques that are capable of capturing such two-dimensional variations of the heat transfer rate as the one on the near-endwall suction side. Figure 2.2 Secondary flows at airfoil-endwall junction (Sharma and Butler 1987) 10

4 2.1.1 Experiments on determining secondary flow effects As noted on the previous page, flow over a single airfoil in a gas turbine engine is highly three-dimensional and the heat transfer at the surface is highly two-dimensional. Many experimental investigations focus on determining heat transfer rates on an airfoil while many others focus on determining the flow field around the airfoil. One of the factors that complicate the flow field and heat transfer situation is the presence of secondary flows in the passage. A type of secondary flow that has significant influence on heat transfer to an airfoil surface is the passage vortex whose impact is the so-called triangular region of high heat transfer rate on the suction side of the neighboring airfoil, beneath the passage vortex in Fig The suction side, or the convex side, of the airfoil is so called because the pressure is lower. Similarly, the concave surface is called the pressure side. Sharma and Butler developed the model shown in Fig. 2.2 based on available information in the literature (1987). Other studies that show such models are by Langston (1980), and Goldstein and Spores (1988). An example of a study on near-endwall, suction-side heat transfer is presented in Chung (1992), and Chung and Simon (1993). The investigation uses various flow visualization techniques to understand the behavior of the passage vortex. When measuring the heat transfer rates in the region influenced by the vortex, liquid crystal thermography is used. Cholesteric liquid crystals are known to be temperature-sensitive (Kasagi et al., 1989). They are discussed in the following sub-section as a type of experimental method. Chung and Simon discuss the application of an endwall fence to prevent the passage vortex from hitting the suction surface. Furthermore, they find that the triangular region becomes smaller in size at a high freestream turbulent intensity (TI) level of about 10%. Analyzing the flow field indicates that turbulent mixing associated with elevated core turbulence suppresses the vortex near the endwall and prevents it from growing and climbing the suction surface (Chung and Simon, 1993). The thesis of Chung 11

5 presents actual photographs of liquid crystal sheets placed on the suction side of the airfoil, which will be important sources of data for the present study. Another example of a study on near-endwall suction-side heat transfer is presented in Chen (1988) and Chen and Goldstein (1992). This investigation employs the naphthalene sublimation technique to an entire surface of an airfoil near an endwall. Therefore, the results show the effects of the suction-side-leg horseshoe vortex and the passage vortex scrubbing the suction-side surface (see Fig. 2.2). The thickness of the inlet boundary layer (Fig. 2.2) is varied in the experiment to see its effect since both vortex systems originated at the leading edges of the airfoils within the endwall boundary layer. The investigators find little difference between the cases of the two different boundary layer thicknesses. They notice that, within the triangular region, mass transfer rates are not quite constant but have a stripe of lower values just above the narrowly located higher values next to the endwall (see Fig. 4.4). The near-endwall values indicate the strong influence of the suction-side corner vortex (Chen and Goldstein 1992). This is not shown in Fig. 2.2, but this small vortex runs within the corner made by the suctionside surface and the endwall, just underneath the passage vortex shown in Fig The resulting mass transfer rates, or mass transfer Stanton numbers, contribute a set of important data to the present study. Yet another example is seen in Wang (1997). His investigation is similar to that of Chen. The major differences are the airfoil shape and the level of freestream turbulence. Wang s experiments involve four different levels of turbulence intensity (TI) while Chen s involve only low-ti cases. Wang s results reveal how the triangular region is suppressed with increasing TI, as noted by Chung and Simon (1993). The first two experimental investigations, Chung s and Chen s, were performed with an airfoil geometry which is representative of the CF6 engine from General Electric. Wang s was with a geometry which represents the GE90 airfoil from General Electric. There are some differences in the surface heat transfer rates between the different geometries of airfoil, but the essential trends, such as the presence of the triangular regions, are similar (Goldstein et al., 1996). 12

6 2.1.2 Experimental techniques To measure a field of temperatures, the following techniques may have advantages over others: liquid crystal and naphthalene sublimation. The high spatial resolution of each of these techniques is advantageous. Liquid crystal is a technique used in heat transfer experiments to resolve a field of temperature distribution. Liquid crystals have been known since 1888, and their color has been recognized to be temperature-sensitive (Fergason 1964). With such a unique name, the characteristics of liquid crystal have interested many researchers in heat transfer, and thanks to their efforts, liquid crystals have established the status of a quality scientific experimental method. In the early 1980s, the technique began to be applied with computers that are capable of digitizing liquid-crystal color maps and calculating fields of temperature distributions. An example of such work is by Simonich and Moffat (1982). The objective of this study was to measure heat transfer on a concave surface with a turbulent boundary layer above it. They also discuss the development of liquid crystal research since the Fergason generation. The procedures that Simonich and Moffat used began with a calibration of a liquid crystal sheets correlating a particular color to a value of temperature within ±0.25 C. They took photographs of liquid crystals and digitized them, using an infrared sensor traversed over the photograph. The calibration data were used to deduce temperature fields. They noted that the uncertainty of this technique in the calculated heat transfer coefficient was 9.5 % and that this was not the most accurate method available to measure heat transfer. Above all, they mentioned that the reason they used the liquid crystal method was its advantage in obtaining fields, as opposed to point measurements of temperature. Jones and Hippensteele used a different method in their measurements (1988). When temperature varies linearly on a sheet of liquid crystal, each location has an associated color. The narrowest color is chosen for the technique, for it is the most distinct and accurate of all. Using only this color for the measurement improves the 13

7 uncertainty of the experiment. They took photographs of the liquid crystal surface while the surface heat flux was varied. They observed the narrowest color band move across the surface with increasing heat flux and took an isotherm for each heat flux. Camci et al. proposed yet another method (1991). They recognized that, when a liquid crystal photograph is color-separated to hue, saturation and intensity, temperature has a linear relationship with hue for a large range of hue. They used this with a broad band of color play for the liquid crystal, so that much of the surface was one color or another. The data reduction process could be made completely automatic with a single photograph of liquid crystal while the previous method provided only one contour line per photograph. Today, with a use of a digital camera, digitization of the photographic films can be eliminated. When a scanner is used to digitize a photograph, one is concerned about the resolution that can be extracted from the original photograph. Another technique used in heat transfer experiments is naphthalene sublimation. If heat transfer data are desired, results from this technique must be converted from mass transfer coefficients to corresponding heat transfer coefficients, using the heat/mass analogy. The topic of heat/mass analogy functions is one of the two main focuses of the present study. Although the problem of converting the mass transfer results to heat transfer results using the analogy function remains, the sublimation mass transfer technique has several advantages over heat transfer experiments. Sherwood number is the mass-transfer equivalent of Nusselt number of heat transfer. Sherwood extensively studied mass transfer phenomena; the Sherwood number was named after him. T. K. Sherwood is discussed in the next section about the history of the analogy function. Goldstein and Karni (1984) applied the naphthalene sublimation experiment to a circular cylinder in a crossflow. The cylinder was placed between flat plates, simulating the endwalls of a leading edge region of a gas turbine airfoil in a channel. The experimental technique allowed them to point out that there were strong vortices forming at the stagnation points. The vortex formations were due to the presence of the endwall 14

8 and were dependent on distances from the endwall. These vortices were washed along the cylinder surface and formed the horseshoe vortex. Details about heat transfer around a circular cylinder are presented in a book by Zukauskas and Ziugzda (1985). Within this book, several experiments are presented. Achenbach s heat transfer experiment was done with a high Reynolds number (1975). Lowery and Vachon performed heat transfer experiments and considered the effects of turbulence on heat transfer from a circular cylinder (1975). An equivalent mass transfer experiment is done by Lee et al. (1994). The naphthalene sublimation technique was applied to the entire airfoil geometry in the study of Chen (1988) and Chen and Goldstein (1992). They observed how the horseshoe vortex proceeded downstream and formed the passage vortex that traveled across the channel from pressure side of an airfoil to the suction side of the neighboring airfoil. Once it hit the suction side, it would climb the suction surface and grow. The vortex traveled from pressure side to suction side of the endwall by the pressure difference between the two. It grew by virtue of the three-dimensionality of the endwall boundary layer. Goldstein and Cho noted several advantages of the sublimation mass transfer technique over heat transfer experiments (1995). They referred to the mass transfer technique as being free of radiation or conduction effects at the boundaries. This allows the experimental apparatus to be free of complex heating and measuring systems, including insulating materials. For this reason, boundary conditions are exact. Thus, the mass transfer technique measures only convection and flow diffusion effects at the surface, according to the authors. 2.2 Gas Turbine Heat Transfer Analytical and Numerical Methods Analytical methods remain useful now that experimental techniques and computer systems are highly advanced. They are much less time-, memory- and data-storageconsuming than expensive experiments and numerical calculations. The present study is 15

9 supported by some very relevant analytical works related to gas turbine heat transfer and boundary layer flow. These are discussed below. Numerical calculations of gas turbine heat transfer and fluid dynamics have advanced to the level at which the fully-compressible, Navier-Stokes, energy and mass transfer equations in turbulent flows can be computed. The present study, however, utilizes a computational code for boundary layer flow and a classic numerical solution of heat conduction in a solid, as discussed below Boundary layer flow Prandtl conceptualized boundary layers in 1904 and reduced the Navier-Stokes equations to those known as Prandtl s boundary layer equations (see the Introduction and Chapter VII in Schlichting, 1979). The simplest example of the application of the boundarylayer equations is what Schlichting calls a flat plate at zero incidence (Chapter VII of Schlichting, 1979). The zero incidence means that the plate is perfectly aligned with the flow and the flat plate has a sharp, or very thin, leading edge. This was the doctor s thesis of H. Blasius (1908), a student of Prandtl, deriving the well-known Blasius velocity profile. This is a basic type of similarity solution. Similarity solutions for laminar boundary-layer flow developed from the fact that velocity profiles at various locations in the boundary layer are similar to each other (Chapter 7 of Eckert and Drake, 1987). This technique can be applied to wedge-type flows in which the flat plate is not aligned to the flow, but is inclined. The resulting surface skin friction variations for various acceleration profiles, or angles of inclination, are tabulated in Kays and Crawford (Chapter 10, 1996). In the 1970s, a computational code for two-dimensional boundary layer flows, called STAN5, was developed. Its updated version is called TEXSTAN. The code is capable of handling turbulent flows with a turbulence model that users choose. The present study uses TEXSTAN for laminar flow calculations in Chapter 3. The code, as applied, simulates two-dimensional boundary layer flow with and without heat and/or 16

10 mass transfer. Users of the code can implement streamwise pressure gradients in the form of velocity distributions. More details about the program are summarized in a report of Crawford and Kays (1976) Heat conduction The present study includes also numerical calculations of steady-state heat conduction done in a unique way. Original algorithms are developed in Chapter 5 and applied in Chapter 6. Steady-state heat conduction is an important problem in gas turbine airfoil heat transfer because the gas side and the coolant side temperatures may be more than a few hundred degrees Kelvin different. One of the works introduced here is directly related to the present study. It provides one-dimensional analyses of the problem while the present study provides two-dimensional computations. Other works introduced have specific relevance to the present study, especially when developing the new algorithms. Eckert et al. (1997) is directly related to the present study. It includes the onedimensional equivalent of Chapter 5 of the present study, which focuses on twodimensional heat conduction within models of a gas turbine airfoil. The one-dimensional analyses focus on the direction from the gas-side to the coolant-side surface with or without surface curvature. This is the main direction of the gas-to-coolant heat transfer. The two-dimensional analyses add the longitudinal direction to the one-dimensional analyses. One of the goals of the present study is to resolve two-dimensional effects. A classic type of an atypical heat conduction problem is a moving-boundary problem. It has a history of more than a hundred years, and several solution methods have been developed (Chapter 10 of Özisik 1980). The present study has only an indirect connection to the moving boundary problem as will be discussed. Crank and Gupta solve moving-boundary problems using cubic splines or polynomials (1972). They developed a finite difference equation in which a cubic spline approximation is embedded. Overcoming computational instabilities is an important step for the present study. Although the author develops an algorithm to overcome instabilities, there are other researchers who have used cubic spline methods. Papamichael and Whiteman used cubic 17

11 spline approximations to a one-dimensional, transient heat conduction problem (1973). The problem is of the same type as the one used as an example in Chapter 4 of the present study. The major difference is that their cubic spline approximations are embedded in the finite difference equations while, in this study, the approximating function is applied after the calculations of temperatures are made using the finite difference equations. Another type of atypical heat conduction problem is proposed by Beck et al. (1985). The authors call this type of conduction problem ill-posed. They explain that these problems come from experiments in which surface heat fluxes are to be determined based on temperatures measured inside a solid. They are able to estimate the surface heat fluxes based on the history of the measured temperatures. Although this is not directly related to the present study, the aspect of having no boundary condition at a surface, with such a boundary condition being the goal of the calculation, is common to the present study. 2.3 History of Heat/Mass Transfer Analogy Analogous behaviors of heat and mass transfer have been long recognized. Lewis (1927) contributed analysis on the analogy. They were simultaneously discussed by Nusselt and Schmidt in the late 1920s (Chilton and Colburn, 1934). In this publication, they discuss that, for the same dimensionless numbers (later to be called Prantdl and Schmidt numbers), heat and mass transfer, and Nusselt and Sherwood numbers, respectively, are analogous (Schmidt 1929 and Nusselt 1930). At this time, however, none of the four non-dimensional parameters were named. Colburn (1930) conducted an experimental investigation. Sherwood, after whom the non-dimensional mass transfer coefficient was later named, published one of his works on sublimation mass transfer in Many of today s mass transfer experiments are based on the sublimation processes, which has allowed the technique to be applied to non-flat surfaces, such as in gas turbine heat transfer. By the late 1960 s, all the four nondimensional parameters introduced above were present. 18

12 Eckert and Goldstein (1970) suggest sublimation mass transfer as an experimental technique for quantitatively measuring heat transfer. This step is remarkable because the experimental technique essentially relies on what is later introduced in the present study as the conventional mass/heat relationships. These conventional relationships are easy to apply and used in many situations. In the 1990s, experimental works have provided more insight into the analogy. In Häring and Weigand (1995), the authors analytically developed analogy functions for airfoil flow which take fluid compressibility into account. They compared favorably with heat transfer measurements (Häring et al. 1995). These results further agreed well with numerical calculations by TEXSTAN (Hoffs et al. 1997). In Goldstein and Cho (1995), the authors compare experimental results on objects with various geometries with the conventional relationships that are derived from the governing equations. It is shown that, except in flow separation zones, the conventional relationships apply quite well. 19

Introduction to Heat and Mass Transfer. Week 12

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