Modelling investigation on the influence of measurement error on performance evaluation of organic Rankine cycle system

The evaluation of the performance of organic Rankine cycle (ORC) depends on the measured data of the operating parameters. The influences of the measurement errors during the evaluation of the system’s performance, which are derived from the measuring instruments, are rarely studied in the literature. In order to explore the effects of measuring instruments’ accuracy on the evaluation of system performance, the error propagation model of a subcritical ORC system was established. The working fluids adapting to different operating temperature range, including R134a, R245fa and R123, were selected. Temperature, pressure and flow rate of fluid were adopted as the operating parameters that were measured. The net output power and the efficiency were used as the performance parameters that were evaluated. Sensitivity analyses of the measurement errors of single parameter and multiple parameters were conducted under five designed conditions. The results show that the key positions of the measurement errors affecting the performance evaluation were at the expander’s inlet, followed by the expander’s outlet, and the inlet and outlet of the fluid pump. To keep the errors of net output power and thermal efficiency within ± 10%, the measurement errors of the temperature at the inlet and outlet of the expander and fluid pump were less than ± 2%, ± 5%, ± 9% and ± 9% in all cases, respectively. To meet the same requirements, the measurement errors of the pressure at the inlet and outlet of the expander were less than ± 10% and ± 100% in all cases, respectively. The fluid adapting to higher operating parameters allowed higher measurement errors for both the temperature and the pressure. To keep the errors of net output power and thermal efficiency within ± 10%, the allowable measurement error of the expander’s inlet temperature increased from ± 0.8% for R134a to ± 2.0% for R123. The allowable measurement errors at the expander’s inlet decreased with the increase in the operating parameters for the same fluid. In most of the operating parameters, the allowable measurement errors of R245fa were larger than R134a and less than R123 under the same operating temperature range. The importance of the accuracy of temperature’s measurement outweighed those of pressure and flow rate. Finally, recommendations were made for the selection of accuracy of the measuring instruments.


Introduction
With the rapid increase in the energy consumption in the last few decades, more attention has been paid to energy-saving and emission reduction. Organic Rankine cycle (ORC) has made great progress as a promising technology to recover waste heat in various fields. Brun et al. [1] investigated the techno-economic potential of ORC to recover waste heat from jacket-water of stationary internal combustion engines. Xu et al. [2] discussed the application of ORC in the recovery of heavy-duty vehicle waste heat from system architecture, selection of equipment and working fluid, control strategy, and performance optimization. Ma et al. [3] explored the cascade utilization of waste heat from exhaust gas and jacket water of internal combustion engine using ORC. Liao et al. [4] proposed an alternative ORC that was based on the combined systems of recovering moderate-to-low temperature waste heat of exhaust flue gas from coal-fired power plant. Liang et al. [5] used ORC as the subsystem for the low-grade power generation in an original oxy-fuel combustion natural gas power plant. Liang et al. [6] combined the ORC with two-stage series evaporation to enhance the heat recovery of flue gas from gas turbine. Baral et al. [7] investigated a small-scale solar ORC system using experimental and thermo-economic analyses. The small-scale solar ORC designed by Taccani et al. [8] achieved a gross electrical efficiency of up to 8%. Wang et al. [9] built a variable-capacity power system by integrating ORC with Flash to match different temperatures of the geothermal resource. Yao et al. [10] proposed a novel power generation system that combined a natural gas expansion plant with a geothermal ORC. Invernizzi et al. [11] applied the supercritical and subcritical cycle configurations in a biomass-powered ORC with the working fluids consisting of pure and binary mixtures of hydrocarbons. Wang et al. [12] proposed a novel domestic combined heat and power (CHP) system that was based on ORC using the biomass fuel. Undoubtedly ORC has been widely investigated and tentatively applied worldwide.
According to the up-to-date studies, one of the research hotspots of ORC was to explore the relationships between the operating parameters and the system performance. The experiments conducted by Wu et al. [13] showed that there existed an optimal load power that was based on the expander's isentropic efficiency, output power, electric power, thermal efficiency and exergy efficiency. Quoilin et al. [14] found that the expander's rotational speed and the mass flow rate passing through the fluid pump took effect on the cycle's performance by changing the evaporating pressure and superheating. The effects of fluid pump's speed and fluid's mass flow rate on the performance of the system were also explored [15]. Declaye et al. [16] evaluated the performance of an expander with the operating parameters including inlet pressure, outlet pressure and rotational speed of the expander. Zhou et al. [17] confirmed that the evaporating pressure had a significant impact on the heat recovery efficiency, the output power of the expander and its exergetic efficiency. Meanwhile, the effect of the degree of superheating of the fluid on the performance of the system was insignificant. The orthogonal experiments done by Xi et al. [18] indicated that the temperature at the inlet of the expander was critical for thermal efficiency and output power, while the plunger travel of fluid pump dominated in the consumption of pump work and the mass flow rate of the fluid.
The evaluation of the effects of operating parameters on the performance of the system relies on the measurement data in the experimental study. The accuracy of the measuring instrument represents the allowable error between the measured value and its true value. Therefore, the accuracy of the measuring instrument directly affects the accuracy of the evaluation. Some studies have already paid attention to this point in recent years. Lecompte et al. [19] evaluated the reliability of the experimental data by investigating the heat balances over the heat exchangers and through error propagation of the measurement uncertainties. Kosmadakis et al. [20,21] considered the accuracy of the calculated parameters that would be influenced by the accuracy of the measured parameters. The results showed that the relative errors of the calculated parameters could be restricted to a small range by adopting the measuring instruments with appropriate accuracy. Wang et al. [22] compared the fluctuation of measurement data and instrument accuracy to examine whether the accuracy of measuring instrument meet the requirements or not. Han et al. [23] used Gaussian law of error propagation [24] to evaluate the relative error of derived quantities such as that in thermal efficiency caused by the measurement error.
Temperature, pressure and flow rate of the working fluid are the main operating parameters of an ORC system. The common measuring instruments to detect these parameters are listed in Tables 1, 2, 3. According to Table 1, for the temperature measurement, PT100 thermal resistance and T-type and K-type thermocouples are widely used. The thermocouples have the characteristics of high sensitivity, fast response and wide temperature range. However, the thermocouples are liable to be disturbed by electromagnetism. The accuracy of the T-type thermocouple is generally higher than that of the K-type. However, the applicable temperature range of T-type thermocouple is inferior to K-type. The P100 thermal resistance has higher measurement accuracy and lower applicable temperature range. The diffused silicon, capacitive and strain gauge pressure transducers are normally used for pressure measurement. The accuracies of these pressure transducers vary from ± 0.075 to ± 0.5%. The diffused silicon pressure transducer has better stability, though it incurs higher price. The capacitive pressure transducer has the advantages including high accuracy, strong anti-overload capability, and adapting to the measurement of very low pressure. The strain gauge pressure transducer has many advantages, such as high sensitivity and accuracy, wide operating temperature range and strong adaptability to environment. It disadvantages are poor anti-interference capability and obvious nonlinearity. Referring to the flow rate measurement, the accuracy of Coriolis mass flow meter is higher than that of the volumetric flow meter. However, the price of Coriolis mass flow meter is about 2-6 times that of the volumetric flow meter. The data presented in Tables 1, 2, 3 indicate that obvious differences exist in the accuracy of measuring instruments selected among these papers. In addition, the instruments installed in different positions selected the same measurement accuracy in most of the papers. The selection of measuring instrument still lacks basis, and involves certain randomness. Too low accuracy of measuring instrument may lead to excessive error in the evaluation of performance. However, too high accuracy of measuring instrument means higher cost. Actually, there are significant differences in the effects of measurement errors on the evaluation of system performance among different operating parameters. It is more reasonable to adopt high accuracy instruments only for the operating parameters, whose measurement error significantly affects the performance, whereas low accuracy instruments should be used for other instruments. To select the appropriate measuring instrument, distinguishing the requirements of measuring accuracy of different operating parameters of ORC system becomes necessary. The paper aims at matching different measuring instruments and the requirement of accuracy to avoid the inaccurate evaluation results and unnecessary expense. The effects on the evaluation of system performance of measurement error of different operating parameters of ORC system were investigated  to obtain the requirement of accuracy. As mentioned above, the pressure, temperature and flow rate were related to the system performance and selected as the operating parameters. Modelling was adopted in this paper. Sensitivity analyses of the performance parameters to measurement errors of both the single and multiple parameters were conducted. Finally, the selection basis of the accuracy of measuring instruments of ORC system was proposed.

Methodology
The error propagation model of a subcritical ORC system was established based on thermodynamic and error propagation analyses. The model was used to analyze the influences of the measurement errors of the operating parameters on the evaluation of the system performance. The measurement errors of the operating parameters and the errors of the performance parameters were used as the input and output of the model, respectively. The operating parameters investigated included temperature, pressure and flow rate. The parameters used for evaluating the system performance included net output power and thermal efficiency. Figure 1 shows the schematic of subcritical ORC system and its T-s diagram. There are three circulations consisting of flue gas loop, working fluid loop and cooling water loop. The superheated vapor of the working fluid drives the expander to generate power through the expansion process 1-2 (for ideal state 2 s). Next, the low-pressure vapor flows into the condenser and is liquefied by the cooling water through isobaric condensation process 2-3-4-5. Then the working fluid is pumped back to the evaporator through the pressurization process 5-6 (for ideal state 6 s). Subsequently, the fluid is directly heated by the flue gas in the evaporator through isobaric evaporation process 6-7-8-1 and then, the new cycle begins.

System description
As shown in Fig. 1a, the measuring points of temperature and pressure are located at the inlet and outlet of the expander (T1-T2, P1-P2) and the fluid pump (T5-T6, P5-P6). The flow meter (V1) is set at the outlet of the fluid pump because the measurement accuracy of liquid state is higher than that of the gaseous state due to the gaseous characteristics of compressibility, high diffusibility and low viscosity.

Error propagation model based on thermodynamic and error propagation analyses
As can be seen in Fig. 1b, the heat transfer in the evaporator can be calculated using Eq. (1).
Similarly, the heat transfer in the condenser is calculated using Eq. (2).
Then, the expander output power is calculated according to Eq. (3).
where the isentropic efficiency η exp is defined by Eq. (4).

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Subsequently, the work consumed by the working fluid pump is defined by Eq. (5).
where the isentropic efficiency of the working fluid pump is expressed by Eq. (6).
Consequently, the net output power and thermal efficiency of the ORC system can be defined by Eq. (7) and Eq. (8), respectively. As given by Eq. (8), the thermal efficiency of the ORC system is theoretically related to the enthalpy differences in the evaporator and condenser.
The detailed heat transfer process in heat exchangers is depicted in Fig. 2. The axis of F in Fig. 2 represents the factions containing preheating section, evaporation section and superheating section for evaporator, and precooling section, condensation section and subcooling section for condenser. The locations of the pinch points in the evaporator and condenser are marked. According to Fig. 2a the pinch point temperature difference in the evaporator means the difference between the outlet temperature of heat source in the evaporation section and the evaporation temperature, and can be expressed by Eq. (9).
The heat balance in the evaporator is calculated by Eq. (10).
As shown in Fig. 2b, the pinch point temperature difference in the condenser is the difference between the condensation temperature and the outlet temperature of the cooling water in the condensation section. Besides, the total temperature increments ( ΔT c ) of cooling water in the subcooling and condensation sections are listed in Table 4. Therefore, the condensation temperature can be calculated using Eq. (11).
As mentioned above, the measurement errors affect the evaluation of the effects of the operating parameters on the performance of the system. Generally, the systematic error and the accidental error both exist in the measuring process. However, the accidental error can be erased by averaging the values of repeated measurements. The ineradicable systematic error is closely related to the accuracy of measuring instruments. Therefore, the propagation of the systematic error is considered in this model. Due to the reason that the enthalpy value is determined by the corresponding temperature and pressure, the absolute error of enthalpy can be calculated according to Eq. (13) [19].
where T and p are the measurement errors of the temperature and pressure, respectively. The measurement error is equal to the measured value minus its true value. The relative error of enthalpy can be expressed by Eq. (14).
Moreover, the density is also affected by the corresponding temperature and pressure. The calculations of the absolute and relative errors of density are similar to enthalpy. The absolute and relative errors of other parameters can be obtained through mathematical operations. The propagation of error can be calculated as follows. If a variable R is obtained by adding and subtracting other variables, for example, R = A + B -C, the absolute error of R can be obtained according to Eq. (15) [43].
Moreover, if a variable R is given by other variables through multiplication and division, for example, R = AB C , the absolute error of R can be calculated using Eq. (16) [43]. Furthermore, Eq. (16) can be rewritten as Eq. (17) according to the relationship between the absolute error and the relative error.
According to the aforementioned thermodynamic model of ORC system, the net output power is the product of mass flow rate and enthalpy difference. Meanwhile, the thermal efficiency is the ratio of enthalpy differences. Therefore, the propagated errors of these two parameters can be obtained using Eq. (16) and Eq. (17). Finally, the flow chart of model solution is illustrated in Fig. 3. Some key parameters referring to flue gas, cooling water and ORC systems are presented in Table 4. In most of the studies, the pressure ratio of the expander is less than 10 [13][14][15][16][35][36][37]. It is used as a constraint in this model. In order to compare the effects of measurement error under different working temperature ranges, three basic cases (cases 1, 2 and 3) were established with different flue gas temperatures. In addition, R134a, R245fa and R123 were selected as the working fluids, which adapted to the corresponding temperature ranges. Meanwhile, R123 was selected as the working fluid adapting to the heat source temperature of 150 °C. Additionally, R123 was commonly used for the experimental ORC systems, although such working fluids with ODP (Ozone Depletion Potential) were banned and being phased out due to regulations and agreements. Some previous studies have indicated that the operating parameters of ORC system using R123 has similar variation tendencies with other working fluids. The study conducted by Wang et al. [44] showed that the variation trends of thermal efficiency, global thermal efficiency, power output and pinch point temperature against the evaporation temperature were similar for the ORC systems using R123, R601, R245fa, hexane and isohexane fluids. From the calculated results of the model built by Xiao et al. [45], the net power output and the total exergy destruction rate of the ORC systems using R123, R245fa, R601a and R601 appeared consistent with the evaporation temperature and the condensation temperature. The ORC systems using R123, R600 and R245fa showed similarity in terms of the relationship among the thermal efficiency, expansion ratio and evaporating pressure [46]. The same variation trends were obtained of the net power output with the pressure and temperature in the turbine inlet and the pinch temperature difference in the evaporator for R123, R245fa and isobutane [47]. Li et al. [48] found that the ORC systems using R245fa and R123 presented similar variation of net power output, maximum net power output and optimal evaporation temperature with respect to pinch point temperature difference in the evaporator. Therefore, the results can be used as a reference for the ORC system using non-R123 fluids.
Two more cases (Cases 4 and 5) with the thermodynamic parameters similar to Case 1 were supplemented to compare the effects of measurement errors of different working fluids under the same temperature range. In order to obtain the optimum operational parameters based on maximum output power, modeling was done using MATLAB along with REFPROP 9.0. The output power of the ORC is affected by the amount of recovery heat, evaporation pressure, evaporator outlet temperature and condensed pressure. The initial evaporation pressure was first assumed, and then, gradually increased. The corresponding operating parameters and output power were calculated using model simulations based on key parameters presented in Table 4. Finally, the maximum output power and the corresponding operating parameters were obtained. The temperatures range of 150-250 ℃ is the common range of heat source temperature of an ORC system. Therefore, the temperatures of 150, 200, and 250 ℃ were selected as the heat source inlet temperature for the model. The pinch point temperatures were assumed to be 5 ℃ for the evaporator and condenser. The optimized thermodynamic parameters under designed conditions are listed in Table 5. It should be noted that the selections of model parameters and working fluids were based on a previous investigation [41,42].

Comparison of the effect of measurement error under different working temperature ranges
The effects of measurement error of operating parameters under different working temperature ranges were compared through Case 1, 2 and 3. The optimal working fluids adapted to the corresponding working temperature ranges were selected. According to the aforementioned model, the simulation values of thermodynamic parameters were considered as the true values. The deviation input of model represented the measurement error between the true value and the measured value. Finally, the effects of measurement error on the evaluation of the performance could be reflected by the influence of the deviation input of the model on the output performance parameters of the model.

Effect of the measurement error of temperature on the performance evaluation of the system
The variations of the relative errors of the performance parameters with the relative measurement errors of the temperature are shown in Figs. 4, 5, 6, 7. In coordinates, " + " and "-" mean the measured value is more and less than the true value, respectively (both here and below). It can be seen that the enthalpy error was positively correlated to the measurement error of temperature at all positions. However, the degree of influence varied. The effects on the net output power and thermal efficiency of temperature measurement error at the inlet of expander and fluid pump were the same as those on the enthalpy. However, at the outlet of the expander and fluid pump, the variation trends were opposite to those for enthalpy. Considering Case 1 as an example, when the measurement error of the temperature was + 2%, the errors of enthalpy at the inlet and outlet of the expander were + 0.87% and + 0.17%, respectively, and those at the inlet and outlet of the fluid pump were + 0.33% and + 0.34%, respectively. The errors of net output power corresponding to the measurement error + 2% of the temperature at the inlet and outlet of the expander were + 25.13% and − 4.78%, respectively, and corresponding to those at the inlet and outlet of the fluid pump were + 5.09% and − 5.34%, respectively. The errors of the thermal efficiency corresponding to the measurement error + 2% of the temperature at the inlet and outlet of the expander were + 22.76% and − 4.78%, respectively, and corresponding to those at the inlet and outlet of the fluid pump were + 5.09% and − 4.95%, respectively. If the errors of net output power and thermal efficiency were to remain within ± 10%, the measurement errors of the temperature at the inlet and outlet of the expander should not exceed ± 0.8% and ± 4.0%, respectively. In order to meet the same requirement, the measurement errors at the inlet and outlet of the fluid pump should both be kept within ± 4.0%. Considering that the temperature values at different locations vary greatly, thus from the perspective of the absolute measurement errors of temperature, they should be no more than ± 0.76 ℃, ± 1.40 ℃, ± 1.08 ℃ and ± 1.15 ℃ at the inlets and outlets of the expander and the fluid pump, respectively. It can be inferred that net output power and thermal efficiency are the most sensitive to measurement error of the temperature at the inlet of the expander. The less sensitive position are the outlet of the expander.

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As for Case 2, in order to keep the relative errors of net output power and thermal efficiency within ± 10%, the relative measurement errors of temperatures should be less than ± 1.6%, ± 4.0%, ± 8.0% and ± 8.0% at the inlets and outlets of the expander and the fluid pump, respectively. The corresponding absolute errors were ± 1.94 ℃, ± 2.66 ℃, ± 2.16 ℃ and ± 2.22 ℃. Moreover, a similar situation occurred for Case 3. In order to keep the relative errors of the performance parameters within ± 10%, the relative measurement errors of temperatures should be less than ± 2.0%, ± 5.0%, ± 9.0% and ± 9.0% at the inlets and outlets of the expander and the fluid pump, respectively. Furthermore, the corresponding absolute errors were ± 2.40 ℃, ± 3.35 ℃, ± 2.43 ℃ and ± 2.48 ℃ at the inlets and outlets of the expander and the fluid pump, respectively. It can be found by comparing Cases 1, 2, and 3 that the fluid operating in higher working temperature range allows larger measurement error for temperature. The sensitivity to the measurement error of the temperature of R245fa is less than that for R134a and larger than that for R123 in their own optimal working temperature.
The T-h diagrams of R134a, R245fa and R123 are illustrated in Fig. 8. The same analysis results can be obtained for three different working fluids. The magnitude of the slope h T indicates the degree of influence of measurement error of the temperature on enthalpy. The positive or negative slope indicates whether the enthalpy error is positively or negatively related to the measurement error of the temperature. As can be observed, the slope of each point on the gas and liquid line was positive. The working fluid was in gaseous state at the inlet and outlet of the expander, and the slope at the expander's inlet was larger than that at the expander's outlet. This confirms the aforementioned conclusion that the measurement error of expander's inlet temperature has a more significant impact on the enthalpy as compared to that of the expander's outlet temperature. Moreover, the slopes h T were similar at the fluid pump's inlet and outlet where the fluid was in liquid state, indicating that the influences of the measurement errors of the temperature on enthalpy were similar at the inlet and outlet of the fluid pump.

Effect of the measurement error of pressure on the performance evaluation of the system
Subsequently, variations of relative errors in the performance parameters, caused by the relative measurement errors of pressure at the inlets and outlets of the expander and the fluid pump, are shown in Figs. 9, 10, 11, 12. The enthalpy errors were negatively correlated with the measurement errors of the pressure at the inlet and outlet of the expander; however, they were positively correlated with those at the inlet and outlet of the fluid pump. The errors of net output power and thermal efficiency were negatively correlated with the measurement errors of the pressure at the expander's inlet and fluid pump's outlet; however, they were positively correlated with those at the expander's outlet and fluid pump's inlet. Considering Case 1 as an example, when the measurement error of pressure was + 10%, the errors of enthalpy were − 3.50% and − 0.45% at the inlet and outlet of the expander, respectively, while the corresponding values were 0.0003% and 0.0086% at the inlet and outlet of the fluid pump, respectively. The errors of net output power were − 100.57% and 12.33% corresponding to the measurement error + 10% of pressure at the inlet and outlet of the expander, respectively, whereas the values were 0.00459% and − 0.13489% corresponding to those at the inlet and outlet of the fluid pump, respectively. The errors of thermal efficiency were − 100.62% and 12.33% corresponding to the measurement error + 10% of pressure at the inlet and outlet of the expander, respectively, while the values were 0.00459% and − 0.12453% corresponding to those at the inlet and outlet of the fluid pump, respectively. When the measurement error of pressure is positive, a negative error of performance parameter means that if the measured value is greater than the real value, the performance parameters obtained will be less than the real value. In order to keep the errors of net output errors of pressure at the inlet and outlet of the fluid pump had little influence on the corresponding enthalpies, which were not sensitive to the pressure changes of liquid fluid. The results indicated that the performance parameters were sensitive to the measurement error of the pressure at the inlet of the expander, followed by the outlet of expander. With regard to Case 2, in order to keep the relative errors of net output power and thermal efficiency within ± 10%, the relative measurement errors of pressures should be less that ± 6.0% and ± 70.0% at the inlet and outlet of the expander, respectively. The corresponding absolute errors were ± 0.11 MPa and ± 0.12 MPa, respectively. In pursuit of the same goal as in Case 3, the relative measurement errors of pressures should be less than ± 10.0% and ± 100.0% at the inlet and outlet of the expander, respectively. Moreover, the corresponding absolute errors were ± 0.11 MPa and ± 0.11 MPa, respectively. Similar to Case 1, the measurement errors of the pressures at the inlet and outlet of the fluid pump had little influence on the corresponding enthalpies in Case 2 and 3. Comparing the Case 1-3 with each other, the allowable measurement error of the pressure for R245fa was larger than that for R134a, whereas it was less than that for R123 in their own optimal working temperature. The working fluid operating in higher working temperature range allowed larger measurement error of pressure.
The p-h diagrams of R134a, R245fa and R123 are illustrated in Fig. 13. The same analysis results can be obtained for three different working fluids. The magnitude of the slope h p indicates the degree of influence of measurement error of pressure on the enthalpy. The positive or negative slope indicates whether the enthalpy error is positively or negatively correlated to the measurement error of the pressure. Obviously, the slopes of both the inlet and outlet of expander were negative, while those of both the inlet and outlet of fluid pump were positive and tended to be zero. Furthermore, the slope at the expander's inlet was greater than that at the expander's outlet, suggesting that the inlet of the expander was the key position of measurement accuracy affecting performance evaluation for both the temperature and the pressure measurements.

Effect of the measurement error of volumetric flow on the performance evaluation of the system
The effect of the relative measurement error of volumetric flow rate on the performance evaluation of the system is taken into consideration. Considering Case 1 as an example, the influences of the measurement error of volumetric flow rate at the outlet of the fluid pump on the relative errors of the performance parameters are shown in Fig. 14.
Assuming that other parameters were measured accurately, the error of net output power was proportional to the measurement error of the volumetric flow rate. However, the error of thermal efficiency was irrelevant to the measurement error of volumetric flow rate. When the measurement error of the volumetric flow rate is + 5%, the errors of net output power and thermal efficiency are + 5% and 0% respectively. If the relative error of net output power did not exceed ± 10%, the relative measurement error of the volumetric flow rate should remain within ± 10.0%. As for the absolute measurement error of the volumetric flow rate, it should be less that ± 1.07 m 3 /h in Case 1.

Comparison of the effects of measurement error on the performance evaluation among different working fluids
The effect difference in measurement error among different working fluids operating within the same temperature range is further investigated based upon Case 4 and 5. The working temperature ranges of Cases 4 and 5 were set similar to Case 1; however, the working fluid was changed from R134a (Case 1) to R245fa (Case 4) and R123 (Case 5). The influences of the relative measurement error on the relative error of net output power in Cases 1-5 are shown in Fig. 15. It is worth mentioning that the measurement errors of the pressure at the inlet and outlet of the fluid pump are not taken into consideration due to their negligible effects on the system performance. Besides, the allowable relative measurement errors of the operating parameters in Cases 1-5 are listed in Table 6 assuming that the relative error of net output power is within 10%. The results show that there is a significant difference in the change of relative error of net output power with the measurement errors of operating parameters among the five cases. The comparison of Cases 1, 4 and 5 shows that the properties of working fluid dominate this huge difference. Figure 15 shows that most of the relative error of net output power caused by the relative measurement errors of the operating parameters of R245fa, is less than that of R134a, whereas it is larger than that of R123 under the same working temperature range (comparison of Cases 1, 4 and 5). This means that the allowable relative measurement errors of the operating parameters of R245fa are larger than R134a and less than R123 under the same working temperature range. The only exception is the expander's output temperature, whose measurement error causes larger error of net output power of R245fa than R134a, as shown in Fig. 15c. Therefore, the allowable relative measurement error of expander output temperature of R245fa was smaller than R134a under the same working temperature range. As mentioned in Sect. 3.1, the fluid adapted to higher working temperature range allowed larger measurement errors of the operating parameters. It can be found that the higher the operating parameter at the inlet of expander, the lower the allowable measurement error for the same fluid (based upon the comparison of Cases 2 and 4 or Cases 3 and 5 in Fig. 15a, b and Table 6). This is due to the reason that, when the fluid was in high-temperature gaseous state, the higher the operating parameters, the more susceptible the enthalpy was to temperature or pressure, as shown in the T-h and p-h diagrams in Figs. 8 and 13. Figure 15 indicates that the critical measurement position of measurement error affecting the performance evaluation is the same for different fluids under the same working temperature range. The changes of fluid type and operating parameters did not change the critical measurement position. However, the change in fluid type and operating parameter affected the allowable measurement error. The pressure and temperature affect the performance parameters mainly through enthalpy. The pressure at the expander's outlet is significantly lower than that at the expander's inlet. As shown in the p-h and T-h diagrams in Figs. 8 and 13, the slope h p at the expander's outlet is far less than that at the expander's inlet and the slope h T at the expander's inlet and outlet. This means the variation of enthalpy caused by the variation of expander's outlet pressure is far less than that of other operating parameters. Thus, the measurement error of expander's outlet pressure has a less impact on the enthalpy as compared to that of other operating parameters. Furthermore, as mentioned in Sect. 2.2, the net output power is theoretically related to the enthalpy differences in the expander. As a result, the allowable relative measurement error of expander's outlet pressure is much greater than that of other operating parameters while keeping relative error of the net output power. Therefore, the allowable relative measurement error for P2 is relatively higher in Table 6. To sum up, the lower accuracy pressure sensors with less cost can be used for the measurement of expander's outlet pressure.

Combined effect of measurement errors of multiple parameters on the performance evaluation of the system
In the previous sections, the influence of the measurement error of a single operating parameter on the performance evaluation of the system was explored. In practice, the measurement errors of multiple parameters exist simultaneously. Figure 16 shows the variation of relative errors of the performance parameters with the combined relative measurement errors of temperature and pressure at the same position. It is worth noticing that the measurement errors of temperature and pressure may have an adverse effect on the performance parameters. Therefore, the absolute value of relative error was used to represent the comprehensive effect without considering the fact if the error was positive or negative. In order to simplify the simulation, the change intervals of the measurement errors of the temperature and pressure at the same position were kept consistent. Considering Case 1 as an example, in order to keep the errors of net output power and thermal efficiency within 10%, the measurement errors of both the temperature and the pressure should not exceed 0.5% each at the inlet of the expander. The corresponding absolute errors were within 0.48℃ and 0.016 MPa, respectively. These allowable measurement errors were stricter than those of the single parameter in Case 1. Similar results were obtained for the other three positions. The allowable measurement errors of temperature and pressure were 3% at the outlet of the expander and 4% at the inlet and outlet of the fluid pump. The result is consistent with the analysis of measurement error of single operating parameter that states that the data accuracy at the expander's inlet should be guaranteed first, followed by the expander's outlet, and the inlet and outlet of the fluid pump.
In practice, measurement error is ubiquitous during the measurement of all the parameters. The relative errors of performance parameters are presented in Table 7 when the relative measurement errors of all operating parameters exist simultaneously. The relative measurement errors of temperature, pressure and volumetric flow rate were assigned different values according to the accuracy of measuring instruments. The color shades were used to indicate the relative errors of net output power and thermal efficiency. The darker the color, the smaller the relative errors, indicating that the combination of measurement errors in this circumstance was more acceptable. It is clearly shown that, based on the influence on system performance evaluation, the importance of measurement accuracy of the temperature outweighs that of the pressure and the volumetric flow rate. When the relative measurement error of volumetric flow rate changed from 0.1 to 1.0%, the relative errors of net output power and thermal efficiency were negligible. Comparing Case 1, 2, and 3 in Table 7, the fluid adapting to higher operating temperature also allowed larger measurement errors in the operating parameters under the circumstance of combined measurement errors.

Selection of measuring instrument accuracy
In this section, recommendations are made about the suggestion about the accuracy of the measuring instruments based on the analysis of measurement errors. As discussed above, the inlet of the expander is the most critical position for measurement error of operating parameters affecting the performance evaluation, followed by the expander outlet, and the inlet and outlet of the fluid pump. Therefore, the highly accurate sensors are required for the measurements of both temperature and pressure at the expander's inlet. In addition, compared to the pressure, the temperature sensors require higher measurement accuracy. The expander's inlet temperature for the ORC system is lower than 400 °C. Therefore, PT100 thermal resistance and T-type thermocouple with high accuracy are a better choice for this purpose. The measurement errors of pressure at the inlet and outlet of the fluid pump have little influence on the performance evaluation of the system. The low accuracy pressure sensors with less cost can be selected. The performance evaluation is significantly influenced by the measurement error of mass flow rate. Volumetric flow meter is a choice considering both price and accuracy. However, the combined measurement errors of temperature and pressure would lead to an error in the density of working fluid, thereby resulting in an error in the mass flow rate. Therefore, Coriolis mass flow meter is preferred in this case. Coriolis mass flow meter measures directly the true mass flow rate with the advantages of high accuracy, rangeability, repeatability and independency on fluid properties. Coriolis mass flow meter works on the principle that fluid generates acting force to the measuring tube, and the size of this acting force reflects the change of mass flow. The acting force is named as Coriolis force which is the combined effect of the linear velocity of the fluid and angular velocity of tube. But Coriolis mass flow meter has been found to underread the mass flow rate in laminar flow region [49]. The tube section experiences a reduced effect of Coriolis force due to overcoming a secondary oscillatory shear force generated in a laminar regime. In addition, the pressure loss of Coriolis mass flow meter is as many as and even more than that of the volumetric flow meter. As mentioned above, the fluid adapting to higher working temperature range allows larger measurement errors of the operating parameters, which reduces the requirement of highly accurate sensor. However, for the same fluid, the allowable measurement errors at the expander's inlet decrease as the temperature and pressure of fluid increase, which increases the requirement for sensor's accuracy.

Conclusions
In order to provide the basis for the selection of the accuracy of measuring instruments, the influences of the measurement errors of the operating parameters on the performance evaluation of the system are investigated. The error propagation model of a subcritical ORC system was established. The measurement parameters studied included temperature, pressure and flow rate. The parameters for the performance evaluation included net output power and thermal efficiency. Five cases were introduced to compare the effect of different measurement errors on the performance evaluation of the system under different working temperature ranges and different working fluids. The key conclusions are drawn as follows.
1) The most critical position of the measurement error affecting the performance evaluation of the system is the inlet of the expander for both temperature and pressure, followed by the expander outlet, and the inlet and outlet of the fluid pump.
2) The importance of measurement accuracy of the temperature outweighs those of the pressure and the flow rate based on the influence on system performance evaluation.
3) The working fluid adapting to higher temperature range allows larger measurement error of both temperature and pressure. Therefore, the sensors with lower accuracy can be selected. For the same fluid, the higher the operating parameters at the inlet of expander, the lower the allowable measurement error, indicating that the requirement of an accurate sensor increases. 4) In most of the operating parameters, the allowable measurement errors of R245fa were larger than R134a and less than R123 under the same working temperature range. 5) The validation of experiment and simulation of the research results will be considered in our future studies.