A New Approach to Modeling Slip and Work Input for Centrifugal Compressors

The flow phenomena that cause slip in centrifugal compressors are qualitatively well understood due to extensive research and analysis over the past century. The theory of the relative eddy, which is the foundation of the vast majority of slip models available in the open literature, was originally presented by Stodola [1] in the 1920s. For the case of two-dimensional, steady, potential flow, there is no mechanism to generate or destroy vorticity. Stodola utilized the potential flow perspective to propose the theory of the relative eddy. Since vorticity can neither be created nor destroyed in potential flow fields, a secondary flow develops in the impeller passages with an angular velocity equal and opposite to the angular velocity of the impeller reference frame. The secondary flow, termed the relative eddy, maintains the condition of irrotationality relative to the inertial reference frame. When superimposed on the meridional bulk flow at the impeller exit, the relative eddy causes a reduction in the exit tangential flow velocity, i.e., slip. Stodola's original discussion was intentionally simplified and qualitative in nature, but numerous researchers later published exact solutions for the potential flow field in impeller passages in which slip was confirmed to be an inviscid phenomenon [2–7].

The reduction in blade loading at the trailing edge is an alternate and equivalent approach often used to explain the phenomenon of slip. Like in axial blades, the blade loading must decrease to zero at the impeller trailing edge to satisfy the Kutta–Joukowski condition. The gradual decrease in blade loading corresponds to a reduction in the pitchwise pressure gradient within the passage, from which a cross-passage velocity opposed to the direction of impeller rotation arises [8,9].

Whether attributed to the relative eddy or relaxation of the cross-passage pressure gradient, the secondary flow within the passage is the root cause of the difference between the actual velocity triangle and the ideal velocity triangle associated with the perfect flow guidance (PFG) condition, Fig. 1. The perfect flow guidance condition is representative of ideal operation where the exit relative flow angle, $β2$⁠, is identical to the impeller exit blade angle, $β2b$⁠, such that the difference in the impeller exit tip velocity, $U2$⁠, and the intended tangential velocity reduction due to backsweep, $Vm2tanβ2b$⁠, is equal to the impeller exit tangential velocity, $Vθ2,PFG$⁠. The perfect flow guidance exit tangential flow velocity corresponds to the maximum possible work input that can be delivered by an impeller at a given operating condition. In actual operation, however, the secondary flow in the passage diminishes the exit tangential velocity by the slip velocity, $Vslip$⁠, and the actual work delivered by the impeller is reduced from the maximum. While the reduction in tangential flow velocity due to backsweep is a known part of the design incorporated for improved stability and efficiency, the slip velocity is unknown.

Reliable prediction of slip is vital to success in the preliminary design of centrifugal compressor impellers, but despite a well-established qualitative understanding of the phenomenon, accurate quantification of slip during design remains difficult. The many models available in the open literature are inconsistent in their ability to predict slip: they may provide good estimates of slip for one impeller family and give large errors for another. Due to the unreliability of models available in the open literature, some organizations have developed correlations for specific impeller families instead of relying on general slip models [10].

Since the publication of Wiesner's model, efforts to develop a reliable method for the prediction of slip factor have continued. Proposals have included experimental curve fits [13,15], potential flow solutions [4,6,16], and multizone techniques [17,18]. The single relative eddy model was proposed by von Backström [14], which was derived by applying the relative eddy to the impeller as a whole rather than to each passage. The single relative eddy method provided a better approximation of Busemann's solution than Wiesner's correlation, though its accuracy in the prediction of slip factor was similar to that of Wiesner's model.

Until recently, most slip models were based only on compressor geometry despite documentation that the slip factor varies with compressor operating conditions [8,11,17,19,20]. Qiu et al. [9] addressed the variation of slip with flow conditions by incorporating the impeller exit flow coefficient into a relative eddy approach. Stuart et al. [18] expanded Qiu's approach to a multizone technique in an effort to improve the off-design utility of the model.

Despite continued investment by the research community, reliable prediction of the slip factor at neither design nor off-design has been attained. Harrison et al. [20] evaluated the efficacy of Wiesner's, von Backström's, and Qiu's models for predicting the slip factor and work input of four high-speed, open literature compressors at design and off-design conditions. Each model overpredicted the slip factor of the high-speed impellers, with the largest overpredictions occurring at design speed. The overprediction in slip factor resulted in a corresponding overprediction in work input scaled by the square of the impeller machine Mach number. Thus, modest overpredictions in slip factor propagated to large errors in work input at or near design speed for high-speed impellers.

Stodola's [1] original method used geometry to predict the size of the relative eddy, and by extension, the slip factor associated with a given impeller design. Most models available in the literature have followed Stodola's approach and primarily rely on the impeller geometry to predict the behavior of a highly unsteady, three-dimensional flow field at the impeller exit [2,9,11,12,14,21]. Due to errors and inconsistencies in predicting the behavior of a complex flow field chiefly from geometry, as well as limited success in predicting slip at off-design conditions, there is significant room for improvement in the prediction of slip factor. This paper seeks to address that challenge by proposing a model that utilizes thermodynamic principles in conjunction with the meanline equations, rather than flow modeling, to predict work input and slip factor at design and off-design conditions.

Equation (7) is mathematically identical to Eq. (5), but the terms have been manipulated to give the actual total temperature rise ratio delivered by the impeller as a function of the perfect flow guidance loading coefficient as well as the ratio of the actual loading coefficient to the perfect flow guidance loading coefficient. While the actual loading coefficient is unknown due to its dependence on the slip factor, at the perfect flow guidance condition the slip factor is, by definition, equal to one. Therefore, the perfect flow guidance loading coefficient can be directly calculated from the meanline equations.

Since Eq. (7) is a mathematically valid relationship, the term that must be modeled, the ratio of loading coefficients, is not required to be the primary driver of the model. Instead, the machine Mach number and loading coefficient inherently approximate both the magnitude and trend of the total temperature rise ratio, and modeling of the ratio of loading coefficients serves as a correction for magnitude.

where $m˙$ is the mass flow rate and $ρ0$$D2$ is the impeller exit diameter. From Eq. (9), the dimensionless parameters incorporate a measure of the impeller shape factor through the radius ratio, which has often been observed to influence slip factor [5,12]. Additionally, the impeller exit tip speed and compressibility at the impeller inlet contribute to the correction from perfect flow guidance conditions to actual conditions. Other combinations of dimensionless parameters also provided satisfactory correlations, but together, the square of the machine Mach number and flow coefficient provided the best combination of minimal scatter throughout the dataset and low error at high machine Mach numbers (i.e., at or near design speed).

The ratio of specific heats is assumed to be constant and is absorbed into the coefficient $A$⁠. The ratio of loading coefficients, $χ$⁠, is not directly present since it is modeled by the product of the inlet flow coefficient and square of the machine Mach number raised to the power of coefficient $B$⁠. The coefficients $A$ and $B$ are determined from a nonlinear regression analysis and, as is discussed in detail in Secs. 3–6, may be prescribed per the recommended general values, customized to fit specific impeller families, or tailored to fit the implementation of the model. Detailed procedures for the regression analysis, incorporation of the model into meanline analysis tools, and calculation of the slip factor itself are given in the appendices.

A generalized meanline model for centrifugal compressors was developed in the Object-oriented Turbomachinery Analysis Code (OTAC) environment of the Numerical Propulsion System Simulation cycle analysis tool [22–24]. The OTAC environment was developed with the intention of functioning as an all-in-one design code to support analysis of both axial and radial compressors and turbines at design and off-design. This versatility was utilized to quickly switch between various compressor geometries, operating conditions, loss models, and slip models. OTAC was used to model four well-documented centrifugal compressor cases available in the open literature: Krain's transonic impeller stage (SRV2-O), NASA's low specific speed impeller (CC3), Came's transonic backswept impeller (Came stage B), and Eckardt's backswept impeller (Eckardt impeller A). Additionally, the Honeywell test Single Stage Centrifugal Compressor (SSCC) facility at Purdue University was utilized to provide an additional high-speed dataset. The primary design point performance and geometric parameters of each stage are given in Table 1. Detailed documentation for each of the stages can be found in Refs. [25–36].

The work input provided by the impeller was predicted through either the new slip model or Wiesner's slip model [12]. Wiesner's model is one of the most widely used models in the open literature and as such was included as a benchmark for evaluation of the new model. Additionally, Wiesner's model has been shown to provide better estimates of slip factor for high-speed compressors than some other models in the literature [20].

The stage efficiency is traditionally predicted by incorporation of loss models into the meanline equations, but loss models often result in misrepresentation of the impeller efficiency. To avoid introducing any confounding errors as a result of misprediction of efficiency, the experimentally measured stage efficiency was prescribed at each operating point. Each of the open literature cases was outfitted with a vaneless diffuser, and the experimentally prescribed stage loss was divided between the vaneless diffuser loss, calculated according to Stanitz's vaneless diffuser model [37], and the impeller. The Honeywell SSCC featured a vaned diffuser, but calculated performance parameters were available at the impeller exit. As such, the same procedure of efficiency matching was repeated for the SSCC but without the need for a diffuser loss model. Based on a sensitivity study of the slip factor to external loss presented by Harrison et al. [20], 80% of the impeller loss at each operating point was allocated to internal loss mechanisms. The remaining 20% of the impeller loss was allocated to external loss sources unless otherwise stated.

Since a directly measured value for slip factor was not available for the SRV2-O, CC3, Came B, or SSCC impellers, OTAC was used to calculate an experimental slip factor from measured stage performance data. The total pressure ratio and efficiency data were used to determine the total temperature rise of each operating point. The experimental total temperature rise was provided as an input to OTAC, and the losses for each operating point were distributed as previously described. The slip velocity necessary to match the experimental work input was used to calculate the experimental slip factor for each data point of the high-speed stages. Although measured values of slip factor were available for Eckardt's impeller, the same procedure was applied to Eckardt A for consistency and validation. The calculated slip factor matched the experimental slip factor reported in Ref. [25] with an error of less than 0.02 in slip factor. For reference, an absolute error of ±0.05 in slip factor has traditionally been considered acceptable in evaluation of slip models.

As a final note, choke occurred in the inducer of the impeller for the vaneless diffuser configurations under investigation. To avoid any confounding effects associated with inducer choke, all points showing evidence of performance decrements due to choke were neglected in the proceeding analysis.

The new model supports various methods of implementation, and thus can be customized to fit the needs of the user. The general and family implementations of the model are considered first, indicated in figures by the GEN and FAM subscripts, respectively. Each implementation uses the definition of the model given in Eq. (11), but the coefficients $A$ and $B$ are calculated differently depending on the implementation.

A weighted nonlinear regression analysis was conducted on Eq. (11) to determine the coefficients $A$ and $B$ for the general application of the model. All the operating points available for the four open literature impellers were included in the analysis, but each stage was given equal total weighting such that stages with smaller data sets had the same influence on the regression as the CC3 impeller, for which many data points were available. The general values for $A$ and $B$ in the first row of Table 2 are recommended as the default coefficients in the model. The family coefficients are also determined through nonlinear regression analysis, but the analysis is performed on each stage individually such that coefficients $A$ and $B$ are unique to each impeller. The values of $A$ and $B$ resulting from individually conducted nonlinear regression analyses on each of the open literature impellers are also given Table 2. The general implementation of the model is intended for use in design cases where slip factor data from a reference impeller are not available for nonlinear regression analysis. The general coefficients are weighted toward high-speed application, but together the four impellers represent a broad range of specific speed and loading. Thus, the coefficients are developed with the intent of applicability throughout the centrifugal compressor design space. The family implementation is recommended when an existing design is being modified and performance data at numerous operating conditions are available for regression analysis.

Throughout the proceeding analysis, the Honeywell SSCC impeller is used as a validation case for general implementation of the new model. The slip factor of the Honeywell SSCC impeller was predicted using the coefficients $A$ and $B$ for the general implementation of the new model as calculated from the open literature compressors and given in the first row of Table 2. Therefore, the open literature cases provide an evaluation of the accuracy of the new model, and data associated with the SSCC impeller provide validation data to evaluate the predictive capability of the model. The results for the Honeywell SSCC validation case are shown alongside that of the open literature cases in most of the discussion. Since the family implementation of the model is customized for each impeller design, a validation case is not appropriate and is not included. The SSCC impeller is included in discussion of the family implementation to provide an additional dataset for assessing the new model.

The differences between the calculated experimental slip factor and the slip factor predicted by Wiesner's model, the general implementation of the new model, and the family implementation of the new model are given in Figs. 2(a), 2(b), and 2(c), respectively. The bar graphs in Fig. 2 show the distribution of the error data color coded by stage, and the black, vertical, dashed line highlights the zero-error location. Overpredictions of slip factor are indicated by negative values to the left of the zero-error line, while underpredictions of slip factor are positive and to the right. The stars above the bar graph indicate the error in predicted slip factor at the design speed peak efficiency (PE) point of each stage. Each stage is given equal visual representation in the bar plots by normalizing the ordinate such that a single stage with a large dataset does not skew interpretation of the data.

Both the general and family implementations of the new model result in a normal distribution of error in predicted slip factor centered around 0 error, while there is broad, irregular scatter in the error distribution resulting from Wiesner's model. Additionally, the range of error for both implementations of the new model is smaller than that of Wiesner's model. The scatter of the error distributions depicted in Fig. 2 is quantified in Table 3. The general and family implementations of the new model predict more than 60% and 90% of the data, respectively, within ±0.025 of the experimental slip factor, while Wiesner's model captures just over 50% of the data within ±0.05. Specifically, for the SSCC validation case, 83% of the data are predicted within the ±0.025 error band, which supports application of the general implementation to impellers not included in the regression analysis.

Traditional geometry-based approaches to modeling slip factor have been shown to exhibit large levels of error at design speed for high-speed compressors [20], and this can be seen for Wiesner's model in Fig. 2(a). Of the high-speed compressors, only the design point slip factor of Came's impeller is predicted within the traditionally accepted ±0.05 error band. The new model shows significant improvement in predicting slip factor at the design speed peak efficiency point in both implementations. Although the slip factor predicted for Eckardt's impeller is outside the ±0.05 band for the general method, the high-speed stages are all within the band. For the family implementation, the design point slip factor is predicted within ±0.025 for all cases.

Since errors in the prediction of slip factor propagate to errors in work input, the ability of the new model to predict work input is also considered, Fig. 3. As in Fig. 2, the zero-error location is highlighted by a black, vertical, dashed line, and stars across the top of the graph indicate the error at the design speed peak efficiency point. However, the abscissa now shows the relative error in total temperature rise ratio rather than absolute error in slip factor. The large error range in slip factor for Wiesner's model (Fig. 2(a)) propagates to larger mispredictions of work input. Less than 35% of the data are predicted within ±5% of the actual work input, and at design speed, errors of at least 5%, and up to 20%, are observed. The errors observed for the new model are also larger, but the distributions remain normally distributed about 0% error. More than 70% and 90% of the predictions are within ±5% of the actual work input for the general (Fig. 3(b)) and family (Fig. 3(c)) implementations of the new model, respectively. Additionally, the design point total temperature rise ratio prediction is within ±2.5% error for the family implementation. The design point error is larger for the general implementation but still represents an improvement relative to Wiesner's model when applied to high-speed impellers.

Prediction of the work input for the SSCC case using the general implementation of the model again supports application of the new model to the broader centrifugal compressor design space. The model predicts the total temperature rise ratio for 68% of the SSCC operating points within ±2.5% relative error, and the slip factor at design point is predicted with an error of less than 1%.

The improvement in prediction of total temperature rise ratio can be observed through comparison of the work plots for Krain's compressor as measured in experiments and predicted by each of the models, Fig. 4. All three models provide good estimates of the work input at low speed, but as speed (i.e., machine Mach number) increases, the prediction provided by Wiesner's model diverges from the experimental data. In contrast, both implementations of the new model provide good estimates of the work input throughout the operating range of Krain's impeller. A similar trend is observed for each of the high-speed impellers.

Numerous investigations have confirmed that the slip factor varies as a function of the compressor operating conditions [8,11,34,38,39]. However, traditional models derived from the relative eddy approach, such as Wiesner's model, are dependent only on the impeller geometry and cannot capture the variation of slip factor with operating conditions driven by Eqs. (14) and (15). Some recent relative eddy approaches have incorporated the impeller exit flow coefficient to account for the variation of slip factor with operating conditions [9,21], but these methods dilute the influence of the machine Mach number in Eq. (15). The variation of the exit flow coefficient throughout the compressor map is usually small relative to the range of machine Mach numbers available to high-speed compressors, and furthermore, the machine Mach number is raised to a larger power than the exit flow coefficient.

Thus, the relationship of slip factor with machine Mach number in Fig. 5 is not captured by relative eddy models, regardless of exit flow coefficient inclusion. As the machine Mach number increases, the slip factor decreases both as a general trend and for each individual design. The dependence of slip factor on machine Mach number and increasing influence of machine Mach number at high tip speeds also drives the increasing overprediction of work input with wheel speed between Wiesner's model and the experimental results observed in Fig. 4. In contrast to geometry-based models, the new model inherent inherently incorporates the influence of the machine Mach number on slip factor through Eq. (15), emulates the trend of slip factor in Fig. 5, and supports accurate characterization of the trend of slip factor with compressor operating conditions. Additionally, significant improvement in the prediction of the slip factor is attained at and near design speed for the high-speed impellers where influence of the machine Mach number is large.

Figure 5 also shows that a range of values for slip factor are possible for a single machine Mach number (i.e., along a speedline). The new model captures this behavior through incorporation of the loading coefficient and flow coefficient, further supporting the capability of the new model to capture the variation of slip factor throughout the compressor map.

Additionally, Fig. 5 provides insight into the spread of the data observed in Figs. 2 and 3. While the slip factor decreases with increasing machine Mach number, the trend is not exactly replicated across all of the impellers. For example, the slip factor for Krain's impeller decreases more rapidly with machine Mach number than for Came's impeller. The differences in the relationship of each impeller with machine Mach number, as well as the inlet flow coefficient, are emphasized in Table 3. The family coefficients are slightly different for the different designs, and the unique relationship between slip factor and the dimensionless coefficients is not exactly captured in the general implementation. Since the family implementation is developed for each impeller design, the family implementation more accurately represents the trend for each impeller and improves the accuracy of the model.

From Eq. (16), the error in slip factor is propagated to total temperature rise ratio scaled by the square of the machine Mach number for traditional slip models. Figure 6 illustrates Eq. (16): as the machine Mach number increases, the same level of error in slip factor is increasingly deleterious for the prediction of total temperature rise ratio. Thus, the errors in slip factor predicted by Wiesner's model in Fig. 2 propagate to large errors in total temperature rise in Fig. 3. The new method uses a predicted total temperature rise ratio to calculate the slip factor, so the direction of error propagation is reversed: the error in predicted total temperature rise ratio propagates to error in slip factor but is divided by the square of the machine Mach number. Consequently, the new method can tolerate larger mispredictions since the error propagation from total temperature rise ratio to slip factor reduces the magnitude of the absolute error in slip factor for machine Mach numbers greater than one.

In addition to providing high levels of accuracy, the new model is ideal for use in the preliminary design phase when basic parameters such as flow areas, velocity triangles, and thermodynamic states are set. At present, all other slip models require the specification of the number of impeller blades and/or backsweep angle, if not blade thickness and other geometric parameters, as well. The new model relies only on dimensionless parameters that are fundamental to the definition of the compressor design. Therefore, no additional parameters beyond those required to solve the meanline equations must be introduced to estimate the slip factor with the new model.

Because the new model is dependent only on design parameters and derived from governing relationships, it provides both improved estimation of the slip factor and ease of use in preliminary design. However, application of the model is unconventional. The perfect flow guidance loading coefficient must be determined to calculate the predicted total temperature rise ratio, and the impeller exit flow coefficient is required to calculate the slip factor from the predicted total temperature rise. A one-dimensional analysis tool is necessary for both of these steps, and therefore, the new model is intended specifically for use in meanline solvers for centrifugal compressors as outlined in the Appendices.

Internal losses only affect the entropy generation of the stage and have negligible bearing on the work input delivered by the impeller. In contrast, external losses (such as disk friction and recirculation) contribute to the overall stage total temperature rise ratio without providing additional total pressure rise. At a given operating point where the impeller wheel speed, mass flow rate, and total temperature rise ratio are constant, increasing levels of external loss correspond to reduced impeller work input. By extension, the slip factor must decrease to support the reduction in impeller work input. External losses must, therefore, be accounted for when relating slip factor, total temperature rise ratio, and loading coefficient. References [40] and [41] are recommended for additional details on internal and external losses.

Exact allocation of external losses is possible in one-dimensional modeling. As such, all references to the total temperature rise ratio and loading coefficient up to this point have been to those parameters as specifically associated with the impeller: the external losses have not been included. Separating the total temperature rise due to the impeller and external losses is recommended since Eq. (7) is defined specifically for the work input generated by the impeller. However, the percentage of the temperature rise delivered by the impeller versus external losses often cannot be rigorously partitioned in experiments.

Fortunately, the new model is robust to the inclusion of external losses, Fig. 7. The error distribution for the total temperature rise ratio predicted by Wiesner's model is shown again in Fig. 7(a) for comparison to the new model. Figures 7(b) and 7(c) show the error distribution of the general and family implementations of the new model including external losses. In these implementations of the model, the nonlinear regression was conducted on Eq. (11) using the stage total temperature rise ratio instead of the impeller alone for each of the open literature impellers. Thus, the SSCC impeller again gives validation for general implementation of the model. The inclusion of external losses in the regression analysis results in more scattered error distributions for both implementations of the model. However, the overall accuracy remains improved relative to Wiesner's model, particularly at design point for the high-speed compressors and for the family implementation of the model.

The relative error distributions in total temperature rise ratio corresponding to Eqs. (17) and (18) are shown in Figs. 8(b) and 8(c), respectively, with the error distribution for Wiesner's model shown again for reference in Fig. 8(a), and the SSCC impeller serving as a validation case. The general implementations for both the specific speed and ratio of flow coefficients are shown in Fig. 8, but both can be utilized in the family implementation with recalculation of the coefficients for specific impeller designs. The error distributions for the general implementations again show improvement in the prediction of total temperature rise ratio relative to Wiesner's model and are comparable to the general correlation with inlet flow coefficient presented previously. When considering only the design points, the correlation with specific speed shows improvement relative to the correlation with inlet flow coefficient in Fig. 3(b). While the correlation with specific speed provides high accuracy at design point, the dependence on the specific speed introduces additional complexities into the model. For the model not to be recursive with the meanline equations, the perfect flow guidance specific speed must be used. Usage of an additional perfect flow guidance parameter introduces further reliance on modeling to the method. The same reasoning applies to the correlation with the ratio of flow coefficients: the impeller exit flow coefficient must be calculated from the perfect flow guidance condition. Thus, the inlet flow coefficient correlation was chosen as the primary method for discussion in this work. The alternate correlations are presented, however, as the additional resources may be of use during the design process.

A new method to model slip and work input for centrifugal compressor impellers has been introduced. Traditional slip models use the theory of the relative eddy to predict the flow physics at the impeller exit based on the impeller geometry. Rather than modeling flow physics, the new approach utilizes dimensionless parameters to predict the actual total temperature rise ratio from the perfect flow guidance condition. The new model is based on dimensionless parameters that are representative of both the operating conditions and impeller geometry. Thus, the model inherently supports prediction of slip factor at design and off-design conditions. The model is applicable to impellers throughout the centrifugal compressor design space by its dependence on the key design parameters of machine Mach number, loading coefficient, and flow coefficient. Additionally, many slip models in the open literature require the specification of geometric parameters beyond what is strictly necessary to establish the velocity triangles of the stage, such as blade number or blade thickness. The new model predicts the slip and work input utilizing parameters that are fundamental to establishing the duty of the compressor stage.

The reliance of the new model on governing relationships rather than modeling techniques, inherent incorporation of design conditions, and favorable uncertainty propagation together provide an improvement in predictive capability for slip factor during centrifugal compressor preliminary design.

The authors would like to thank Honeywell Aerospace for support during this study. Thanks are also extended to Scott Jones at NASA Glenn Research Center for his assistance in the utilization of OTAC. Finally, the support of Dr. Fangyuan Lou, Ms. Amelia Brooks, Mr. Matthew Fuehne, and Mr. William Brown in conducting experiments at the Purdue Compressor Research Laboratory is gratefully acknowledged.

$a$ =

speed of sound

$A$ =

new model coefficient

$B$ =

new model coefficient

$cp$ =

specific heat at constant pressure

$D$ =

diameter

$h$ =

enthalpy

$m˙$ =

mass flow rate

$MU2$ =

machine Mach number

$NS$ =

specific speed

$r$ =

radius

$R$ =

gas constant

$T$ =

temperature

$TTR$ =

total temperature rise ratio

$U$ =

impeller velocity

$Vm$ =

meridional velocity

$Vslip$ =

slip velocity

$Vθ$ =

tangential velocity

$w$ =

work input

$Z$ =

number of blades

Greek Symbols

$β$ =

relative flow angle

$β2b$ =

impeller exit blade angle

$γ$ =

ratio of specific heats

$ϵ$ =

limiting radius ratio for Wiesner's slip model

$ρ$ =

density

$σ$ =

slip factor

$ϕ$ =

flow coefficient

$χ$ =

ratio of actual loading coefficient to perfect flow guidance loading coefficient

$ψ$ =

loading coefficient

Subscripts

$PFG$ =

perfect flow guidance condition

$0$ =

stagnation condition

$1$ =

impeller inlet state

$2$ =

impeller exit state

For cases where experimental data for the external losses are unavailable, as is common, the external losses can be modeled according to best practices. An alternative to the use of external loss models is to assign a portion of the overall loss to the external losses, as outlined in Sec. 3 of this paper. As a final option, the new model is robust to the inclusion of external losses as discussed in Sec. 5, and the regression analysis can be conducted with overall total temperature data with a slight penalty to the accuracy of predictions. It must be noted that while the method is not overly sensitive to external losses, separation of the diffuser loss from impeller loss is necessary. Thus, the efficiency data provided to the method should either be the impeller efficiency, or the stage efficiency must be used in conjunction with a diffuser loss model to isolate the impeller loss from the diffuser loss.

Before proceeding with the regression analysis, the perfect flow guidance condition corresponding to each operating point in the dataset must be calculated to determine the perfect flow guidance loading coefficient. The perfect flow guidance-specific speed and impeller exit flow coefficient can also be determined from the perfect flow guidance condition if one of the alternative implementations of the model given in Sec. 6 is used. The perfect flow guidance condition is calculated by using the mass flow rate and rotational speed associated with each operating point as inputs to a meanline analysis tool. A slip factor of one is used to calculate the perfect flow guidance condition, and, in this work, the efficiency at the perfect flow guidance condition was assumed to be the identical to the actual operating conditions. Additionally, the proportion of external loss to overall loss was assumed to be identical between the actual operating condition and the perfect flow guidance condition. For example, 20% of the impeller loss was attributed to external loss at both the actual and perfect flow guidance conditions in this work. Matching the experimental efficiency was accomplished in this work by iterating on the impeller loss until the total loss between the impeller and diffuser loss model resulted in an efficiency equal to the measured stage efficiency.

Nonlinear regression is then conducted on Eq. (11), Eq. (17), or Eq. (18) by providing a nonlinear regression analysis tool with the appropriate parameters from the chosen equation. The matlab function nlinfit was used in this work [42]. The nonlinear regression analysis provides the user with the coefficients $A$ and $B$ to be implemented in the model of choice in a meanline analysis tool.

The steps to introduce the new model into a meanline analysis tool are not rigid, and the procedures given herein are intended as guidelines rather than as imperatives. The two primary steps to incorporate the new model into an existing code are calculation of the slip factor from the predicted work input and integration of the model into the solver.

Without the meanline equations, the impeller exit meridional velocity is unknown and the slip velocity cannot be determined. Thus, the model must be used in conjunction with meanline analysis tools to complete the system of equations. Once the slip velocity is determined from the model in conjunction with the meanline equations, the slip factor can be calculated according to its definition in Eq. (2).

An iterative procedure is recommended to incorporate the new slip model for predictive use in development of a new impeller design (rather than matching experimental data). To initialize the iteration, a guess of the efficiency is made to calculate the perfect flow guidance condition at the operating point of interest. The appropriate perfect flow guidance parameters are then provided to Eq. (11), (17), or (18) from the perfect flow guidance operating point to calculate the slip factor. The slip factor and appropriate loss models are used to calculate the actual operating condition. The newly calculated efficiency is then used to recalculate the perfect flow guidance conditions and a new value of slip factor, and so on, until the efficiencies of the perfect flow guidance condition and actual operating condition converge.