1. Introduction
Supercritical CO
2 (S-CO
2) power cycle technology has received interest from a wide range of energy industries because of its identified advantages. One of the advantages is that the S-CO
2 Brayton cycle can achieve competitive efficiency (~50%) even with moderate temperature heat sources (450~750 °C) [
1]. High efficiency can be achieved by compressing CO
2 near the critical point (30.98 °C, 7377 kPa) because the high density of CO
2 near the critical point makes compression work small and maximizes heat recuperation [
2]. The system also has compact components and a simple configuration due to the high average density and having no phase change processes in the cycle. These advantages make the S-CO
2 power cycle suitable for distributed power generation [
3].
In order to successfully commercialize the S-CO
2 power conversion system, the design of components involves optimization. However, the strong real gas effects of S-CO
2 near the critical point make the design of an S-CO
2 cycle component challenging. Turbomachinery is the key component for expansion and compression processes. It has a significant impact on the efficiency of an S-CO
2 power system. Therefore, turbomachinery design should be able to predict the performance of a real component with small uncertainty to have minimal impact on the design of an S-CO
2 power cycle. Generally, utilizing appropriate loss models increases the prediction capability of turbomachinery design since losses in turbomachinery affect geometry, operation point, and the associated design of a motor or generator. Good loss models for the S-CO
2 turbomachinery design should be able to predict the losses even under the strong real gas effects of S-CO
2. For this reason, many studies have focused on developing loss models of S-CO
2 turbomachinery [
4,
5,
6].
The turbomachinery losses can be classified into two types: internal and external losses. The internal loss mainly affects the design of the primary flow path, while the external loss determines losses occurring in the secondary flow path, which typically translates to additional work loss in turbomachinery. Thus, auxiliary cooling systems for turbomachinery are mainly affected by external losses. To minimize the size of the auxiliary system, the external loss should first be estimated with good precision.
Windage loss is one of the most significant external losses in S-CO
2 turbomachinery. It is by far the dominant loss for the S-CO
2 conditions which contrasts with turbomachinery operating with air. This is due to S-CO
2 having a higher density and S-CO
2 turbomachinery having a higher rotational speed for the same capacity. To demonstrate how S-CO
2 conditions increase windage loss in turbomachinery, the windage loss model proposed by Vrancik is used to compare the windage losses between the S-CO
2 condition and air condition, which is shown in
Figure 1.
The S-CO
2 recompression test conducted by Sandia National Lab (SNL) [
7] provides a useful illustration of how windage loss is handled in practice. A Turbo-Alternator-Compressor (TAC) used in this test features a single shaft that connects compressor and turbine impellers. SNL constructed a turbomachine with rotor cavity depressurization to reduce its windage loss by utilizing a scavenging pump. However, the minimization of an auxiliary system such as a scavenging pump or a cooling system is necessary to utilize S-CO
2 for the distributed power generation because the auxiliary systems can overwhelm the advantage of an S-CO
2 power system being compact. Therefore, accurately predicting windage loss not only reduces the prediction uncertainty of cycle performances, but also impacts the economy of the system by reducing uncertainty in the design of auxiliary systems. Thus, understanding the windage loss mechanism in an S-CO
2 turbomachinery and developing a good model for the windage loss prediction are important.
Most of the existing empirical models for windage loss are based on data obtained from air, water, and glycerin–water mixture tests. These models must first be validated regarding whether they can reflect the real gas effects of S-CO2. Furthermore, the Taylor number or the Reynolds number range of the existing models are too narrow to apply to the typical S-CO2 power cycle conditions. Therefore, a new windage loss measurement performed under the S-CO2 condition is necessary to evaluate the validity of the existing empirical models.
In this paper, S-CO2 windage loss experiments conducted by a research team at Korea Advanced Institute of Science and Technology (KAIST) are first presented. The existing empirical windage loss models are then compared to the data obtained from experiments performed under S-CO2 conditions. A method is proposed to improve the existing models by reflecting the real gas effects better. Finally, a new windage loss model which explains the data better is proposed and compared.
3. Comparison of Data to Existing Windage Loss Models
The obtained windage loss data are compared to the existing empirical windage loss models summarized in
Table 2 [
12,
13,
14]. The measured windage loss is compared with the windage loss value obtained from the models by using inlet conditions measured during experiments. The results are shown in
Figure 4,
Figure 5,
Figure 6 and
Figure 7. The plotted points in these figures show the windage loss measured by a power analyzer at the S-CO
2 rotor test rig as the y value. The x value of the points is the windage loss value calculated with each empirical model. The range of the points shows its uncertainty calculated at
Section 2.2. The linear fitting via regression analysis is shown by a pink line. The blue line shows when the calculated windage loss and the measured windage loss are the same (i.e., y = x). If the empirical models explain the experiments well, the pink line is closer to the blue line. The two green lines represent a 20% error band. Thus, the top green line is y = 1.2x, and the bottom green line is y = 0.8x. How well the empirical model predicts the experimental results can be evaluated with the number of the plotted points inside the region between two green lines, which denotes how many data points can be predicted using the model with a 20% error. The R
2 value of each model can also be compared to evaluate the fitness of the model. The R
2 value, which is an indicator of the regression model performance for each model, is shown in
Table 3. From the figures, it is concluded that empirical models based on liquid tests all underestimate the S-CO
2 windage loss. Mack and Vrancik’s models, which are based on the air test, show smaller differences from the experimental measurement than other models, but they all overestimate the S-CO
2 windage loss. It is clear that the Wendt, Yamada, Bilgen and Boulos, and Nakabayashi’s models are unsatisfactory for explaining the S-CO
2 windage loss test results because the R
2 value is negative. Mack and Vrancik’s models have positive R
2 values. However, an R
2 value close to unity cannot be found amongst the tested models, which implies that a new model development is necessary.
Table 2.
Existing empirical windage loss models.
Table 2.
Existing empirical windage loss models.
Author | Empirical Equation | Note |
---|
Mack [12,13] | | | is obtained with assuming continuous change of at Air test |
| |
Vrancik [14] | | | Couette velocity profile in laminar and air test for in turbulence regime Air test for |
| |
Wendt [15,16] | | | Pure water, water-glycerin mixture Gap size: 0.95~4.7 cm |
| |
Yamada [15,17] | | Spindle oil test |
Bilgen and Boulos [15,18] | | | Pure water, water-glycerin mixture Gap size: 0.32~4.45 cm |
| |
Nakabayashi [15,19] | | | Freezer oil, pure water, water-glycerin mixture Roughness test |
| |
| |
One of the major reasons why the existing models fail to predict the data successfully is due to the negligence of property variations from inlet to outlet due to viscous heating. The viscous heating from windage loss changes the CO2 properties dramatically from inlet to outlet in the secondary pass of turbomachinery. However, it was originally suggested that the existing empirical models should only use inlet conditions, and therefore, this property change effect of CO2 cannot be reflected properly.
Figure 8 shows the work loss due to windage loss estimated with inlet conditions and outlet conditions, respectively, for each windage loss model to illustrate how strong this effect can be. The inlet condition is 80 bar and 36 °C, and the outlet condition is 79.5 bar and 53 °C, which are selected conditions from the actual tests conducted in the KAIST S-CO
2 rotor test rig [
8]. As the figure shows, the existing empirical models cannot predict the windage loss under S-CO
2 conditions accurately by using only inlet conditions. The inlet-to-outlet density ratio is shown in
Figure 9 to explain the reason for this. As shown in the figure, the density ratio is less than unity during the test. This is because the density is decreased substantially due to viscous heating, and this in turn significantly affects windage loss. It is noted that the variation of the density ratio over time is due to testing many conditions in a single test, and the test section reaches a steady state quite quickly when the test conditions are changed.
4. New S-CO2 Windage Loss Model Development
A new windage loss model is developed by modifying the existing model and the validity of the newly proposed model is presented in this section. The Taylor number is first examined to determine whether it alone is enough to describe the windage loss test data. The measured windage loss is plotted with respect to the Taylor number, as shown in
Figure 10. Since the data have wide dispersion without a clear pattern with respect to the Taylor number, it is concluded that other variables are required to describe the windage loss data better. The heat capacity ratio
is therefore considered to be used next in conjunction with the Taylor number to better explain S-CO
2 windage loss test data. The measured windage loss data are now plotted with respect to
and
in
Figure 11. By correlating the data with the Taylor number and
, the data can now have one-to-one correspondence. Therefore, these two variables are used to correlate S-CO
2 windage loss data.
Since Mack’s model has the best regression result as shown in
Table 3, Mack’s model is selected to correct with
and
. The modified model should produce identical results with the original model when the
value becomes equal to air, since Mack’s model is developed from data obtained from air tests. The assumed skin friction coefficient with this constraint can thus have the following relations shown in Equations (15) and (16). The least square method is used to find the value for exponent
. The newly proposed model is shown in Equation (17) for the supercritical phase and in Equation (18) for the gas phase.
Even though the developed model is based on one S-CO2 test rig, it is expected that this model could be applied to the design of other turbomachinery. This is because the developed model is based on non-dimensional numbers, and as long as the similitude holds, the model can be applied to the design of other equipment operating under S-CO2 conditions. The newly developed model is a model modifying the base model (i.e., Mack’s model) by multiplying the term to reflect the change in properties of the working fluid due to viscous heating. Therefore, this model can reflect not only S-CO2 conditions but also other fluids with similar characteristics if the fluid is in a superheated gas or supercritical phase. Other conditions such as geometry or rotation speed are different in the designed machinery, the developed model will follow the trend of Mack’s model, which is based on a larger test data set. In summary, the newly suggested model is generally applicable.
The new models are compared with test data as shown in
Figure 12 and
Figure 13. The R
2 values are 0.9772 and 0.9673 for supercritical and gas phases, respectively, which are near unity, so the new model is better than Mack’s model for predicting the windage loss under S-CO
2 conditions.
However,
Figure 12 and
Figure 13 still show that the measured windage loss can be different, with the predicted windage loss being more than 20% in some cases. The newly suggested model is based on the existing empirical model, and to evaluate if there are any missing factors for the S-CO
2 working fluid for the new model, the Pi theorem is used. Since the heated S-CO
2 flows through the axial direction, the added factor is the axial flow velocity,
.
There are two dimensionless variables ( and ) and six variables with dimensions (). The dimensions used in this equation are mass, length, and time. Based on the Pi theorem, two dimensionless variables are required because there are five variables and three dimensions (mass, length, and time).
Let
,
, and
be repeating variables. Then, the dimensionless numbers are listed as below.
To make
dimensionless,
and
are zero and
is −1. Similarly,
becomes dimensionless when
,
are −1 and
is −2. It can be shown that
is
and
, inverse of
, is
. However,
becomes
, which is not a frequently used term. For a familiar term,
is modified as
which is
. Therefore, third dimensionless number becomes
which is
. The newly suggested form of
is shown in Equation (23).
and
are obtained with the least square method again. The results are shown for the supercritical condition in Equation (24) and the gas condition in Equation (25). They are compared with the test data as shown in
Figure 14 and
Figure 15. The R
2 values are now 0.9800 and 0.9763 for the supercritical and gas phases, respectively, which are closer to unity. Based on the R
2 value, the newly suggested model is better than the existing models to explain the S-CO
2 test results. However, the exponent of the Reynolds number is quite low, which indicates that the Reynolds number does not play too important a role in the windage loss.