1、 Christian K. Bach is an Assistant Professor, Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK. Howard Cheung is a Postdoctoral Research Fellow, Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN. Mapping of Vapor Injec
2、ted Compressor with Consideration of Extrapolation Uncertainty Christian K. Bach, PhD Howard Cheung, PhD Associate Member Associate Member ABSTRACT A companion paper (Bach et al., 2015) introduced a mapping for a dual port vapor injected compressor, based on a non-dimensional, -type approach. The pa
3、rameters in that model were chosen based on best fit to the data, and the accuracy of the mapping was reported based on the model predictions for the taken data. The vapor injection flowrates were limited for the heat pump dataset due to the coupling to the three-staged refrigerant expansion process
4、 from condenser to evaporator. It is likely that researchers and engineers will apply the obtained mapping to their applications, with higher or lower injection mass flowrates, which poses the question of accuracy. This paper investigates inter- and extra-polation accuracy of the mappings for the pr
5、ediction of the overall isentropic efficiency of the compressor in a rigorous fashion. This includes the different sources of uncertainty (inputs, training data, model random error, and output) and their effects onto the prediction results. Actual test data from a different experimental setups was e
6、mployed to investigate the behavior of the method. It was found that the main sources of uncertainty for predicting data outside of the training data range are model random error and uncertainty from training data (Maps 1 and 2). A reduction of the number of coefficiencts in the model lead to a redu
7、ction of the uncertainty from training data, with an increase in model random error and in maximum deviation between measured and predicted value. Uncertainty from input was much smaller than all other contributions to the uncertainty. Increasing the training data range to include the points that ar
8、e mapped decreased these uncertainties significantly, while the uncertainty from the outputs remains approximately constant (Map 3, companion paper). This also led to a significant reduction in deviation between measured and predicted value, despite using fewer coefficients than for Map 1. INTRODUCT
9、ION AND LITERATURE REVIEW Mappings are widely used as a compact and computationally inexpensive source for compressor performance data. One common type of mapping used for conventional compressors is a bivariate polynomial curve fit, defined in AHRI Standard 540 and European Standard EN 12900. Jhnig
10、 et al. (2000) investigated the application of this type of mapping to different datasets of hermetic compressors for domestic refrigerators/freezers. For a test case with only ten data points, good prediction of the combined efficiency defined in the paper was achieved, but its reliability at evapo
11、rating temperature lower than the minimum in the training data is questionable as the predicted efficiency does not change with the saturation temperature monotonically during extrapolation. The authors did extra tests to define the extrapolation accuracy of the map to be within 10% of the measureme
12、nt when the evaporating and condensing temperature lied within 10 K (18R) from the extremes in the training data, but it is unknown how the magnitude of the temperature difference 10 K (18R) is coupled with the range of the saturation temperature in the training data. While a number of deterministic
13、 models (e.g. Wang et al. 2008, Bell 2011) were developed for vapor injected compressors that lead to quantitatively good results, they also require a large computational effort to run. Navarro et al. (2013) developed a simple correlation for the injection flowrate of a single port vapor injected co
14、mpressor, which included suction mass flowrate and the pressure ratio between suction and injection port. His correlation had an R2 of greater than 0.99 for the tested compressor. Dardenne et al. (2015) developed a 10 coefficient semi-empirical model for power, injection flowrates, suction flowrates
15、, and discharge temperature. The power consumption (discharge temperature) of 59 (56) of the 63 experimental data points was predicted within 5% (5 K). The -type mapping introduced in the companion paper (Bach et al., 2015) is much simpler to apply than the semi empirical approach introduced in Dard
16、enne et al. (2015). While the -type approach is simpler, it is more empirical and is more unreliable at extrapolation. However, existing compressor maps do not include a method to quantify the uncertainty of map outputs at extrapolation other than conducting more tests at extrapolation data points.
17、To quantify the extrapolation uncertainty without additional laboratory testing, this paper focuses on the investigation of uncertainty for the mapping approach introduced in the companion paper, Bach et al. (2015). REVIEW LINEAR REGRESSION AND NATURAL LOGARITM The mapping introduced in the companio
18、n paper (Bach et al., 2015) is based on a -type approach, e.g. = 0 ( )1 ( , )2( ,)3(,)4 (,)5 ()6 (,)7()8= 0 = 0 11 22 33 88 (1) There are two options to find the coefficients in (1): an iterative minimization procedure, or linear regression (equivalent to ordinary least squares method, OLS). The ben
19、efit of the OLS is that it is more computationally efficient since it does not require iteration. However, equation (1) first needs to be transformed to be used with linear regression. Recall that log( ) = log() + log() , and (2) log() = log(). (3) Applying the rules (2) and (3) to (1), results in l
20、og(0) = log (0)+ 1 log(1)+2 log(3)+3 log(3) +8 log (8). (4) For a linear regression problem, the required form is = =0 = 0 0 + 1 1 + 2 2 + .+ , (5) where is the estimate of output y, U is the number of model parameters, is the estimate of the real model parameters , and is the independent input for
21、model parameter . Comparing equations (4) and (5), it is visible that they are quite similar, = log(0), = , and log () = . For = 0, this does not work, therefore 0 in equations 1 and 4 need to be replaced by log (0), and = 1. After the transformation, the well-known OLS estimate can be used for the
22、unknown parameter vector , = ( )1 = 0 1 2 , (6) Where is the vector of the outputs in the training data (e.g. =log(,1)log(,2)log(,3) log (,), and is the matrix of the training data inputs from the training data, = log (), with i being the number of the data point (e.g. 1 for data from first steady s
23、tate test) and U being the parameter (e.g. 1 for ). For i = 0, x is 1 for all U. UNCERTAINTY CALCULATION OF TRAINING DATA There are two kinds of uncertainty in the training data used to generate (1). The first one is the uncertainty of the measurement that includes two components - the uncertainty o
24、f the measurement apparatus and the uncertainty of sampling. The second one is the uncertainty of the equation of state used to calculate the enthalpy difference in (1). For simplicity, the second uncertainty is assumed to be constant at 0.5% that equals to the uncertainty of specific heat capacity
25、according to Lemmon (2003). These uncertainties are propagated and summed to the variables in (1) according to Kline and McClinktock (1953). For instance, the uncertainty of the enthalpies of vaporization in (1) can each be calculated by (7). () = ()2 + (0.05100 )2, (7) where () is the uncertainty o
26、f the enthalpy of vaporization and is the uncertainty of measurement and uncertainty of sampling of suction pressure. SOURCES OF UNCERTAINTY OF THE PREDICTED ISENTROPIC EFFICIENCY To quantify the effect of extrapolation on the compressor map with uncertainty calculation, four different sources of un
27、certainty to the output of the map are considered: the uncertainty from inputs, the uncertainty from model random error, the uncertainty from training data, and the uncertainty from outputs. This was done in Cheung and Bach (2015) to understand how the range and amount of training data affects the u
28、ncertainty of AHRI Standard 540 compressor map output. However, the study was only conducted with one single linear regression equation, and it did not discuss how the extrapolation uncertainty changed with different maps as illustrated in this paper. Uncertainty from inputs Uncertainties from input
29、s are the uncertainties propagated from the map inputs to the outputs of the map. Input variables to the map may carry uncertainties such as uncertainty from measurement instruments. These uncertainties propagate to the map output according to an expression from (8) according to Kline and McClinktoc
30、k (1953): , = ()2, (8) where , is the uncertainty from inputs to the map output, xi is the ith input to the map such as mass flow rate and compressor such pressure in (1), and is the uncertainty of xi. Uncertainty from model random error Uncertainty from model random error quantifies the inability o
31、f a linear regression model to account for all relationships between the inputs and outputs by linearity. A linear regression model as shown in equation (5) always carries an uncertainty related to the nonlinearity between the outputs and inputs or random effects of variables other than the variable
32、s considered in the model. According to Montgomery (2005), this uncertainty can be quantified by equation (9). = ,1+ ( )1, (9) where is the uncertainty from model random error of in equation (5), is the mean square error of the linear regression model equation (5), is the t-statistics of Student dis
33、tribution with degree of freedom and confidence level , and is a column vector of entries in equation (5). However, results from equation (9) cannot be used directly in the estimated result of isentropic efficiency because the estimated isentropic efficiency is not an output of the linear regression
34、 model. Extra calculation is needed to convert the uncertainty from (9) to the uncertainty of model random error of the isentropic efficiency, and the conversion is given by (10). , = max (exp( + ) exp (),exp () exp( ), (10) Uncertainty from training data Since the parameters in equation (5) are est
35、imated from training data which carry uncertainties, these uncertainties also propagate to the parameters which in turn propagate to the output of the map. The propagation is calculated by equation (11). , = ()()2), (11) where si is the ith values used in the training data such as the refrigerant ma
36、ss flow rates, isentropic efficiency from measurement data, etc., and the partial derivatives of the coefficients with training data are calculated by using finite difference methods with equation (6). Uncertainty from output Since the training data uses the isentropic efficiency from measurement to
37、 create the map, the map only estimates the isentropic efficiency from measurement, and the uncertainty calculated in the previous sections only helps to calculate the uncertainty between the predicted isentropic efficiency to its corresponding measured value. However, the uncertainty of the map pre
38、diction should be the uncertainty between the predicted isentropic efficiency and the corresponding true value. To achieve this goal, an additional component of uncertainty called uncertainty from output needs to be considered. This component can be predicted by considering the uncertainty between m
39、easurement and the corresponding true value in addition to the uncertainties in the previous sections. This value is estimated by averaging uncertainty of isentropic efficiencies in the training data as equation (12). , = 1 (,)=1 , (12) Overall uncertainty The overall predicted value of the uncertai
40、nty of the estimated isentropic efficiency is given by equation (13). = (,)2 + (,)2 + (,)2 + (,)2. (13) RESULTS AND DISCUSSION Experimental setups introduced in the companion paper (Bach et al., 2015) were used to create 3 isentropic efficiency maps. Figure 1 shows the prediction results for Map 1 w
41、ith data collected from the heat pump setup in the companion paper only, where all coefficients of eqn. (1) were used. The map predicts the experimental data within 0.02 of the measured value. However, if the map is applied to the data from Song (2014), the maximum deviation increases to 0.19 as sho
42、wn in Figure 2. The sum of the predicted uncertainties is more than 2 times the sum of the absolute deviation between measured and predicted values, i.e. the proposed method predicts an uncertainty that is larger than its actual occurrence in the data set. The main source of uncertainty stems from t
43、raining data as well as from the model because the map extrapolated from the limited training data range. A particularly large risk of map 1 is the injection superheats of the training data its maximum injection superheat is 1.1 K (2.0R) which is lower than 11 injection superheat values in Song (201
44、4) with a maximum at 7.5 K (13.5R). Injection superheats in the heat pump data is a result of the employed vapor seperatpors and within the uncertainty of the measurement. Therefore injection superheat could not be considered in that map. Map 2 was created by removing injection mass flowrates, since
45、 it was found that injection mass flowrates are mainly a results of the injection pressure ratios. However, Figure 3 shows that Map 2 has reduced accuracy when compared to Map 1. The maximum deviation doubled from 0.02 to 0.04 for the training data, and Figure 4 shows that map 2 leads to only slight
46、ly less accurate predictions for the data of Song (2014) all values are predicted within 0.21 of the measured value. The predicted uncertainty is much smaller than for Map 1. The sum of the predicted uncertainties is around 50% larger than the sum of the absolute deviation between measured and predicted value. Map 3 was created with data from 3 different test setups listed in the companion paper (Bach et al., 2015) and the same mathematical form as Map 2, except that normali
