Theoretical Analysis of Heat Transfer Through an Idealized Gas Turbine Blade Model with Thermal Barrier Coating

In this study, a simpli®ed gas turbine blade model is considered in order to investigate the eectiveness of a new thermal barrier coating material. The model consists of two concentric cylinders with high temperature gas ̄ow in the annular space and with low temperature liquid in the inner cylinder whose exterior surface is coated with thermal barrier coating material having high emissivity. The new coating material, which was developed by the Noritake Co. Ltd., is silica-based glass ceramic. A control volume approach is used for gas and liquid ̄ows, while surface energy balance is used for the surfaces of the inner and outer cylinders. Results obtained for the new coating material are compared with those obtained for the materials with lower emissivities and zero emissivity, that is, the case in which radiation eect is neglected. It is disclosed that a gas turbine blade coated with the new thermal barrier coating material, glass-ceramic, results in higher heat transfer rates, which in turn enhance power output.


INTRODUCTION
The gas turbine is a primary energy deliverer not only for vehicular propulsion of such as air, land and water, but also for power generation.Several major factors aect thermal eciency or speci®c fuel consumption of a gas turbine plant (Yang, 1997).These include: (i) Increase in the turbine inlet temperature, namely ®ring temperature.(ii) Reduction of cooling air usage.(iii) Improving component eciency.(iv) Cycle enhancement.0 The ®rst factor is the concern of this study, and the others are not considered.The turbine inlet temperature is a measure of gas turbine power output.It can be increased in three dierent ways: (i) Aerodynamic cooling techniques.(ii) Thermal barrier coating.(iii) High performance materials for rotors, nozzles, and buckets.0 Gas turbine blades operate under severe stress conditions induced by high gas temperatures and high rotating speeds.There are three methods of protecting the blades: (i) Internal cooling (ii) External cooling (iii) Thermal barrier coating (TBC)0 As can be seen, thermal barrier coatings are very important for both increasing the turbine inlet temperature (and therefore turbine power output) and protecting the turbine blades (and therefore increasing their safety and life).Recently, a new TBC material, silica-based glassceramic has been developed by Noritake Co. Ltd.This material is characterized by a nearly constant high emissivity of over 0.80 up to a temperature of 2000 C. It has been experimentally demonstrated that a metal coated with such material can withstand a very high temperature without melting, while the original material (without the coating) begins to melt.
In this study, a simpli®ed analysis is performed in order to simulate a gas turbine blade placed in a high tempera-ture gas ¯ow passage.The results obtained for the new material are compared with the results obtained for two other materials that are commonly used in high-temperature applications, and with results obtained for the case without radiation.

ANALYSIS
The geometry considered in this study is shown in Figure 1.
It consists of two concentric tubes of diameters d 1 and d 2 .
Flue gases ¯ow through the annular space, while the cooling liquid, water, ¯ows through the inner tube.The system geometry given in Figure 1 is a simpli®ed model that simulates an internally-cooled gas turbine blade situated at the turbine inlet.The exterior surface of the outer cylinder is insulated while the exterior surface of the inner cylinder is coated with a thermal barrier coating material of emissivity, ".Conduction resistance through the inner and outer tubes, and eects of gas absorption and scattering on heat transfer, are neglected.
For the thermal simulation of the system, the control volume method is used.In x-direction, 100 control volumes are used for space discretization.Then, the model is divided into four parts for thermal simulation: the outer wall, the inner wall, the annular space through which the hot gases ¯ow, and the inner space through which the cold water ¯ows.In the following, each part is treated separately.

The Outer Wall
From a surface energy balance on the wall of the outer cylinder, which says that where Q h and Q r are the convection heat rate representing heat transfer from the hot ¯uid to the outer cylinder surface, and the radiation heat rate representing the radiative heat exchange between the outer cylinder surface and the inner cylinder surface, respectively.If their values are substituted into Eq.
[1], one obtains "cT 1 ÿ "c d r h 2 T 2 d r h 2 T m;2 0 3 where T 1 , T 2 and T m,2 stand for the surface temperature of the inner cylinder, the surface temperature of the outer cylinder, and the mean temperature of the hot ¯uid, respectively.Throughout each of the control volumes, these temperatures are assumed to be constant.In Eq. [3], ", and h 2 denote the surface emission, the Stefan ± Boltzman constant and the convection heat transfer coecient of the hot gas ¯ow, respectively.The term c is used to linearize the radiation term, which is given as 2.2.The Inner Wall Similar to the case for the outer wall, a surface energy balance formulation can be developed for the inner wall.Such an analysis yields q h2 q r ÿ q h1 0 5 where q h1 and q h2 are the convective heat ¯uxes representing the convection heat transfer from the inner wall surface to the cold liquid (water), and the convection heat transfer from the hot ¯uid to the inner wall surface, respectively.
The q r represents the radiative heat ¯ux between the outer and inner wall surfaces.Substituting the related values into Eq.
[5] yields ÿh 2 "c h 1 T 1 "cT 2 h 1 T m;1 h 2 T m;2 0 6 where h 1 and T m,1 denote the convection heat transfer coecient and the mean temperature of the water ¯owing through the inner tube.As stated earlier, the temperatures through each control volume are assumed to be constant.Similar to the case for the hot gas, applying the conservation of energy, Eq.where m 1 , c p,1 and 1 stand for the mass ¯ow rate, the speci®c heat at constant pressure, and the density of the water, respectively.
The space and time derivatives seen in Eqs.
[8] and [9] are discretized.The convective heat transfer coecients for the inner and annular space are determined from an empirical correlation for a fully developed, turbulent ¯ow (Incropera & DeWitt, 1996): Nu 0:023Re 4=5 Pr n 10 where Nu, Re and Pr are the Nusselt number, the Reynolds number, and the Prandtl number, respectively, which are de®ned as: Nu hd c =k Re Ud c = Pr = 11 Here k, and denote the conduction heat transfer coecient of ¯uid, the kinematic viscosity, and the thermal diusivity, respectively.The power n of the Eq.[9] is 0.4 for the annular space (where the gas temperature is higher than the surface temperature), and 0.3 for the inner space (where the water temperature is lower than the surface temperature).In Eq.
[11], d c denotes the characteristic dimension which is taken as d 1 for the inner space.For the annular space, the Reynolds number de®nition is where, is the dynamic viscosity.In evaluating the Nusselt number, d c is d 2 ± d 1 .

RESULTS AND DISCUSSION
The set of linear equation systems, Eqs.surface temperature, T s as shown in Figure 2, which can be approximated as " 0:982 ÿ 9 2 10 ÿ5 T s 13 which is T 1 here.Two other materials which are commonly used in high-temperature applications are also examined for comparison: Al 2 O 3 and ®re-proof clay.For these materials, the emissivities are determined using:  The length of the concentric tubes is L 1 m.The inlet temperatures for the hot gas and cold water are 3000 K, and 300 K, respectively.The gas properties used are: c p,2 2726 J/kgK, 2 955 2 10 ÿ 7 N s /m 2 , 2 0.32 kg/m 3 , Pr 2 0.536 and k 2 0.486 W/mK, and the water properties are: c p,1 4179 J/kgK, 1 997 kg/m 3 , and U 1 0.3 m/s.
In addition to the eect of the surface emissivity, that of the mass rate of the hot gas ¯ow, or Reynolds number, on the heat transfer are also predicted.The gas mass ¯ow rate is varied as 0.25, 2.5 and 25, which are equivalent to Re of 9.512 2 10 3 , 9.512 2 10 4 and 9.512 2 10 5 , respectively.The steady-state solutions are achieved when the temperature dierences in each control volume become less than 10 ÿ5 .At m 2 0.25 kg/s for dierent coating materials, Figure 3 show the changes of the hot gas mean temperature, T m,1 , and the cold water mean temperature, T m,2 , in the xdirection at steady-state.It is seen that all three materials give better results than the uncoated surface and that the new material, the glass-ceramic, has a higher exit mean water temperature and a lower exit mean gas temperature than the other two surface coating materials, Al 2 O 3 and ®reproof clay.This is an indication of higher heat transfer enhancement in the new surface coating material.A similar eect is observed for m 2 2.5 kg/s (Re 9.512 2 10 4 ) and for m 2 25 kg/s (Re 9.512 2 10 5 ), as illustrated in Figures 4 and 5, respectively.It is found that an increase in the mass ¯ow rate or the Reynolds number reduces the signi®cance of using the surface coating materials because of an increase in the convection heat transfer eect.Therefore, at a very high Reynolds number ( for example, Re 9.512 2 10 5 ), the eect of the surface radiation heat transfer is suppressed by an increase in convection heat transfer contribution, as depicted in Figure 6.

CONCLUSIONS
A simpli®ed heat transfer analysis is performed on two concentric tubes with hot gas ¯owing through the annular space and cold liquid (water) ¯owing in the inner tube.The outer surface of the inner tube is coated with a special thermal barrier coating material.The in¯uence of the Reynolds number of the hot gas ¯ow on the heat transfer is determined by changing the mass ¯ow rate of the hot gas ¯ow.Results are obtained for the spatial variations of the mean temperatures of the hot gas and cold water at steady state.The study has revealed that the use of coating materials on the inner surface having high emissivities has FIGURE 2 Variation of the emissivity, ", with temperature for dierent surface coating materials.

FIGURE 3
FIGURE 3 Variation of the T m,1 and T m,2 along the tube at Re 9.512 2 10 3 for dierent surface coating materials.

FIGURE 4
FIGURE 4 Variation of the T m,1 and T m,2 along the tube at Re 9.512 2 10 4 for dierent surface coating materials.

FIGURE 5
FIGURE 5 Variation of the T m,1 and T m,2 along the tube at Re 9.512 2 10 5 for dierent surface coating materials.