Estimation of cooling flow rate for conceptual design stage of a gas turbine engine

Development of a gas turbine engine starts with optimization of the working process parameters. Turbine inlet temperature is among the most influential parameters that largely determine performance of an engine. As typical turbine inlet temperatures substantially exceed the point where metal turbine blades maintain reasonable thermal strength, proper modeling of the turbine cooling system becomes crucial for optimization of the engine’s parameters. Currently available numerical models are based on empirical data and thus must be updated regularly. This paper reviews the published information on turbine cooling requirements, and provides an approximation curve that generalizes data on all types of blade/vane cooling and is suitable for computer-based optimization.


Introduction
The primary trend in improving efficiency of gas turbine engines is the systematic increase of the working process parameters: turbine inlet temperature (TIT) and overall pressure ratio (OPR), bypass ratio (BPR) together with increasing the efficiency of engine's elements. Engine's reliability throughout its lifetime plays a crucial role as well.
Maximum TIT has reached 1600-2000 K while a metal turbine blade maintains reasonable thermal strength at much lower temperatures around 1200 K, so the TIT increases at a substantially higher pace (see [1,2] for more details). That means that engine's lifetime and reliability could be guaranteed only by cooling turbine blades and vanes, or by switching to new materials like ceramics.
Numerous algorithms for gas turbine cooling design are available from simpler ones [3,4,5,6] to more sophisticated and detailed calculations of all elements of the cooling systems [1,2,7]. This paper will focus on the conceptual design stage with an inherently high level of uncertainty that makes more detailed approaches unsuitable.
Traditional algorithms to estimate required cooling air flow rate use the cooling intensity (θ). Unfortunately, numerous publicly available models connecting cooling intensity with relative cooling flow rate [7][8][9][10][11][12][13][14] show dispersion up to 3.5 percent for various cooling types, which is too high for reliable calculations. At the same time, increased cooling flow rate impacts both turbine efficiency and its exhaust temperature resulting in increased specific fuel consumption (SFC) [15]. Accuracy of cooling flow rate estimation at the conceptual design phase determines effectiveness of the 2 optimization of working process parameters. For this reason it is important to analyse the available models connecting cooling intensity with relative cooling flow rate.

Literature search and analysis of the cooling efficiency of the aircraft gas turbines
The following types of turbine blade cooling are currently available:  convective cooling,  convective film cooling,  porous cooling. Cooling flow rate includes the following constituents: where .
-flow rate for cooling blades, kg/s; .
-flow rate for cooling turbine disk, kg/s; .
-cooling air flow leaks, kg/s. Required cooling flow rate is a function of cooling type and cooling intensity θ: where 4 * -stagnation temperature of the combustion chamber exhaust, K; -blade temperature, K; * -stagnation temperature of the cooling air, K. Cooling flow diagram is on figure 1.     For each cooling type (figures 2-4) an envelope curve was established. These envelope curves are shown on figure 5. Next, a universal approximation curve was determined for all types of cooling. This approximation curve is described in the next section.   Based on the data shown in table 1-3 the share of film cooling of 0.7 may be used for the conceptual design phase. Figure 6 shows the relative cooling flow rate for the turbine disc cooling vs TIT [12]. Figure 6. Influence of maximum turbine inlet temperature on the relative cooling flow rate for the turbine disc cooling. Figure 7 shows the relative cooling air flow leaks vs maximum turbine inlet temperature.
As it was mentioned before, higher cooling flow rates increase the specific fuel consumption because of the reduced efficiency of turbines. For this reason it is important to accurately estimate turbine efficiency reduction as a function of cooling air flow rate. Open sources of information on turbine blades and vanes cooling were analyzed and the data was generalized as shown on figure 8.

Calculation of required cooling flow rate for an aviation gas turbine
Overall cooling flow rate is a sum of flow rates for cooling each turbine spool: where i=1, n -number of turbine spools (high pressure, medium pressure, low pressure, free turbine), -cooling air flow rate calculated using the equation (1).  Figure 7. Influence of the maximum turbine inlet temperature on the relative cooling air flow leaks.  6) Required relative air flow rate for blades cooling is determined using equations (3). Analysis shows that relative air flow rate for vanes and blades may be determined using the same graphs or equations. 7) Cooling air flow rate for blades cooling: . 9) Overall cooling flow rate for the turbine охл is calculated using the equation (1).

Determining required cooling flow rate for variable turbine inlet temperature ( 4 * = )
The algorithm for determining required cooling flow rate is similar to previously described. Instead of equations (3), the universal approximation curve (generalizing various types of cooling as a function of cooling intensity) should be used (see figure 5). Using separate curves for various cooling types may impede the optimization process as there would be efficiency leaps while switching from one cooling type to another resulting in false design points. Universal approximation curve, established for a wide range of turbine inlet temperatures solves this problem. This approach is accurate enough for the conceptual design stage and improves the efficiency of optimizational calculations: = 12,423 3 − 6,378 2 + 4,273 − 0,0225.

Conclusion
Not theoretical functions are available for the conceptual design stage, while the empirical data shows high dispersion of over 3.5 percent. This study analysed and generalised published empirical data on gas turbine cooling efficiency. Constant improvements to the turbines' designs and their materials require regular updates to the regression models of cooling efficiency.
The models developed in this study take into the account the air for cooling of vanes, blades and disks, as well as cooling air leaks. These models laid foundation for two algorithms for estimation of cooling air flow rates: for a fixed turbine inlet temperature and for the optimization purposes (when TIT is variable). Variable TIT required generalization of the curves for various types of cooling. Use of the universal approximation curve eliminates efficiency leaps and thus false design point calculations and is suitable for computer-aided design systems.