2.7.1. Compressor Performance Map

The working characteristic of a compressor is usually represented by the performance map as shown in Figure 2 with the mass (or volume) flow rate and pressure ratio as the horizontal and vertical coordinate, respectively.

As shown in Figure 2, most of the compressor performance maps provided by the manufacturers only contain several discrete iso-speed and iso-e fficiency curves in the design operating zone. However, for marine turbocharger compressor, its actual rotational speed may be lower than the lowest rotational speed presented in the performance map, or higher than the highest one; in addition, its pressure ratio may approach to unity under certain operating conditions, such as slow steaming, activation of the auxiliary blower and ship maneuvering [14]. Therefore, it is necessary for the developed compressor model to be capable of extrapolating to these o ff-design operating zones accurately and robustly. In addition, the developed compressor model is also required to accurately interpolate within the unknown areas between these discrete curves.

**Figure 2.** Compressor performance map.

### 2.7.2. Compressor Mass Flow Rate Model

In a previous study, the first author of this paper carried out an applicable and comparative research of compressor mass flow rate empirical models to two marine large-scale compressors (A270-L59 and TCA88-25070 marine compressor) [15]. The range of applicable and comparative analysis included both the predictive accuracy in the design operating area and the extrapolative ability to the off-design operating areas. These off-design operating areas include the area with rotational speed lower than the lowest speed available in the performance map, the area with rotational speed higher than the highest speed as well as the area to the right of the curve that connects the points with maximum flow rate at each iso-speed curve. These off-design operating areas are named as LS (Low Speed), HS (High Speed) and LPR (Low Pressure Ratio) area, respectively, for convenience of expression. By analyzing the applicable and comparative results, it can be found that none of these compressor empirical models was able to achieve satisfactory predictive and extrapolative accuracy in the whole operating area simultaneously. To solve this problem, a zonal compressor mass flow rate model is proposed in this paper, which selects the model with the best accuracy for each operating area.

Based on the applicable and comparative results presented in Shen et al. [15], it can be found that for the A270-L59 turbocharger compressor, GuanCong model achieves the best predictive accuracy in the design operating area, Leufvén and Llamas ellipse model is capable of capturing the compressor choking phenomenon accurately, whereas the Karlson-II exponential model is able to extrapolate to the LS and HS area robustly and reliably. The detail description of the three models can be found in Shen et al. [15], which will be not introduced in this paper.

For implementing the zonal compressor mass flow rate model, it is necessary to define the zone division standard firstly. As shown in Figure 3, the lowest and highest iso-speed curve available in the compressor performance map is used as the border between the LS and HS area and the design operating area, respectively, whereas the curve connecting the points with maximum flow rate at each iso-speed curve is used to divide the LPR area from the other areas.

**Figure 3.** Zone division standard of the compressor performance map.

The changing trend of pressure ratio Πlb and mass flow rate .*m*lb with rotational speed on this curve can be represented by using Equations (17) and (18), respectively. As shown in Figure 4, the exponential function is capable of capturing the changing trend of Πlb with rotational speed accurately, which increases slowly under low rotational speed conditions and then rapidly under high rotational speed conditions; on the other hand, the changing trend of .*m*lb can be described satisfactorily with the arc-tangent function. It is also found that the changing trend of the pressure ratio Πsur and mass flow rate . *m*sur with rotational speed on the surging line can be captured satisfactorily by using Equations (17) and (18) as shown in Figure 4.

$$
\Pi\_{\text{Ilb}} = a\_1 \mathbf{e}^{a\_2 N\_{\text{I}c}} + a\_3 \tag{17}
$$

$$
\dot{m}\_{\rm lb} = b\_1 + b\_2 \text{arctan}(b\_3 \text{N}\_{\rm fc} + b\_4) \tag{18}
$$

**Figure 4.** Measured pressure ratio and mass flow rate at the boundary of surging zone and low pressure ratio zone and corresponding fitting result: (**a**) pressure ratio; (**b**) mass flow rate.

To increase the reliability of Equations (17) and (18) when extrapolating to the LS area, two additional data points are added when parameterizing the two equations, that is the value of Πlb, Πsur, .*m*lb and .*m*sur when the rotational speed is equal to zero. It is assumed that Πlb = Πsur = 1 and . *m*lb = . *m*sur = 0 in this paper.

For the area to the left of the surging line, it is assumed that the mass flow rate is equal to .*m*sur if the current pressure ratio is larger than the respective Πsur under the current rotational speed condition.

Note that the mass flow rate predicted by the GuanCong model (or Karlson-II exponential model) when the pressure ratio is equal to Πlb is usually different from the one predicted by the Leufvén and Llamas ellipse model. Therefore, to prevent the possible discontinuity phenomenon during the

simulation process when the operating point enters into the LPR area, a simple curve blending method is proposed in this paper with the following steps:

(1) Estimate the mass flow rate by using the GuanCong model (or Karlson-II model) and the Leufvén and Llamas ellipse model, respectively, that is .*m*GuanOrKarl and .*m*ell;

(2) Calculate the weighting coefficient *z* according to Equations (19) and (20):

$$z = 3q^2 - 2q^3\tag{19}$$

$$q = \frac{\Pi - 1}{\Pi\_{\text{lb}} - 1} \tag{20}$$

where the range of *z* and *q* is between 0 and 1.

(3) Blend .*m*GuanOrKarl and .*m*ell by using the weighting coefficient *z* according to Equation (21) to obtain the mass flow rate .*m*LPR under the current pressure ratio and rotational speed condition.

$$
\dot{m}\_{\rm LPR} = z \cdot \dot{m}\_{\rm CuarıOrkırıl} + (1 - z) \cdot \dot{m}\_{\rm ell} \tag{21}
$$

By applying this curve blending method, it can guarantee the smooth transition of the iso-speed curve when the operating point enters into the LPR area from the other areas; in addition, it can fully take the advantage of the GuanCong model (or Karlson-II exponential model) and the Leufvén and Llamas ellipse model, which presents satisfactory predictive capability at the operating point with pressure ratio equal to Πlb and 1, respectively.

Figure 5 presents the blending results in the LPR area as well as two extrapolated iso-speed curves in the LS and HS area, respectively. As can be observed from this figure, the zonal compressor mass flow rate model is capable of not only predicting the available measured data points accurately but also extrapolating to the off-design operating area robustly and reasonably, which can effectively improve the steady and transient simulation accuracy of the MVEM developed in this paper. In addition, as also can be observed in Figure 5, under low and medium speed conditions, obvious difference exists between the blended curve and the original curve, indicating the deficiency of GuanCong model (or Karlson-II exponential model) in capturing the choking phenomenon; however, the difference gets less obvious with the increase in rotational speed and this is because that under high speed conditions, the measured data points available in the compressor map are very close to the choking point or have already choked, which can be estimated accurately by the GuanCong model (or Karlson-II exponential model), so the blending manipulation is unnecessary.

**Figure 5.** Prediction and extrapolation result of the zonal compressor mass flow rate model.
