Calculating Cooled Turbine Efficiency With Weighted Cooling Flow Distributions

However, the definition of actual and ideal power contributions becomes increasingly complex with the addition of cooling streams in a modern highly cooled turbine.

For the many cooled turbine efficiency formulations available in the literature, stage exit conditions are often represented by area- or mass-weighted (MW) averages. However, the spatial distribution of cooling flows at the stage exit locations are known to be non-uniform, including radial variations due to injection near the hub (i.e., purge flows), circumferential variations due to injections from stationary hardware (i.e., trailing edge flows from the vanes), and general film cooling distributions on the airfoils or endwalls. Despite the known non-uniformity of cooling flows in the annulus, a fully mixed approximation is often used to simplify efficiency calculations.

The present study provides a modified approach that accounts for non-uniform spatial distributions of cooling flows in the efficiency calculations. This modified approach is applied to a single thermal efficiency equation, although it could be applied similarly to other efficiency formulations. Measurements from a public dataset are used to show that efficiency variations approaching half a point of efficiency can be realized depending on a number of factors: cooling flow distribution, stage exit profiles, and overall stage total pressure ratio (TPR). The results of these analyses show that careful consideration of cooling flow distributions in efficiency calculations may be particularly important when comparing turbine designs that result in different distributions of cooling flow or a shift of radial pressure and temperature profiles at the stage exit plane.

The effects of cooling flows on turbine efficiency have been discussed extensively throughout the open literature, but there is a distinct lack of agreement on which method is most appropriate [1]. In many cases, different gas turbine manufacturers may utilize separate equations to represent an approach most aligned with prior experiences. Kurzke [2] identifies this as a particular challenge because the same highly cooled turbine could be characterized by efficiency numbers that vary by multiple points, depending on the equation used for calculation.

Despite the disparity when defining efficiency equations, efficiency remains as a valuable tool for assessing design merits. Building upon early assessments of cooled turbine efficiency and preliminary loss modeling approaches [3–5], Louis et al. [6] introduced a simple enthalpy-based equation for calculating efficiency of a cooled turbine stage with a primary focus on understanding overall cycle efficiency when including different cooling methods. Similar analyses have also been performed by El-Masri [7] and Horlock [8]. In addition to traditional efficiency modeling approaches, a series of studies have analyzed cooled turbine efficiency with considerations for mixing and loss development, including work by Jordal [9], Horlock et al. [10], Young and Wilcock [11,12], and Wilcock et al. [13].

In some studies, focus has been placed on whether specific cooling streams should be included in the calculation of actual and ideal power for the efficiency equation. As part of the energy efficient engine (E³) program, Timko [14] defined η_GE, which neglects all cooling flow in the calculation of the ideal power term. Walsh and Fletcher [15] proposed that any vane or platform cooling entering downstream of the vane throat does not contribute to power calculations for its own stage, but it does count for downstream stages; the same argument is made for all cooling flows upstream and downstream of the rotor, and film cooling on the rotor itself. With exception to vane cooling upstream of the throat, the Walsh and Fletcher methodology is similar to Timko. As a slight modification, Kurzke [2] notes that all vane cooling air contributes to power extraction in the rotor.

In opposition to these prior discussions, Young and Wilcock [12] showed that rotor coolant does contribute to the actual power term in the numerator of the efficiency equation. In an effort to build equivalence, Young and Horlock [1] recommended industry-wide adoption of an approach to include all coolant streams in the ideal power term, regardless of whether or not they are included in the actual power term. Aside from these many discrepancies, prior authors have also disagreed over the inclusion or exclusion of the coolant pumping power in the actual power term. Young and Horlock [1] proposed removing the pumping power from the actual power in the numerator, whereas earlier presentations prefer to include the pumping power when calculating actual and ideal power terms [2,14].

The method by which actual and ideal power terms are determined can be broken down further beyond these initial nuances. Hartsel [16] utilized a kinetic energy approach for cascade analysis that was later adopted by Young and Horlock [1]. The extension by Young and Horlock represents thermodynamic efficiency of a cooled turbine as a simple expansion of all individual flows (main gas path and discrete cooling streams) from their respective inlet conditions to a shared output condition at the stage exit. This simplistic enthalpy formulation parallels similar equations presented by Leach [17]; the same formulation is identified by Jordal [9] and Kurzke [2] as a “black box” approach due its objective approach toward the source of individual cooling streams.

Beyond the equations used for efficiency, Young and Horlock [1] also emphasize the importance of providing coolant supply stations as supporting information to accompany efficiency values. In this way, Young and Horlock identify that the boundaries of the selected control volume have meaningful impacts on the outcome of calculated efficiency. As an example, this can be important whether rotor cooling is defined on-board with the rotating hardware (i.e., at the blade attachment) or on the stationary hardware (i.e., near or upstream of the preswirler).

Of similar importance, the manner in which non-uniform flows are averaged can also have significant effects on the calculated parameters contributing to efficiency calculation. The process by which discretized experimental measurements are converted to representative values at specific measurement planes has been addressed at length in the literature. Skiles [18] outlined several methods, but identified that area averaging inlet and exit parameters for turbine efficiency calculations were most common among several testing organizations. Similarly, Pianko and Wazelt [19] presented results for weighting turbomachinery flows based on area, mass, entropy, and momentum.

Further discussions on averaging approaches were evaluated by Cumpsty and Horlock [20], including assessments of availability-average, thrust-average, and work-average. As a primary outcome, Cumpsty and Horlock emphasized the importance of selecting averaging techniques that are most appropriate for the outcome. Whereas mass averaging is typically most appropriate for stagnation pressure and temperature; the authors also conceded that area averaging is often the easiest approach to implement for discretized measurements (i.e., profile rakes). More complex methods for calculating average properties at turbomachinery measurement planes have also been introduced recently using statistical approaches [21,22]. Each of these techniques has relative merits presented by the authors and needs to be considered relative to the desired outcome, as proposed by Cumpsty and Horlock.

Non-uniformity of cooling flows has also been identified in studies investigating rim seal ingestion characteristics. To analyze re-ingestion of purge flow injected upstream of a turbine rotor into a downstream cavity, Scobie et al. [23] performed a radial traverse measuring sealing effectiveness, which represents an analog for gas concentration. Through this approach, the authors identified mass fraction of injected cooling flow in a radially traversed profile downstream of the rotor.

Similar measurements were also conducted by Monge-Concepción et al. [24]. In their study, Monge-Concepción et al. evaluated multiple concurrent cooling flow injections through the use of superposition to show independent mass fraction in the annulus. Although the authors reported radial profiles, the profiles represented circumferentially averaged results captured across one vane pitch. The results with multiple cooling flows showed quantitatively how the distribution of cooling flow mass fraction in the annulus can differ substantially based on the nature of the cooling flow injection—i.e., hub-injected purge flows versus vane trailing edge (VTE) cooling flows.

Several studies have specifically evaluated the efficiency impacts of injected cooling flow rate and location using isolated research turbine environments [25–27]. Of these references, only Girgis et al. [25] studied a high-work turbine stage design; many studies have generally been performed in facilities with notably simplified efficiency equations due to low stage pressure ratio and temperature ratio. Along these lines, relatively few authors have also included spanwise efficiency as a metric for assessing cooling flow impacts [28–30]. Despite identifying that cooling flow injection migrates into the main gas path, the distribution of cooling flow in the annulus was not explicitly measured or included as part of the efficiency equations in these prior studies.

The current study builds upon the body of literature encompassing the definition of efficiency equations and management of non-uniform flow distributions. For the first time, this analysis combines known cooling flow distributions to mass weight individual contributions of the cooled turbine efficiency equation and account for the non-uniform split of primary main gas path flow and multiple cooling flows.

To demonstrate the effect of cooling flow distributions on efficiency, a one-stage turbine design with well-documented experimental test data was selected for analysis. The one-stage high-pressure turbine (HPT) was designed by Pratt & Whitney (PW) as part of the NASA energy efficient engine (E³) program. A general overview of the E³ program is provided by Batterton [31], and a series of design and test reports for the PW turbine were published documenting nearly every facet of the turbine at various stages in the design process [17,32–35]. The cooled turbine component test rig details presented by Leach [17] were specifically selected to define the framework for this study. Although the PW E³ turbine is a design more than four decades old, the wealth of detail provided in the reports and the single-stage design make it an ideal vehicle for the present analysis.

In Eq. (2), the inlet and outlet measurement locations are defined according to the cross-sectional schematic, Fig. 1.

In addition to outlining the geometry for the one-stage turbine, Fig. 1 defines the injection locations for five different cooling streams prescribed by Leach [17]. As an accompaniment to the cross-sectional cooling flow locations, Table 2 defines the boundary conditions for the cooling flows based on the measurement locations and control volume defined by Leach.

The five cooling streams are defined corresponding to different regions of the turbine: {I} the secondary “mini” tangential on-board injection (TOBI) system provides pre-swirled cooling flow to the front rim cavity region; {II} the main TOBI provides pre-swirled cooling air to the blades; {III} a bore cooling line flows from an upstream plenum through an air gap separating the disk and the shaft before passing through a series of holes in the disk on the downstream side (not shown in Fig. 1). Following this circuitous path, the bore cooling flow effectively leaks into the main gas path on the aft side of the turbine disk; {IV} the vane inner diameter (ID) flow supplies the ID platform cooling and a portion of the vane cooling in the forward internal cooling passages; {V} the vane outer diameter (OD) and blade outer air seal (BOAS) flow supply the OD platform cooling, a portion of the vane cooling in forward passages, all of the cooling for the vane trailing edge, and additional cooling entering through the BOAS over the blades.

For averaged thermodynamic properties, calculations throughout this study were performed using both AW and MW quantities. The results yielded trends that are both qualitatively and quantitatively similar, matching the conclusions from Ref. [20]. Mass averaging was selected for presentation, and area-averaged results are excluded for brevity.

Many equations for cooled turbine efficiency have been presented for consideration, as outlined in the literature review. For this study, a straightforward thermodynamic efficiency was selected as the basis equation with no consideration for the location of the cooling flow injection. This method follows the “black box” approach [2,9] and is a direct match for the efficiency equation defined in the E³ comparison study [17].

In Eq. (6), each of the “j” cooling flows has a supply mass flow rate, $m˙c,j$⁠, and supply total enthalpy, h_t,c,j. The total enthalpy parameters are calculated based on the total pressure and total temperature using the thermodynamic equation program, refprop [36], with standard dry air as the working fluid. The enthalpies achieved through isentropic expansion, h_t,out,s and h_t,out,s,j, are calculated based on constant entropy relative to the respective initial condition, which is also calculated via refprop.

Note all of the enthalpies in Eq. (7) are marked mass-weighted averages for completeness, although the uniform values applied to main gas path inlet and cooling supply values in the present study could warrant a simplification.

A few critical notes are required to accompany the spanwise efficiency definition in Eq. (12). Most importantly, the inlet conditions for each flow stream (primary main gas path and individual cooling streams) are considered by their mass-weighted averages using Eq. (8). Due to potential radial redistribution of flow through the stage, it is not appropriate to track Kiel-to-Kiel values paired at the inlet and exit of the machine without prior knowledge of particle tracking. However, if the inlet conditions are represented by an averaged value prior to implementing Eq. (12), then particle tracking assumptions are not required. In the case of the present study, the inlet values are assumed to be constant values at the outset, so Eq. (12) is valid without additional processing required.

To assess the relative effect of these different efficiency equations, each was applied to the single-stage E³ turbine test data. The results of this comparison are shown in Fig. 3 as deltas relative to the primary efficiency definition, η_MWTP, from Eq. (7). When applied to the selected single-stage experimental data, the range of efficiency values agrees within 0.1 points. It should be noted that the absolute value of efficiency using the MWTP method yields an output here of 87.47 points, which is slightly lower than the 88.37 points reported in by Leach [17]. The efficiency equation documented by Leach reflects Eq. (6), but other details about the management of inlet profiles, circumferential variation, and exact calculations of enthalpy (thermodynamic lookup tables, for example) are not shared in the report. Based on these potential sources of variation, the discrepancy is not a particular concern, but the primary focus remains on efficiency deltas for this comparison and throughout the remainder of this study.

Scobie et al. [23] collected radial profiles of cooling flow concentration downstream of the rotor in a 1.5-stage turbine operating at 4000 rpm. The cooling flow was injected axisymmetrically at one inboard location upstream of the disk. For comparison, Monge-Concepción et al. [24] collected measurements using engine-representative HPT airfoils and rim seal geometries at rotational speeds approaching 11,000 rpm. Cooling flows were injected through two different pathways—one as a purge flow from an inboard location and a second through slots spaced radially across the trailing edge of the vanes (the VTE flow). The profile of VTE flow shown in Fig. 4(c) was calculated using a proven superposition technique relative to the purge-only profile shown in Fig. 4(a); this superposition technique is also discussed by Monge-Concepción et al. It is possible that the different rotational speeds and geometry definitions documented by Scobie et al. and Monge-Concepción et al. may affect the mixing driving secondary flows and, accordingly, the shape of the profiles in Fig. 4. Ultimately, the different shapes shown here will provide guidance of possible effects on efficiency.

For all three experimental datasets shown in Fig. 4, the E³ rake Kiel positions are superimposed as data markers on the measured profiles. Although the measured data from Scobie et al. [23] and Monge-Concepción et al. [24] are independent of the E³ turbine stage geometry introduced for this study, the results from Scobie et al. and Monge et al. are being used as representative cooling flow distributions that may occur for different types of turbine cooling flow injection.

It should be noted here that the sealing effectiveness, ɛ, is analogous to the mass fraction, x, for a binary composition of main gas path flow and one cooling flow injection. To connect the profiles from Fig. 4 with the desired Kiel locations, a discretized area-weighting approach was applied. Area-weighting is appropriate here because the parameter of interest (mass fraction, x) already represents a mass quantity, so further mass-weighted averaging would be redundant.

The corresponding discrete cooling flow weights are shown in Fig. 5 for all three datasets from Fig. 4. Because the input values of sealing effectiveness represent a mass fraction of cooling flow at the specified measurement location, the output of the weighting process shown in Fig. 5 is a new type of mass weight: the cooling flow mass weight, w_mc. Similar to the mass weighting approach described in Eq. (4), w_mc represents the portion of total cooling flow that is attributable to each rake Kiel. As for other weighting methods discussed in this study, the discrete cooling flow mass weights must sum to unity.

With knowledge of how much of each cooling flow is present at a given Kiel, a new formula for efficiency is proposed that accounts for these weighted cooling flow distributions. The cooling flow weighted enthalpy (CFWH) approach is defined by

Although the approach presented here considers only radial profiles of pressure, temperature, and cooling flow distribution, a deeper application of this method could be similarly applied to radial-circumferential surveys. This simplification was chosen for demonstration purposes, but cooling flows that are injected through stationary airfoils, in particular, may yield unequal distributions in the circumferential direction.

With the definition of cooling flow weighted enthalpy efficiency, its effect will be assessed using the E³ turbine design as a known baseline. The five cooling flow injections defined for the E³ turbine and documented by Leach [17] are representative of an engine-realistic test case, but a simplified case was first considered that includes only the mini TOBI purge flow, {I}. This simplified cooling flow case (case 1) is shown with the Fig. 6 cross section for direct comparison with the original layout in Fig. 1 (case 2).

Single-injection cooling flow configurations have been studied extensively in the open literature for the purpose of understanding rim sealing performance and optimizing cavity geometries [23,24,37–39]. A small number of studies have specifically focused on the efficiency impacts of single-injection cooling flows [28–30], but weighted cooling flows were not included for efficiency. For these reasons, the single-injection option represents a valuable step toward understanding implications of CFWH efficiency in the context of prior single-injection cooling flow studies.

Beyond defining the number of cooling flows, the shape of cooling distributions in the annulus is also of critical importance (Fig. 4). Although these profiles will be assessed throughout the present study, their application to the simplified purge flow configuration, case 1, is particularly relevant because it closely matches the original test cases by Monge-Concepción et al. [24] and Scobie et al. [23]. For this reason, two subcases are defined in Table 3: case 1A implements the radial distribution of cooling flow measured by Monge-Concepción et al. (a), and case 1B applies the distribution measured by Scobie et al. (b).

Table 3 also outlines two subcases for the case 2 configuration that includes all five cooling flows. While considering the hub-skewed profiles (a) and (b), a third profile is also considered that represents a more uniform distribution, as measured by Monge-Concepción et al. [24], (c). Case 2A applies the hub-skewed profile (a) to cooling flows that are primarily sourced from the inboard side of the turbine. Although it is likely that some portion of the main TOBI {II} leaks past the forward knife edge on the disk and merges with the mini TOBI, it is assumed here that the phenomenon is negligible and the entirety of the main TOBI {II} exits the blade cooling features resulting in an equal distribution about the annulus. A unique approach was also applied to the vane OD and BOAS cooling stream {V} for case 2A by inverting the shape of the purge distribution (a) from Fig. 4 to yield a tip-skewed profile with similar shape.

Case 2B addresses alternate perspectives for three of the cooling flows relative to case 2A. In particular, the main TOBI {II} is represented by the hub-skewed profile (a) due to the fraction of flow that leaks forward of the disk and between blades. The vane ID flow {IV} and vane OD flow {V} are represented by a uniform distribution (c) from Fig. 4 due to alternate interpretation of the internal vane cooling architecture and associated platform cooling strategies for the E³ design.

The total cooling flow mass fractions were calculated for each of the cases from Table 3 by applying a superposition of the discretized distributions from Fig. 5 as weights against the individual supply flow values from Table 2, and subsequently normalizing by the total cooling flow for the case. These results, Fig. 7, show different radial shapes resulting from simulated cooling flow injection methods and how they influence stage exit flow distribution. In particular, the mass fraction profiles for case 1A and case 1B directly mirror the mass weights from Fig. 5, but the combinations of all five cooling flow injection locations for case 2A and case 2B emphasize the location of more cooling flow at the outer diameter and the inner diameter, respectively. These distribution plots in Fig. 7 will provide further reference for subsequent discussion.

For purposes of this comparison, the MWTP efficiency formula, Eq. (7), was selected as the baseline value to which each of the other approaches will be referenced. The results from applying different efficiency calculations are shown in Fig. 8. In Fig. 8, case 1 results are shown with solid bars, and case 2 results are represented by hatched bars. The initial comparison for case 1 of MWH, and MWE (referenced to MWTP) yields results very similar to those previously shown for case 2 in Fig. 3 (and shown again in Fig. 8 for comparison). Based on how the CFWH efficiency is calculated on a Kiel-by-Kiel basis, it may also be compared with the MWE results.

For case 1, the two different cooling flow profiles extracted from cooling flow case (a) and (b) yield similar outputs in terms of cooling flow weighted efficiency, CFWH-1A and CFWH-1B, respectively. When referencing the MWTP as a standard baseline in Fig. 8, both efficiency demonstrations CFWH-1A and CFWH-1B show a very small decrease. Both cases, however, show a smaller efficiency delta relative to the MWE comparison.

While the inclusion of all five cooling flows for case 2 in Fig. 8 shows similar qualitative behavior compared with the single cooling flow for case 1, the magnitude of the effect is reduced. The relative change of CFWH-2A and CFWH-2B is negligible relative to the MWTP baseline and nearly 0.1 points increase relative to MWE. Recalling that the turbine exit conditions are equivalent for case 1 and case 2, and only the amount and type of cooling flow are changing, the results of Fig. 8 point to the importance of different cooling flow distributions when efficiency is calculated using η_CFWH.

The importance of observations associated with integrated efficiency presented in Fig. 8 may be disregarded by some readers. For instance, experimental accuracy and repeatability of efficiency less than 0.1 points is difficult to achieve. Furthermore, the proposed approach presented for η_CFWH is only changing the calculation method and the potential pursuit of the “actual” efficiency value. While these claims are valid, subsequent analysis will continue to show the utility of the proposed cooling flow weighted efficiency approach.

To begin, Fig. 9 presents the spanwise efficiency profiles associated with integrated efficiency calculations in Fig. 8. In Fig. 9(a), the efficiency is shown as a function of each measurement Kiel at the turbine stage exit. Here, readers are reminded that η_SPAN represents the radially discretized values of efficiency, which are subsequently mass-weighted to determine η_MWE from Fig. 8. Because case 1 represents an artificial modification of the original E³ operating condition, the absolute values of efficiency in Fig. 9(a) are not shown, but the relative differences are valid for comparison.

The differences of efficiency, Fig. 9(b), are shown for both CFWH-1A and CFWH-1B relative to SPAN. The changes in efficiency profiles are noted by their similarity to the inverse of the cooling flow distributions from Figs. 5 and 7. In this case, the primary driver of efficiency change is due to the accurate accounting of how much main gas path flow is present at each Kiel. When more cooling flow is present (e.g., near the hub), that cooling fluid displaces hot main gas path fluid to outer radii. Although there are thermodynamic differences associated with the different Kiels (Fig. 2), the differences quantified in Fig. 9(b) are primarily associated with the main gas path mass flowrate because it carries influence that is an order of magnitude greater than the individual cooling flows.

Figure 9(b) highlights the importance of cooling flow weighted efficiency. The comparison of results for CFWH-1A and CFWH-1B yields the same integrated efficiency and spanwise efficiency when calculated through traditional approaches such as MWTP. As shown in Fig. 8, the integrated values of CFWH efficiency are also unchanged between the two cases. However, the different distributions of cooling flow at the stage exit contribute to an unequal spanwise efficiency using the cooling flow weighted approach, which can be interpreted as a value-added improvement when comparing different designs.

A similar analysis of case 2 extends the integrated efficiency values from Fig. 8 to a spanwise efficiency assessment in Fig. 10. In Fig. 10(a), the spanwise efficiency is shown and compared with the results from Leach [17] for validation purposes. The spanwise profile extracted from Leach was adjusted to match the integrated value reported in the same publication. Slight discrepancies are noted in the midspan region of up to one point at some Kiel position when compared with the “η_SPAN” values calculated in this study. As preliminarily addressed in prior sections, these minor discrepancies relative to Leach are likely associated with differences of thermodynamic property calculations, undocumented averaging approaches for different measurement quadrants, and adjustments introduced cursorily in the E³ report. Regardless of these contributors, there is very good agreement between spanwise efficiency profiles in Fig. 10(a).

The spanwise efficiency in Fig. 10(a) is shown on the same relative scale as case 1 results in Fig. 9(a), and greater magnitudes of differences can be readily identified for CFWH results when compared to the SPAN calculation. Indeed, these efficiency deltas for case 2 configurations, Fig. 10(b), are up to five times the magnitude of case 1 configurations.

Analysis of Fig. 10 is also supported by the temperature profile from Fig. 2. Following a traditional efficiency approach such as MWTP, cooler temperatures near the hub are attributed to main gas path power extraction. However, if a hub-skewed cooling flow distribution is applied through CFWH, the lower temperature near the hub is attributed more to the presence of low-temperature cooling flow. This effect, in combination with redistributed main gas path flow, shows a more pronounced efficiency increase near midspan. In particular, case 2A in Fig. 10(b) includes the tip-skewed cooling profile for the vane OD flow {V} which shifts main gas path flow away from the tip (Fig. 7), leading to a more noticeable efficiency profile change at both the OD and midspan when compared to case 2B.

Building upon the cooling flow weighted efficiency assessments using the predefined E³ test case, further analysis was conducted to determine which turbine operating parameters can influence the magnitude of observed efficiency deltas using CFWH. Several parameters were permuted including cooling flow rates relative to main gas path (i.e., $%m˙in$ from Table 2), cooling flow-to-main gas path density ratio (independently assessed through cooling flow supply pressure and temperature), radial distribution of stage exit pressure and temperature, and stage total pressure ratio. Relative cooling flow rate variations of ±60% relative to the values in Table 2 showed negligible sensitivity for the application of CFWH efficiency. Similarly, cooling flow density ratio variations of −40% to +20% relative to the values in Table 2 also yielded negligible sensitivity. As a result, the outcomes of these assessments are not discussed further. However, stage exit radial profiles and stage pressure ratio showed sensitivities that are worthy of further analysis.

The single-stage turbine used for demonstration in this study includes exit pressure and temperature profiles showing higher work extraction at the hub than the tip (Fig. 2). To supplement these experimentally measured profiles, a series of simulated distributions were created representing skew values of −10% to +10% (Fig. 11). In this case, the skew value is associated with the gradient between 0% and 50% span (or equivalently, 50–100% span). With the linear skew value defined, the overall profile was shifted by biasing the defined profile higher or lower as needed to match the predefined mass-weighted values of exit total pressure and total temperature. An iterative approach was utilized to ensure a match of the desired constraints. For comparative purposes, it is instructive to note that the E³ profiles shown in Fig. 2 are qualitatively similar to the +5% skew profile from Fig. 11, except for mild deviations near midspan.

Each skew profile from Fig. 11 was propagated through the CFWH efficiency calculation approach with mass-averaged stage exit pressure and temperature matching the original E³ test data. In these cases, the total pressure and total temperature were both defined using the same relative profile shapes from Fig. 11. In other words, no mixed profiles were evaluated combining different skew levels for pressure and temperature.

For each skew profile assessment, cooling flow configuration cases 1A, 1B, 2A, and 2B were independently applied. The spanwise efficiency outputs from this analysis are shown separately for each case in Fig. 12 and compared with an overlay for the baseline E³ pressure and temperature profiles from Fig. 2. Comparing Figs. 12(a) and 12(b), there is negligible difference for the skewed profiles between single cooling flow injection cases 1A and 1B, respectively. However, the skewed profiles of pressure and temperature expectedly correlate inversely with efficiency such that the positively skewed pressure and temperature profiles result in negatively skewed efficiency profiles (higher efficiency at the hub).

In contrast to the similarity of cases 1A and 1B, the comparison of results for cases 2A and 2B in Figs. 12(c) and 12(d), respectively, shows more substantial differences. As previously identified, the difference comparing Figs. 12(c) and 12(d) is due to the redistribution of main gas path fluid away from the OD casing in case 2A that is not present in case 2B. Indeed, the outer 50% span for case 2B is qualitatively similar to the results for cases 1A and 1B. Throughout all four parts of Fig. 12, the similarity of CFWH results applying E³ profiles from Fig. 2 shows continued similarity to the +5% skew level with the largest deviations identified in the midspan regions.

Figure 12 further indicates the value of CFWH efficiency when comparing different turbine designs. CFWH is particularly suited to accurate characterization of relative design improvements for comparisons that include novel designs pursuing spanwise work redistribution while also incorporating alternative cooling architectures for reduced cooling needs.

Through this approach, the overall uncooled stage performance is maintained relative to the original E³ test data when simulated pressure ratio variations are applied.

For these defined stage inlet and exit conditions, the skewed profiles from Fig. 11 were subsequently applied, and the integrated CFWH efficiency was assessed relative to the integrated MWTP efficiency as a baseline. The relative similarity of efficiency behaviors identified for cases 1A and 1B in Fig. 12 (and corresponding similarity for cases 2A and 2B) persists to the total pressure ratio sensitivity assessment. For this reason, only the results from case 1A and case 2A are shown for comparison in the pressure ratio analysis (Fig. 13).

Figure 13 addresses a range of stage total pressure ratios from 1.5 to 6.0, which covers a broad spectrum of experimental research turbines in the open literature. In particular, Hura et al. [40] reviewed highly loaded turbine stages and assessed the performance of a single-stage cooled HPT design with a pressure ratio of 5.5; the lower boundary includes designs with lower loading parameters and shows trends toward what might be expected in low-speed test facilities.

Considering first the single cooling flow configuration, case 1A, Fig. 13(a) shows that the CFWH approach contributes more significantly to integrated efficiency as total pressure ratio decreases. Furthermore, more highly skewed exit profiles (i.e., ±10% following the definition in Fig. 11) lead to greater efficiency deltas relative to the MWTP baseline. While turbines with stage pressure ratios exceeding 5.0 may exhibit a calculated efficiency delta of 0.1 points for ±10% skew, a turbine operating with ±10% exit profile skew at a stage total pressure ratio of 1.5 would yield an efficiency delta exceeding 0.4 points.

An associated analysis of case 2A in Fig. 13(b) identifies a reduced band of influence at lower stage total pressure ratios relative to case 1A. With the addition of all cooling flows, a turbine with a ±10% exit profile skew and a pressure ratio of 1.5 would have a corresponding efficiency delta of only 0.3 points, but the behaviors identified at higher pressure ratios are consistent relative to the single cooling flow case. A primary difference is noted in the asymmetry of case 2A, Fig. 13(b), for positively skewed exit profiles relative to negatively skewed profiles that are less pronounced in case 1A. In particular, the addition of weighted cooling flow at both sides of the annulus (ID and OD) leads to the reduction of efficiency sensitivity for positively skewed profiles. As a result, the superimposed data point for CFWH-2A from Fig. 8(b) which most closely follows the +5% skew profile is shifted to a reduced magnitude of efficiency delta compared to its counterpart from Fig. 13(a), as originally observed in Fig. 8.

Also in Fig. 13(b), a series of three cross sections are included to add context for specific turbine designs across a breadth of operating conditions and how they could be impacted by the application of a CFWH efficiency analysis. First, the 4.01 pressure ratio is identified for the current single-stage E³ design assessment. As a more modern reference, the pressure ratio of 3.75 is marked representing the first stage of the high impact research turbine design [41]. Finally, a value of 2.25 is more indicative of the first stage for a two-stage HPT design—in this case specifically matching the first stage parameters outlined by Halila et al. [42]. Ultimately, these cross sections show that whereas the highly loaded single-stage E³ design outlined by Leach [17] may be mildly affected by CFWH analysis, relevant turbine designs at lower pressure ratios are prime candidates for considering the cooling flow weighted approach in turbine design performance assessments.

This study outlines a new approach for calculating cooled turbine efficiency that accounts for non-uniform distributions of cooling flow at the stage exit. Whereas cooling flow distributions are largely known to be non-uniform, some forms of efficiency calculation assume they are fully mixed and uniformly distributed at the turbine exit. For the first time, a methodology is laid out to properly account for expansion of individual flow streams from their supply conditions to discrete measurements at the turbine exit plane.

Test data from a one-stage turbine design for the E³ program was used as a demonstration vehicle for calculations assessing the new method, and comparisons were drawn relative to other calculation methods for cooled turbine stage efficiency. Different distributions of cooling flows and different numbers of cooling flow injection locations were independently evaluated. For the calculation cases applied to the E³ design with a stage pressure ratio of 4.01, the effect of incorporating cooling flow weighted efficiency was typically less than 0.1 efficiency points. Despite this low-magnitude influence on integrated efficiency, however, much more significant differences were identified for spanwise efficiency profiles. The fully cooled E³ design assessment demonstration showed spanwise distribution effects in excess of five efficiency points at some span locations, depending on the selected cooling flow distribution patterns.

Because the adoption of cooling flow weighted efficiency outputs a spanwise efficiency accounting for contributions of cooling flow distributions in the annulus, it is ideal for secondary flow loss assessments. A significant portion of losses in the endwall regions are typically attributed to secondary flows. However, the demonstration in this study showed that cooling flows biased near the endwall may also exhibit decreased efficiency near the wall when applying a cooling flow weighted efficiency approach. Following standard efficiency formulations, losses that are thermodynamically attributable to cooling flows may be partially attributed to secondary flows inaccurately.

Additional assessments were also performed to evaluate the effect of cooling flow weighted efficiency calculations on modifications of the original E³ design, including distribution of stage exit total pressure and temperature, and stage total pressure ratio. Paralleling the observations of spanwise distribution effects, the measured exit pressure and temperature profiles contribute significantly to the magnitude of observed effects. As a result, the cooling flow weighted efficiency approach can provide meaningful information for assessing the merit of new turbine designs, especially when aggressive changes are applied to radial work distribution in the stage or cooling architectures that could reposition cooling flow at the stage exit plane.

Finally, the influence of cooling flow weighted efficiency was shown to be heavily dependent on the stage pressure ratio. Although stage efficiency is known to trend directly with pressure ratio, the associated effect of weighting cooling flows is more sensitive to the change than a traditional fully mixed mass-weighted approach. Efficiency deltas in excess of 0.4 points were identified for stage pressure ratio of 1.5, depending on the corresponding cooling flow configuration in the stage. As a result, cooled turbine efficiency assessments performed on lower pressure ratio machines—for example, the first stage of a two-stage HPT design—would be influenced more by the cooling flow weighted efficiency approach than a highly loaded design.

Regardless of the application, the cooling flow weighted efficiency formulation represents a new consideration in an already large library of efficiency equations. Although many engine manufacturers likely have a preferred efficiency equation built upon decades of historical knowledge, the approach in this paper can be used as a supplementary analysis through which to evaluate design benefits. Particularly, as new turbine designs are sought that utilize complex three-dimensional blade shapes and unique cooling strategies, it is likely that comparisons will also include redistributions of parameters such as stage exit pressure and cooling flow mass fraction. The new approach presented here offers the first opportunity to fully address those changes by quantifying efficiency changes that might otherwise go unnoticed.

The author would like to thank many colleagues for their insightful feedback and challenging questions, including Dr. Eric DeShong, Joel Wagner, Dr. Karen Thole, Dr. Michael Barringer, Gary Zess, and Dr. Thomas Praisner. This material is based upon work supported by the NASA Aeronautics Research Mission Directorate (ARMD) University Leadership Initiative (ULI) under cooperative agreement number 80NSSC21M0068. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Aeronautics and Space Administration.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

c =

gas concentration

$m˙$ =

mass flowrate

w =

weight

x =

mass fraction

A =

area

$P˙$ =

power

Q =

generic quantity

h_t =

total enthalpy

P_t =

total pressure

T_t =

total temperature

 V_x =

axial velocity

γ =

ratio of specific heats

$ε$ =

sealing effectiveness

η =

efficiency

ρ =

density

A =

area weight

actual =

actual condition

c =

cooling condition

ideal =

ideal condition

in =

stage inlet condition

k =

discrete Kiel position

m =

mass weight

mc =

cooling flow mass weight

out =

stage outlet condition

SPAN =

spanwise

U =

uncooled configuration

 $Q=$ =

mass-weighted average

$Q~~$ =

area-weighted average

$Q=^$ =

calculated value using mass-weighted inputs