## Abstract

Efficiency is an essential metric for assessing turbine performance. Modern turbines rely heavily on numerical computational fluid dynamics (CFD) tools for design improvement. With more compact turbines leading to lower aspect ratio airfoils, the influence of secondary flows is significant on performance. Secondary flows and detached flows, in general, remain a challenge for commercial CFD solvers; hence, there is a need for high-fidelity experimental data to tune these solvers used by turbine designers. Efficiency measurements in engine-representative test rigs are challenging for multiple reasons; an inherent problem to any experiment is to remove the effects specific to the turbine rig. This problem is compounded by the narrow uncertainty band required to detect the incremental improvements achieved by turbine designers. Efficiency measurements carried out in engine-representative turbine rigs have traditionally relied upon assumptions such as constant gas properties and neglecting heat loss. This research presents an uncertainty framework that combines inputs from experiments and computational tools. This methodology allows quantifying uncertainty for high-fidelity efficiency data in engine-representative turbine facilities. This paper presents probabilistic sampling techniques to allow for uncertainty propagation. The effect of rig-specific effects, such as heat transfer and gas properties, on efficiency is demonstrated. Sources of uncertainty are identified, and a framework is presented which divides the sources into bias and stochastic. The framework allows the combination of experimental and numerical uncertainty. Gaussian regression models are developed to obtain speed-lines for the turbine map using the uncertainty of the measured efficiency.

## 1 Introduction

Small core turbines in aero-engines enable higher bypass ratios, conducive to higher propulsive efficiency. In power generation, smaller engine cores provide compact units with low starting times enabling integration with renewable energy power units to support power surges. Compactness leads toward low aspect ratio axial turbines, causing large secondary flow structures. Detached flows remain a challenge for commercial computational fluid dynamics (CFD) solvers employed during the design phase; therefore, experimental data are required to anchor the turbulence models in Reynolds averaged Naiver–Stokes and unsteady Reynolds averaged Naiver–Stokes simulations. The need for extensive high-fidelity experimental data has motivated efforts over the past years to enhance accuracy in exploratory testing at engine-representative Reynold's number and Mach number, while respecting engine temperature ratios. In recent years, an improvement in efficiency higher than one percentage point has been considered disruptive in turbomachinery [1,2]. However, from a testing and characterization view, this fractional increase poses a difficult task for the experimentalist. Achieving an absolute uncertainty level below 5% represents a significant challenge in fully rotating rigs, augmented in testing at engine-representative conditions. To confirm that a design works better than the baseline, the test engineer needs to ensure that the uncertainty of the efficiency measurement is less than the fractional increase in performance. Because efficiency is derived from the measurement of aero-thermo-mechanical quantities, a deep dive into the uncertainty of the individual properties and the propagation of measurement uncertainty into the derived magnitude is thus required.

Moffat [3] described quantifying experimental uncertainties and their combination using a truncated Taylor series expansion of the derived variables and neglecting higher-order terms. This has been the standard for most literature, as described by Atkins et al. [4], Mc Lean et al. [5], and Keogh et al. [6,7]. Treatment of gas properties in the uncertainty analysis becomes complicated due to the non-closed form of equations. Denos et al. [8] and Yasa et al. [9] tackled this by doing a sensitivity analysis of efficiency with a change in gas properties around the test point. This method suffers due to the variation in gas properties with the temperature; hence, the sensitivity is dependent on the operational temperature of the turbine. The effect of pressure on the gas properties was not considered. There is no clear consensus in the literature on which statistical metric to use for uncertainty calculation (standard deviation and standard error) and how it changes for a different formulation of efficiency. Beard et al. [10], Mc Lean et al. [5], and Atkins et al. [4] reported both precision and absolute uncertainty levels, but the efficiency calculation's basis are different; Keogh et al. [7] reported only precision uncertainty estimates, while Yasa et al. [9] reported only an absolute level of uncertainty for adiabatic efficiency. Due to the many assumptions made in the definition of the “basis” for the efficiency calculation, which cannot be independently verified, Haldeman et al. [11] argued that it is impossible to prove that the measured or mechanical efficiency is the true efficiency. Hence, the basis for the calculation must be specified to quantify uncertainty properly. The effect of the data system parameters, such as window size and sampling frequency, on statistical metrics for a time series also needs to be explored. Commonly in the literature [4,7–10], the parameters in the definition of efficiency are treated as independent, which results in a conservative estimate of the uncertainty, while in reality, the parameters are correlated to each other. Hudson and Coleman [12] presented this in the form of covariance between the parameters.

This paper explores the effect of assumptions and formulation of efficiency, the data system parameters, and the basis for mechanical and adiabatic efficiency. Numerical sampling enables precise uncertainty propagation in non-closed form equations, providing a simpler treatment of gas properties, and respects local sensitivities. The sampling method allows for the combination of *bias* and *stochastic* uncertainty obtained from numerical and experimental sources, as presented by Bhatnagar et al. [13]. This approach allows for uncertainty propagation from complex non-linear sources, which has required some level of approximation to propagate before in the literature [8,9]. The sampling method used along with turbine characteristics can use the co-dependence of efficiency parameters to predict trends and aid in the identification of test operation points.

Through turbine testing, researchers intend to gather the turbine performance over a wide operating range, commonly obtained through a parametric fit of the measured data [14,15] using the *maximum likelihood estimation* approach. However, experimental literature documents uncertainty estimation for a single operation point as shown in Refs. [4,5,7–9], but there is lack of literature on how this uncertainty is propagated into the turbine operation map. Though uncertainty into interpolation between measured points using simple regression such as linear or quadratic can be provided [16], these regressions are assumptions and are not based on the trends in the data. Our paper explores using Bayesian regression to deliver a probability estimate of the parametric fit for turbine speed-lines, which incorporates point uncertainty. Hence, a basis, such as polynomial, is provided to the regression, but no assumption is made on its order. The Bayesian regression analysis also provides an estimate of total uncertainty in the measured data which can be used to validate the uncertainty analysis of each measurement point. This allows the identification of outliers in the data.

The methodology presented in this paper enables the prediction of turbine performance points with an uncertainty band essential for optimizing future experimental turbine characterization.

## 2 Effect of Efficiency Formulation

*m*is the mass flow,

*h*

_{01}is the enthalpy of the flow at the turbine inlet, and

*h*

_{03ss}is the ideal enthalpy at the turbine exit following an isentropic expansion. Assuming that all the power from the actual expansion process is used to produce shaft power, the mechanical efficiency of a turbine can be defined as

*τ*is the shaft torque and

*ω*is the shaft rotational speed. To simplify the isentropic power calculation, the enthalpy is often assumed to be exclusively dependent on temperature and a calorically perfect gas with constant gas properties. Hence, efficiency could then be expressed as

The error due to this assumption may be further reduced using the *C*_{p} and *γ* values at the thermodynamic condition corresponding to the average value between inlet and outlet conditions, as utilized by Yasa et al. [9]. The impact of these assumptions depends on the inlet condition and pressure ratio of the turbine.

Figure 1 shows the difference in calculated efficiency between the formulations of efficiency found in Eq. (2) versus Eq. (3) for a range of inlet total temperature and total pressure, typical of turbine test rigs, for dry air at three different pressure ratios. The enthalpy state in Eq. (2) is calculated using the National Institute of Standards and Technology (NIST) refprop database [17] and *γ* for Eq. (3) is averaged for the two states. The difference between the approximations and the database for enthalpy increases with the pressure ratio as seen in Fig. 1 left to right. The difference is lower for high temperature and low pressure of the inlet condition where the gas behaves closer to an ideal gas, top left of contours in Fig. 1. This illustrates the importance of assumptions in the formulation of efficiency for rig-level testing.

The mechanical efficiency provides information about the turbine performance in a specific rig or engine test bench. To isolate the gas-path efficiency for comparisons independent of the specific rig environment, it is necessary to understand rig-specific effects like heat transfer and parasitic losses (windage, bearing loss, etc.) on the measured mechanical efficiency and its uncertainty.

*x*is a parameter used to define efficiency, then sensitivity will be written as

The flow enthalpy cannot be directly measured and is a function of the flow total temperature and pressure. Table 1 summarizes the parameter sensitivity of the different measured flow conditions. Each parameter was varied by 1%. The effect in the *η*_{ad} and a parameter sensitivity are reported. The total temperature was changed at three different levels (0.2%, 5%, and 1%), showing that although the enthalpy does not vary linearly with temperature, the change in sensitivity is negligible. The sensitivity analysis indicates that any attempt to improve the uncertainty of the measured efficiency should first focus on improving the measurement of inlet total temperature, followed by the torque, shaft speed, and mass flow.

Parameter | Relative change (%) | Δη_{ad} (%) | Sensitivity |
---|---|---|---|

T_{01} (K) | 0.19 | 0.21 | 1.09 |

T_{01} (K) | 5.00 | 5.25 | 1.05 |

T_{01} (K) | 1.00 | 1.09 | 1.09 |

τ (N m) | 1.00 | 1.00 | 1.00 |

ω (rpm) | 1.00 | 1.00 | 1.00 |

m_{inlet} (kg/s) | 1.00 | 0.95 | 0.95 |

P_{03} (Pa) | 1.00 | 0.50 | 0.50 |

P_{01} (Pa) | 1.00 | 0.42 | 0.42 |

Parameter | Relative change (%) | Δη_{ad} (%) | Sensitivity |
---|---|---|---|

T_{01} (K) | 0.19 | 0.21 | 1.09 |

T_{01} (K) | 5.00 | 5.25 | 1.05 |

T_{01} (K) | 1.00 | 1.09 | 1.09 |

τ (N m) | 1.00 | 1.00 | 1.00 |

ω (rpm) | 1.00 | 1.00 | 1.00 |

m_{inlet} (kg/s) | 1.00 | 0.95 | 0.95 |

P_{03} (Pa) | 1.00 | 0.50 | 0.50 |

P_{01} (Pa) | 1.00 | 0.42 | 0.42 |

The effect of heat transfer on the efficiency is also assessed. In this example, the heat transfer is assumed to be 5% of the shaft power as observed in the literature [18,20–22]. A review of Eq. (5) shows that the total heat transfer and the location of heat transfer are both important. This is corroborated by the sensitivities reported in Table 2, where the greatest sensitivity to heat transfer is observed in stator 1.

Parameter | Relative change (%) | Δη_{ad} (%) | Sensitivity |
---|---|---|---|

Q_{total} (W/m^{2}) | 1.00 | 0.02 | 0.02 |

Q_{stator1} (W/m^{2}) | 1.00 | 0.02 | 0.02 |

Q_{rotor1} (W/m^{2}) | 1.00 | 0.01 | 0.01 |

Q_{stator2} (W/m^{2}) | 1.00 | 0.00 | 0.00 |

Q_{rotor2} (W/m^{2}) | 1.00 | 0.00 | 0.00 |

Parameter | Relative change (%) | Δη_{ad} (%) | Sensitivity |
---|---|---|---|

Q_{total} (W/m^{2}) | 1.00 | 0.02 | 0.02 |

Q_{stator1} (W/m^{2}) | 1.00 | 0.02 | 0.02 |

Q_{rotor1} (W/m^{2}) | 1.00 | 0.01 | 0.01 |

Q_{stator2} (W/m^{2}) | 1.00 | 0.00 | 0.00 |

Q_{rotor2} (W/m^{2}) | 1.00 | 0.00 | 0.00 |

*Z*, which is a function of

*x*and

*y*, the uncertainty in the calculation is given by Eq. (7) [23].

As an illustration, the overall uncertainty in Eq. (3) is assessed using representative measurement uncertainties based on previous campaigns as shown in Refs. [13,24]. The values are shown in Table 3. The uncertainty of each parameter is added to obtain the total uncertainty of efficiency *η* using Eq. (6). It is initially assumed that the specific heat (*C*_{P}) and specific heat ratio (*γ*) are constant and computed based on averages of the turbine inlet and exit conditions. The resultant uncertainty in efficiency is 0.25 as shown in Table 3. Since the specific heat (*C*_{P}) and specific heat ratio (*γ*) are temperature dependent, an uncertainty on those values should be estimated based on the uncertainty in the temperature. This is shown in Table 3, where an uncertainty value for specific heat (*C*_{P}) and specific heat ratio (*γ*) is included, and the total uncertainty on the efficiency increases to 0.33. This is contrasted with the uncertainty assessed from the enthalpy-based equation (2), which does not rely on specific heat (*C*_{P}) and specific heat ratio (*γ*). This is shown in Table 4 where an overall uncertainty of 0.24 is calculated for the same level of uncertainty in the measured parameters.

Parameter | Uncertainty | Units | Sensitivity |
---|---|---|---|

τ | 0.40 | N m | 1.00 |

ω | 0.05 | rad/s | 1.00 |

m | 0.005 | kg/s | 0.998 |

T_{01} | 1.00 | K | 0.99 |

P_{01} | 103.4 | Pa | 0.41 |

P_{03} | 206.8 | Pa | 0.43 |

Δη_{2 overall} | 0.25 | ||

Uncertainty on gas properties | |||

C_{p} | 2.01 | J/kg K | 0.9980 |

γ | 0.001 | — | 1.99 |

Δη_{2 overall} | 0.33 |

Parameter | Uncertainty | Units | Sensitivity |
---|---|---|---|

τ | 0.40 | N m | 1.00 |

ω | 0.05 | rad/s | 1.00 |

m | 0.005 | kg/s | 0.998 |

T_{01} | 1.00 | K | 0.99 |

P_{01} | 103.4 | Pa | 0.41 |

P_{03} | 206.8 | Pa | 0.43 |

Δη_{2 overall} | 0.25 | ||

Uncertainty on gas properties | |||

C_{p} | 2.01 | J/kg K | 0.9980 |

γ | 0.001 | — | 1.99 |

Δη_{2 overall} | 0.33 |

Parameter | Uncertainty | Units | Sensitivity |
---|---|---|---|

τ | 0.40 | N m | 1 |

ω | 0.05 | rad/s | 1 |

m_{inlet} | 0.005 | kg/s | 0.999 |

h_{01} − h_{03ss} | 376 | J/kg | 0.998 |

Δη_{mech overall} | 0.24 |

Parameter | Uncertainty | Units | Sensitivity |
---|---|---|---|

τ | 0.40 | N m | 1 |

ω | 0.05 | rad/s | 1 |

m_{inlet} | 0.005 | kg/s | 0.999 |

h_{01} − h_{03ss} | 376 | J/kg | 0.998 |

Δη_{mech overall} | 0.24 |

The assumption of constant gas properties and incorporation of uncertainty in their values changes the level of total uncertainty. Based on this analysis, to minimize uncertainty, the isentropic power should be computed using enthalpies computed from a reference gas table or another data source rather than based on the assumption of constant specific heat. In Table 4, uncertainty in enthalpy is calculated based on uncertainty in total temperature alone. However, enthalpy is dependent on both total temperature and total pressure. The direction of relative change in total pressures at the inlet and exit conditions affects the uncertainty in total enthalpy. This is shown in Table 5. In case 1, the uncertainty in total pressure is assumed to change in the same direction (added to the mean value) for both the inlet and exit conditions. In this case, the uncertainty in the total enthalpy reduces the total efficiency uncertainty to 0.19 from 0.24 from Table 4. In case 2, the uncertainty in total pressure is changed in different directions in the inlet and exit total pressure (added to the inlet pressure and subtracted to the exit pressure). This increases the uncertainty in total enthalpy, increasing the total uncertainty in efficiency to 0.32 from 0.24. Based on the results of this analysis, it is recommended that the uncertainty in pressure should be incorporated into the uncertainty of total enthalpy.

## 3 Uncertainty

The measurement of a stochastic process gives a distribution with some statistical parameters describing the distribution. For most physical processes, the distribution of noise is Gaussian. Figure 2(a) shows two samples of size *n*_{1} and *n*_{2} with their estimated means *μ*_{n1}(*X*) and *μ*_{n2}(*X*) and a distribution characterized by their estimated standard deviation *σ*_{n1} and *σ*_{n2}. The aim is to characterize the population distribution given by the true mean, *μ*_{true}(*X*), and true standard deviation, *σ*_{true}, shown in black. The true mean and true standard deviation are also referred to as the population mean and population standard deviation. As illustrated in Fig. 2(a), the means of the samples may not be the same as the true mean. The difference between the sample and true mean is called the bias of the measurement. The spread of the sample and population occur due to randomness in the data. Since sampling selects only some part of the population, the mean of the sample may not be the same as the population mean as shown in Fig. 2(b). Because the samples are a finite set of data, the mean value is just an estimation with its own distribution as seen for samples *n*_{1} and *n*_{2}. The mean value of the sample has a distribution characterized by the standard error of the mean, *σ*_{mu}. Increasing the number of samples will decrease the error on the estimation of the sample mean, but any error due to a fixed bias will remain.

Performance metrics, such as efficiency, are derived rather than directly measured quantities. However, since they depend on random variables with a Gaussian distribution, they also follow a Gaussian distribution, and the calculation process can be treated as a sampling process.

### 3.1 Effect of Signal Acquisition and Window.

*V*is the signal amplitude and

*N*(0, 1) is a normal distribution of mean 0 and standard deviation 1. The true signal has a frequency of 1 kHz. This signal is then sampled at different frequencies, namely, 1, 10, 25, 50, 100, 150, and 200 kHz. For each sampling frequency, different window sizes are used consisting of 10,000, 25,000, 50,000, 100,000, and 200,000 sample sizes. The effects of this are shown in Fig. 3. Figure 3(a) shows the error in mean value compared to the true signal. This constitutes the bias error. It is seen that the bias remains fixed for most of the range of sampling frequency and the magnitude increases below 0.1 of the true frequency. Similar trends are observed in Fig. 3(d), where the error deviates only at the lowest frequency and remains unchanged with sample size at higher frequency. Figure 3(b) shows that the standard deviation of the signal is dependent on the sampling frequency when the sampling frequency is lower than the true frequency. It is weakly dependent when the signal is sampled faster than the true frequency. This highlights why standard deviation alone should not be used a metric of uncertainty. For the same system, it is dependent on the data acquisition settings. The standard deviation is largely independent of the sample size which is also seen in Fig. 3(e) where the standard deviation is almost constant with sample size and is dependent only on the sampling frequency. The standard error of the mean (Fig. 3(c)) remains independent between 0.01 times the true frequency and the true mean, increasing below 0.01 times the frequency and decreasing when sampled above the true frequency. The standard error of the mean decreases with sample size as seen in Fig. 3(c), except when sampling frequency is 0.01. Figure 3(f) shows that the standard error on the mean decreases with increasing sample size proportional to $1/(n)$ but differs for each sampling frequency since the standard deviation is dependent on the sampling frequency.

An appropriate data window size and sampling rate are required to obtain good statistical data estimates. An absolute minimum on the window size comes from the time it takes for information to travel from the inlet to the exit of the turbine. Some information such as pressure fluctuations travel at the speed of the sound, giving a sonic time response of the turbine. This is generally much smaller than the advective time response, the time in which advected properties such as temperature travel. Hence, the data window should be based on the advection time response. For the turbine analyzed in this paper, the time responses are 0.55 ms and 2.3 ms for sonic and advection time response, respectively.

### 3.2 Uncertainty Sources.

Uncertainty can be divided into two sources: *stochastic*, the random variation from one measurement to the next, and *bias*, which is a systematic variation. Bias sources can be corrected, but there is uncertainty in the correction, designated the bias uncertainty.

*t*-statistic. The raw measurements have several bias errors that need to be evaluated using other sources than the collected data. These sources have been studied in the literature. The bias errors correspond to calibration errors for the probes [20], thermocouple corrections due to the effect of heat transfer [25,26], corrections for the effects of spatial and data averaging technique [13,27,28], and the effects of probe-turbine coupling [13,29]. These corrections often rely on numerical methods and are also prone to numerical uncertainty, a source of stochastic uncertainty. Correcting for the bias, a corrected mechanical efficiency is obtained. The total standard error is computed based on the ASME standard of combining uncertainty as

*B*is the bias uncertainty. The corrected mechanical efficiency is still a performance metric of the test rig where the data are collected. Compensating for the effect of shaft mechanical loss and heat loss yields the adiabatic efficiency, which can be compared across test rigs and provides an easy boundary condition for CFD. These corrections bring their own sources of bias and stochastic uncertainty depending on the methodology used for the corrections. This general outline can be modified as needed. For example, the shaft mechanical loss can be added into the corrected mechanical efficiency, providing the blade path efficiency with heat loss. This outline and identification of uncertainty sources is important to present with performance data to ensure that efficiency from different sources is comparable.

### 3.3 Uncertainty Propagation.

There are many sources of uncertainty that contribute to the overall uncertainty in efficiency. To accurately capture the overall uncertainty, individual uncertainties need to be propagated through the calculation. This can be challenging at times. For example, the Taylor series approximation shown in Eq. (7) has limitations when the function derivatives are not easily available. This is observed when numerical-based corrections are incorporated to account for measurement bias. An additional common example is observed when the derivatives for gas properties are not constant and depend on the operating conditions of the test rig [17].

Since analytical estimation of the sensitivity to each parameter is cumbersome to calculate for numerical corrections and gas properties, a numerical scheme is proposed to propagate uncertainty instead of Eq. (4). Linear independence of the parameters is assumed. Since the flow conditions are related to each other and the output of the shaft and the losses are related, this assumption will always provide a conservative estimate of the uncertainty.

*I*

_{n}, computed from

*n*samples, for a function

*f*of random variable

*X*, approaches the mean or expected value of function

*f*shown as

To demonstrate this, samples of each parameter used in Eq. (2) are taken from distributions with the mean value as some representative number and the uncertainty level as shown in Table 3. The mean mechanical efficiency is calculated for each sample. Figure 5(a) shows the estimate of mean mechanical efficiency calculated from Eq. (2) with increasing number of samples based on Eq. (9). It is seen that the mean efficiency estimate remains unchanged after 2000 samples. Addition of more samples does not improve on the estimate, hence the summation of Eq. (9) approaches the population mean. The variance on the estimated mean is also stabilized as seen in Fig. 5(b). The estimated mean efficiency (*η*_{ad}) stabilized with a standard error on the mean of ±0.01 computed from Eq. (2). The efficiency spread has a standard deviation of 0.273%, higher than that obtained from the analytical formulation in Table 3. The varying derivatives of the parameters are considered with the numerical sampling.

### 3.4 Uncertainty Through Data With Covariance.

An inherent assumption in uncertainty propagation is that all parameters are independent. That assumption is not always true. For a turbine, the relationships between parameters are governed by the design of the turbine. If information about the turbine is available, this can be treated with a covariance between measurement parameters, as developed by Hudson and Coleman [12]. This concept can be illustrated with a turbine map based on equivalent parameters, as defined in Appendix, Eqs. (A1)–(A8).

In this scenario, to determine an operating point, some of the variables must be defined (the input or control variables). Then, using a given turbine map, relationships can be derived for the output parameters. The input and output variables can change depending on how the turbine is operated but their relationships are governed by the turbine map.

As an illustration, during a test at the Purdue Experimental Turbine Aerothermal Lab (PETAL), the mass flow (*m*), inlet total temperature (*T*_{01}), and shaft speed (*N*) are typically independently controlled. To run a test, a target point is selected on the turbine map by selecting a pressure ratio and an equivalent speed. In addition, an inlet total temperature is selected. Once those input parameters have all been determined, the turbine map then defines the mechanical speed of the shaft, the equivalent mass flow, the torque, and inlet and exit total pressures (output parameters).

The dependence of the output parameters on the turbine map allows the uncertainty of the output parameters to be based solely on the Gaussian uncertainty of the input parameters. The derived output parameters will still have a Gaussian uncertainty distribution. This is shown in Fig. 6, where the uncertainty in the input parameters, mass flow (*m*), inlet total temperature (*T*_{01}), and shaft speed (*N*) (in top row), creates a distribution of the derived parameters, input and exit total pressure and shaft torque (in bottom row). The turbine map is generated using Rolls-Royce proprietary tool Q70. Q70 is a mean-line solver with a loss model in the family of Ainley and Mathieson, Dunham and Came, and Kacker and Okapuu [32–34]. It has proprietary extensions for size effects, tip clearance, Reynolds number, and cooling flow mixing losses. It also has a proprietary solver that is robust for turbines with multiple, choked blade rows.

In an experiment, the derived parameters will have their own measurement noise apart from those induced by the uncertainty in the input parameters. This is illustrated in Fig. 7, where the left image shows measured mechanical efficiency and uncertainty between a baseline and incrementally improved turbine where only input noise and covariance are considered. Figure 7, right, shows the same measurement with the addition of measurement noise in the derived parameters along with the input noise and covariance in the parameters. In this example, the mean difference of the baseline and improved efficiency turbines are the same but the output measurement uncertainty makes it much harder to quantify the difference in actual performance.

Figure 8 further illustrates the impact of considering covariance in the data over the entire turbine map. This figure shows a comparison of uncertainty level for efficiency calculation for a given turbine map with noise only in the input parameters and noise in both input and derived parameters. Inclusion of derived parameter uncertainty in the data results in both an overall increase in uncertainty level and different trends in the equivalent torque uncertainty. With input noise alone (Fig. 8(a)), the uncertainty in equivalent torque is uniform across the turbine map. In contrast, with noise in the input and derived parameters (Fig. 8(b)), the uncertainty has significant gradients with equivalent torque and the pressure ratio.

Table 6 shows a comparison of uncertainty calculated through the different methods described above by assuming constant specific heat through the Taylor series approximation (Sec. 2), by sampling with assumed independent parameters (Sec. 3.3), by sampling with noise only in input parameters and covariance, and by sampling with noise in both input and output parameters (Sec. 3.4).

Independent sampling of the data gives a higher level of uncertainty than compared to the Taylor series approximation due to the assumptions on the specific heat made for the latter. The addition of information from the turbine map and reliance on covariance significantly reduces the variation in the derived parameters. This leads to a tighter spread in the efficiency estimation. This increases when measurement uncertainty is added to the derived quantity. The uncertainty is still lower than that obtained assuming independent parameters, hence this gives a more conservative estimate of uncertainty. When applicable, it is recommended that covariance of the data be applied to reduce the level of uncertainty on the efficiency mean.

As has been shown already, covariance between measurement parameters can be a powerful ally in the reduction of overall uncertainty. There are also additional benefits to data covariance relating to the selection of input variables for a test. Prior to testing in a rig environment when an experimental representation of the turbine is not yet available, a preliminary representation of the turbine can be generated using a mean-line solver. A mean-line-based turbine map provides the experimentalist a tool for designing experiments to minimize overall uncertainty.

For example, Figs. 9(a) and 9(b) show the expected value of efficiency and uncertainty for a given operating point based on a mean-line solution using the Rolls-Royce proprietary tool Q70. The efficiency and uncertainty are calculated for a specific operating condition on the turbine map (constant pressure ratio, equivalent speed, and equivalent mass flow) but at different inlet total temperatures. The precision uncertainty is assumed to remain the same for all three cases. A comparison of Figs. 9(a) and 9(b) illustrates that due to higher inlet temperature, the mechanical efficiency decreases due to higher heat losses. However, the uncertainty spread for the mean mechanical efficiency has decreased since the precision uncertainty does not change with temperature. Repeating this analysis for different operating conditions allows the experimentalist to target conditions which have a favorable balance of target accuracy and uncertainty.

This can also be used to assess the change in measured mechanical performance when two designs of different adiabatic performance are compared at the same operating point. The difference in the adiabatic efficiency and measured mechanical efficiency may not be the same due to heat transfer effects and rig parasitic losses (bearing and disc windage). The change in mechanical efficiency between baseline and design with 1% and 0.5% adiabatic efficiency improvement is shown in Figs. 9(c) and 9(d), respectively. The improvement in mechanical efficiency is lower than the improvement in adiabatic efficiency. Mechanical efficiency increases only by 0.98% (Fig. 9(c)) and 0.49% (Fig. 9(d)). This highlights that due to measurement uncertainty and heat loss, the data will show a smaller increase in mechanical performance than what is present in the adiabatic performance.

### 3.5 Turbine Map Uncertainty.

*m*:

*R*

^{d}→

*R*is the mean function and

*k*:

*R*

^{d}

*XR*

^{d}→

*R*is the covariance function. Therefore, with some data

*D*= (

*x*

_{1:n},

*y*

_{1:n}), where

*y*

_{i}is corrupted with some Gaussian noise defined by standard deviation

*σ*

^{2}, it can be written as

*k*(.,.), is defined by some hyperparameters,

*θ*and the Gaussian noise or likelihood variance,

*σ*

^{2}. The posterior of the hyperparameters is then

The posterior is solved through maximum a posteriori (MAP) estimate of the hyperparameters which is based on the optimization of log(*p*(*θ*, *σ*|*D*)) [35]. This allows for the finding of the required hyperparameters of the basis function based on the data alone, and it also allows quantification of measurement noise. This can be compared against the expected measurement noise to see if another source of noise was missed during the quantification process.

A turbine map is generated using the Rolls-Royce proprietary tool Q70 for the different operating conditions shown as the data points in Figs. 10(a) and 10(b) for a state-of-the-art Rolls-Royce high-pressure turbine. A targeted level of measurement uncertainty, shown in Table 3, is added to the mean value from the solver and the speed-lines are constructed for the normalized torque in Fig. 10(a). For mechanical efficiency, a simple model is used which imposes a 5% of shaft power lost as heat transfer typical of rig testing [20,36,37]. The uncertainty in each parameter is propagated through sampling as presented in Sec. 3.3 and the mechanical efficiency is quantified along with an overall uncertainty level on the mean. This is used in Fig. 10(b) along with the credible interval of the mean and the predictive interval from Eqs. (15) and (16), respectively.

### 3.6 Speed-Line Prediction.

Regression fits are generally carried out for speed-lines, but regression fits can also be used to estimate trends between speed-lines. A good regression fit can also reduce the number of speed-lines that need to be measured to obtain the full turbine map. Speed-lines are generally constructed with a single independent variable. In addition, the information present with multiple speed-lines can be used to define a function to interpolate between speed-lines. This has two purposes, first it reduces significantly the number of tests required to characterize a turbine map. Second, it allows trend verification of the data if the measured speed-line is similar to the predicted based on turbine performance across other speed-lines. To demonstrate all the data from Sec. 3.5 except the speed-line at 100%, normalized equivalent speed is considered, shown as the second line from the bottom in Fig. 10, and a multivariate GP model is constructed. The method works similarly to the GP model described in Sec. 3.5. Figure 11 shows the contour prediction from the GP model for (a) efficiency and (b) normalized equivalent torque along with the data used.

The obtained model is then used to predict performance values in between the speed-lines. Figure 12(a) shows the prediction of equivalent torque at the 100% speed-line versus the numerical data, and Fig. 12(b) shows the prediction of efficiency of the data and the model for an individual speed-line. It is seen that there is a difference in the predictions between the speed-line regression of Sec. 3.5 and the multiple speed-line regression of Sec. 3.6 but both fall within the uncertainty of the speed-line model.

## 4 Experimental Demonstration

The methods presented in this paper are applied to efficiency assessments performed in Purdue's Small Turbine for Aerothermal Rotating Rig (STARR). Figure 13 shows the schematic of the pressure-driven PETAL facility. Air from pressurized tanks can be passed through a heater, bypass the heater, or a fraction of the air can be passed through the heater and then mixed with air that bypassed the heater to provide a range of uniform temperature flow. A settling chamber conditions the flow and a sonic valve downstream sets the back pressure. Air can be passed into the test section or purged to the ambient through a series of fast actuation valves. The STARR two-stage turbine rig was designed to support advanced small core turbine technology development, testing components at engine-representative conditions. Testing in the STARR rig is akin to NASA technology readiness level (TRL) 5–6 [38] corresponding to large-scale rig testing to full-scale system demonstration [39]. It incorporates an extensive instrumentation suite. The rig was designed to be extremely modular, allowing the testing of multiple configurations, tip clearances, etc. The turbine is set to a given speed using a dyno system before air is introduced. Once air is introduced, the dyno system then absorbs the generated power. Figure 14(a) shows the rotating test section installed in the facility and the dyno motor. Efficiency data are taken over the operation map of the turbine including idle, performance, and overspeed conditions. The facility provides engine-representative test conditions for a high-pressure turbine, with a pressure ratio ranging from 2 to 6 and shaft speed of up to 21,000 rpm. Engine Mach and Reynold's number are matched along with all the equivalent parameters, Eqs. (A1)–(A8), at testing temperatures of up to 600 K.

Figure 14(b) contains the cross section of the rotating turbine test section (flow travels from left to right). There are 2 × 4 head total pressure and total temperature rakes at the inlet and exit of the turbine. A sonic Venturi upstream of the turbine provides mass flow measurement and shaft torque and speed are measured using a torquemeter and photodiode. Three 2 head five-hole probes at the exit are used for radial and circumferential traverses. Resistance temperature detectors embedded in the endwalls are used for estimation of heat flux. The details of the rig are described in Ref. [39].

### 4.1 Single Point Estimation.

Table 7 shows the relative uncertainty of each parameter used for the calculation of mechanical efficiency based on Eq. (2) for a single operational condition in the STARR rig. As mentioned, the torque and shaft speed are measured with a torquementer and a photodiode system mounted on the shaft. The inlet total temperature and pressure are area averaged using four-head thermocouple and Kiel probes at the rig inlet. The mass flow is measured using a calibrated sonic Venturi upstream of the test rig, similar to Refs. [13,40,41]. The exit total pressure is averaged using a traverse of 2 × 4 head Kiel probes. It is seen that the precision/stochastic source of uncertainty is lower than that coming from bias sources. This is because the rig is able to hold stable conditions where the test data are taken for several minutes (∼7 min for a full area traverse). The bias sources arise from calibration uncertainty of the sensors. Bias sources also include thermocouple conduction and velocity errors, these were corrected in the methodology presented in Ref. [42], and area averaging effects due to spatial non-uniformity of conditions at the exit of the turbine for the exit total pressure [13]. The heat flux for the calculation of adiabatic efficiency was estimated using wall temperature readings of the endwalls coupled with CFD; the method is documented in Ref. [42]. The method has high uncertainty of around 18% for the full two-stage turbine. The uncertainty results for a single design point estimate are tabulated in Table 8 for a given operating point. In this real-world example, the uncertainty on mechanical efficiency $\sigma \eta q$ is 0.07% and is representative of the precision of the measurement. Adding sources of bias correction on the temperature measurement and the total pressure increases the uncertainty to 1.28%. The bias corrected mechanical efficiency is 0.07% higher than the mechanical efficiency. The data were taken once conditions inside the turbine had achieved thermal stability, and the correction for the total temperature is small. However, the correction methodology combines numerical and experimental sources, and hence, the uncertainty of the method is high. This leads to an increase in the uncertainty of efficiency to 1.28%. Adding estimation of heat and uncertainty leads to an estimation of adiabatic efficiency 2.36% higher than the mechanical efficiency with an increase in uncertainty to 1.41%. Using the pre-test predicted turbine map obtained using the proprietary Rolls-Royce mean-line solver Q70 and correcting for the measured running tip clearance, the mean prediction is 0.18% lower than the mechanical efficiency with an uncertainty of 0.73%. In this example, the noise in both input and derived parameters was calculated from the method of Sec. 3.4. Figure 15 shows a visual representation of the results, with respect to the measured mechanical efficiency. The 95% confidence interval on mechanical efficiency is presented first on the left, the corrected mechanical efficiency next, and the adiabatic efficiency next to it. The rightmost efficiency is the pre-test prediction from the Q70 solver with input and derived parameter noise, as presented in Sec. 3.4. This analysis highlights that to obtain better levels of uncertainty in the efficiency calculation, correction methodology for total temperature and heat load measurement need to be improved.

Parameters | Relative uncertainty (%) | ||
---|---|---|---|

Precision | Bias | Total | |

ω | 0.000 | 0.010 | 0.010 |

τ | 0.009 | 0.501 | 0.502 |

m | 0.006 | 0.250 | 0.250 |

T_{01} | 0.009 | 0.546 | 0.546 |

P_{01} | 0.001 | 0.499 | 0.499 |

P_{03} | 0.010 | 2.505 | 2.505 |

Parameters | Relative uncertainty (%) | ||
---|---|---|---|

Precision | Bias | Total | |

ω | 0.000 | 0.010 | 0.010 |

τ | 0.009 | 0.501 | 0.502 |

m | 0.006 | 0.250 | 0.250 |

T_{01} | 0.009 | 0.546 | 0.546 |

P_{01} | 0.001 | 0.499 | 0.499 |

P_{03} | 0.010 | 2.505 | 2.505 |

Efficiency | Mean value with respect to mechanical efficiency | Uncertainty (%) |
---|---|---|

η_{mech} | 0 | 0.07 |

$\eta mechcorrected$ | 0.07 | 1.28 |

η_{adiabatic} | 2.43 | 1.41 |

η_{control+measurement} | −0.18 | 0.73 |

Efficiency | Mean value with respect to mechanical efficiency | Uncertainty (%) |
---|---|---|

η_{mech} | 0 | 0.07 |

$\eta mechcorrected$ | 0.07 | 1.28 |

η_{adiabatic} | 2.43 | 1.41 |

η_{control+measurement} | −0.18 | 0.73 |

### 4.2 Turbine Map Characterization.

Experimental data were taken at four equivalent speeds to make the turbine performance map for equivalent torque. The uncertainty on each measured performance point was characterized using sampling for propagation as shown in Sec. 3. The data were fit using a GP regression, using the methodology presented in Sec. 3.6 for each equivalent speed to give four speed-lines. The measurement uncertainty quantified is imposed as the likelihood variance and is shown in Fig. 16(a) along with the credible and predictive interval for the equivalent torque. The imposed noise compares well against the MAP estimates for the data except for the 80% normalized equivalent speed-line shown in Figs. 16(b) and 16(c), where the MAP noise estimate is 0.846 compared to the imposed 0.118 obtained from accounting for uncertainty sources from Sec. 3.2. This occurs due to the data taken at a normalized pressure ratio of 0.4, which do not seem to lie in the same family as the rest of the data.

Using the methodology of Sec. 3.6, a multi speed-line GP regression is carried out with the same uncertainty levels used for Fig. 16. This regression uses data from all speed-lines except at 100% normalized equivalent speed. Then predictions for both equivalent torque and mechanical efficiency are made for the 100% speed-line compared to the measured data, as shown in Figs. 17(a) and 17(b), respectively. It is seen that the predicted values for both equivalent torque and mechanical efficiency lie within the predictive interval of the single speed-line model, though outside the credible interval at lower pressure ratio. This shows that the presented regression can be utilized to reduce the number of points measured to construct the turbine map and that performance can be predicted in between measured speed-lines within the predictive interval of the measurement.

## 5 Conclusion

A framework for overall uncertainty propagation is presented for turbine efficiency. Assumptions on the gas properties lead to an error in the efficiency value calculated. The error depends on inlet conditions and can be higher than 5% at inlet pressures above 5 bar, which is closer to engine conditions. The treatment of gas properties also affects how uncertainty is propagated in the calculation and can lead to a 0.8% overprediction of uncertainty. Hence, isentropic power should be calculated using total enthalpy drop based on the inlet and exit pressure and not making assumptions on the specific heat capacity (*C*_{p}) and specific heat ratio (*γ*) as is common in the literature [8,9]. Parametric sensitivity identifies inlet total temperature, inlet mass flow, and the exit total pressure as the aerothermal parameters which have the highest impact on efficiency determination. Uncertainty propagation through sampling is proposed to consider the effect of change in gas properties, enabling consideration of non-linear relationships between gas enthalpy and entropy with total temperature and total pressure and incorporation of numerical uncertainty sources rising from numerical approaches to correct measurements.

An approach to account for correlation between parameters is developed that allows turbine performance prediction using data from a numerically generated turbine map. The model also allows propagation of uncertainty in each parameter. This allows the study of parameter interaction and not treat them independently as done in the literature [6,8–10,18]. Using this model, predictions on the turbine efficiency and its uncertainty are made using control parameters. This provides an estimate on how controlled a given experiment is, and by adding in measurement noise, experimentally measured efficiency and uncertainty are estimated. This is used to validate experimentally measured data, and the comparison between model predictions and actual measurements allow for the identification of deficits in the numerical solvers. Gaussian process regression is used to make speed-lines and turbine map for equivalent torque and efficiency. This regression allows a more generalized fit of the data trend and not force it to fit a specific polynomial. These regression models are used to predict performance at interpolated operating points and estimate the uncertainty.

These methods are demonstrated in a TRL 6 [38] rotating two-stage turbine test section at engine-representative conditions. Uncertainty in a single point measurement is estimated through numerical sampling. It is seen that the 95% confidence interval uncertainty on the mechanical measurement is 0.26%. However, when the mechanical measurements are corrected for bias sources, the uncertainty increases to 1.2%. The corrected efficiency is 0.05% higher than the mechanical. To reduce uncertainty, the correction methodology for total temperature needs to be improved. Correcting heat loss, the adiabatic efficiency is estimated to be 2.41% higher than the mechanical efficiency, with an increased uncertainty of 1.35%. This highlights the importance of improving methods to correct and accurately measure heat loss. The effect of each parameter is assessed for each performance point, and a GP model is built for the turbine map, which is used to predict turbine performance at different operating conditions. It is seen that a few performance points were outside of the MAP estimate based on the uncertainty from the point estimates. The multivariate GP model can predict turbine performance which lie within the MAP estimate of the experimental data. The multivariate GP model is also able to identify potential outliers in the data and is a useful tool for data validation.

## Acknowledgment

The authors would like to acknowledge our sponsors Rolls-Royce Corporation for their financial support. They will also like to acknowledge the efforts of Dr. Papa Aye N. Aye-Addo, Dr. David Liliedahl, Mr. Antonio Castillo, Maj. John Paulson, Lt. Connor McQueen, Mr. Michael Butzen. and Mr. Diego Sanchez for their help during the preparation and execution of the experimental campaign.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*f*=function

*k*=covariance function

*m*=mass flow

*n*=number of samples

*p*=probability function

*q*=heat load

*t*=student-

*t*distribution factor*x*=input data

*y*=output data

*B*=bias uncertainty

*D*=data

*K*=covariance matrix

*N*=shaft speed

*V*=signal amplitude

**X**=random variable

*h*_{0}=total enthalpy

*C*_{p}=specific heat capacity

*P*_{0}=total pressure

*T*_{0}=total temperature

*mf*=mean function

*ss*=property assuming isentropic process

- true =
population or true statistic

*E*() =expected value/mean

*N*(*μ*,*σ*) =Gaussian distribution

*γ*=specific heat ratio

- Δ =
total uncertainty

*η*=efficiency

*η*_{ad}=adiabatic efficiency

*θ*=hyperparameters

*μ*=mean

*σ*=standard deviation

- $\sigma \mu $ =
standard error of the mean

*τ*=torque

*ω*=angular velocity

- 1 =
property at turbine inlet

- 2 =
property at rotor inlet

- 3 =
property at turbine exit

### Acronyms

- CFD =
computational fluid dynamics

- CI =
confidence interval

- GP =
Gaussian process

- MAP =
maximum a posteriori

- PDF =
probability distribution function

- PI =
predictive interval

- rpm =
revolution per minute

- RANS =
Reynolds averaged Naiver–Stokes

- TRL =
technology readiness level

- URANS =
unsteady Reynolds averaged Naiver–Stokes

## Appendix: Equivalent Parameters

*γ*) with the change in total temperature

*T*

_{0}. This is shown in Eq. (A1)

*V*

_{cr}and $\epsilon $ are used to consider change in gas specific heat ratio (

*γ*).The parameter $\epsilon $ is a reference to the actual specific heat ratio (

*γ*) against a reference specific heat ratio (

*γ*

_{reference}) at standard air temperature and pressure (SATP). The corrected temperature ratio is now changed to Eq. (A3)