2.7.3. Compressor Isentropic Efficiency Model

The compressor isentropic efficiency model developed in this paper is based on the one originally proposed by Hadef et al., which is referred to as "Hadef model" herein [16]. The theoretical foundation of the Hadef model is the Euler equation for turbo-machinery, meanwhile, two assumptions are made: (1) air is only accelerated when flowing through the compressor impeller without being any compressed, thus, the air density at the impeller inlet and outlet can be assumed to be identical; (2) under given rotational speed condition, the airflow angle at the impeller outlet does not change with the mass flow rate. Based on the two assumptions, it can be concluded that under given rotational speed condition, the actual specific enthalpy change varies linearly with the mass flow rate as can be observed in Figure 6. Consequently, the Euler equation can be re-written as:

$$
\Delta \text{h}\_{\text{act}} = b(\text{N}\_{\text{tc}}) - a(\text{N}\_{\text{tc}}) \dot{m}\_{\text{c}} \tag{22}
$$

$$a(N\_{tc}) = 0 + a\_1 N\_{tc} + a\_2 N\_{tc}^2 \tag{23}$$

$$b(\mathcal{N}\_{tc}) = 0 + b\_1 \mathcal{N}\_{tc} + b\_2 \mathcal{N}\_{tc}^2 \tag{24}$$

where *a* and *b* are the slope and intercept, respectively.

**Figure 6.** Relationship between actual specific enthalpy change and mass flow rate at each rotational speed condition (interval between each iso-speed curve is 600 RPM).

By applying Equations (22)–(24) with the model parameters calibrated by using Least Square Method (LSM), the actual specific enthalpy change can be calculated under the given rotational speed and mass flow rate condition. On the other hand, the specific enthalpy change under the ideal isentropic process can be derived by Equation (25). Subsequently, the compressor isentropic efficiency can be calculated according to its definition (η*c* = Δ*his*/Δ*hact*).

$$
\Delta l\_{\rm is} = c\_{p,a} T\_{c,in} \left( \Pi\_c^{\frac{\gamma\_a - 1}{\gamma\_a}} - 1 \right) \tag{25}
$$

As can be observed from Figure 6, under high rotational speed conditions, distinguishable non-linear decreasing trend in the actual specific enthalpy change occurs as the mass flow rate gradually approaches to the choking point. As a result, the model's predictive accuracy will be polluted if the model parameters are calibrated by using all the available measured data points in the compressor map. To solve this problem, a zonal modeling approach based on the Hadef model is proposed, which is referred to as "zonal isentropic efficiency model" in this paper. The basic idea is to divide the whole Δ*hact* − .*mc* plane as shown in Figure 6 into several zones depending on the iso-speed curves available

in the performance map, and then each zone is calibrated individually by using the measured data points at the current zone's upper and lower iso-speed curve.

To compare the predictive accuracy of the Hadef model and the zonal isentropic efficiency model proposed in this paper, five error evaluation criteria, including *R*2*c* , *RD*max, *MAPE*, *PEB*±5% and *PEB*±10%, are adopted, the definitions of which can be found in Shen et al. [15].

Figures 7 and 8 presents the predictive results and the relative error distribution for the two isentropic efficiency models, respectively, whereas Table 2 presents respective error evaluation criteria results. Note that for depicting the predictive results clearly, only five iso-speed curves (7200, 9600, 12000, 14400, 16800 rpm) are presented in Figures 7a and 8a. As can be observed from Figures 7a and 8a, both models are capable of capturing the changing trend of the isentropic efficiency with mass flow rate. Nevertheless, it can be found from Figures 7b and 8b as well as Table 2 that the predictive accuracy of the zonal isentropic efficiency model is better than the Hadef model with lower *RD*max and *MAPE* and higher *R*2*c* ; in addition, the relative errors of all the predictive results are within ±5%. The better predictive accuracy of the zonal isentropic efficiency model mainly benefits from the zonal modeling approach. With this approach, the working characteristic of the compressor within different speed range can be captured much more accurately.

**Figure 7.** Predictive result and error distribution for Hadef isentropic efficiency model: (**a**) predictive result; (**b**) error distribution.

**Figure 8.** Prediction result and error distribution for compressor zonal isentropic efficiency model: (**a**) predictive result; (**b**) error distribution.


**Table 2.** Error evaluation criteria result for Hadef model and zonal compressor isentropic efficiency model.

As similar with the compressor mass flow rate model, the isentropic efficiency model is also required to extrapolate to the off-design operating area robustly and accurately. When extrapolating to the LS and HS area, the model parameters belonging to the first and last zone is adopted, respectively. As the definition of isentropic efficiency is directly adopted, the isentropic efficiency value will necessarily be zero when the pressure ratio is equal to 1, which guarantees the model's LPR area extrapolative accuracy to a certain extent.

For the compressor isentropic efficiency model, besides the pressure ratio and rotational speed, the mass flow rate is also required as the input variable. By incorporating the compressor mass flow rate model developed in Section 2.7.2, which is capable of extrapolating to the off-design operating area robustly and accurately, the extrapolative ability of the zonal isentropic efficiency model can be investigated. Figure 9 presents the corresponding isentropic efficiency extrapolative results. As can be observed from this figure, when extrapolating to the LPR area, each iso-speed curve is capable of achieving a smooth transition to the operating point with pressure ratio and isentropic efficiency equal to 1 and 0, respectively; when extrapolating to the LS and HS area, the changing trend of the extrapolated iso-speed curve is similar with the other iso-speed curves available in the compressor performance map, respectively, which verifies the rationality of the extrapolative strategy adopted in this paper to a certain extent.

To further verify the extrapolative ability of the zonal isentropic efficiency model in the LS and HS area, the first and last iso-speed curves available in the performance map are removed firstly, and then the model is calibrated with the remaining measured data points and the removed iso-speed curves are extrapolated by using the model, finally the removed iso-speed curves are compared with the extrapolated results to evaluate the model's extrapolative ability. Figure 10 shows the extrapolative results in the LS and HS area, respectively. As can be observed from Figure 10b, satisfactory HS area extrapolative results are achieved with an *MAPE* of only 0.8422%; on the other hand, although it is slightly inferior with respective to that in the HS area, the model's LS area extrapolative accuracy is still satisfactory with an *MAPE* of 2.0318%.

**Figure 9.** Prediction and extrapolation result of the zonal compressor isentropic efficiency model.

**Figure 10.** Extrapolation result of the zonal compressor isentropic e fficiency model: (**a**) LS (Low Speed) area extrapolation result; (**b**) HS (High Speed) area extrapolation result.

### **3. Model Calibration and Results**
