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- April 25, 2011, 04:08:30 AMR410A and R407C simplified calculation of thermodynamic properties
Abstract: implicit polynomial fitting of the three main alternative refrigerant R22 R410A and R407C of the thermodynamic properties, given the form of a unified thermodynamic properties of refrigerant simplified model to analyze the implicit fitting process appears bifurcation problem and proposed solutions to further improve the model of consistency and stability. Compared with the reference model, the model in the saturation region the maximum absolute relative error is 0.19, the average error is 0.07, overheated zone the maximum absolute relative error of 0.61, the average error is 0.18, calculated the average speed an order of magnitude for product design based on computer-aided design and optimization. Keywords: thermodynamic properties of refrigerant R410A R407C a variety of refrigeration and air conditioning industry, exploration and research shows that mixture in the substitution of refrigerant has great potential, which R410A and R407C as a substitute for R22 is much attention. In order to study their impact on the existing refrigeration system, computer simulation is a good means. and thermal properties of the foundation program as part of the simulation program, the simulation efficiency and results have a considerable impact. but the domestic little research in this area, the general equation of state directly to the use of complex iterative calculations, so a certain extent, reduce the simulation speed and stability. To compensate for this shortfall , we use the implicit explicit calculation of the fitting method, refer to DuPont's data, the thermodynamic properties of R410A and R407C re-fitting the results to avoid iteration, a significant increase in calculation speed while ensuring the required accuracy . 1 In this paper, based on fitting model literature [1,2] prepared to simplify the thermodynamic nature of procedures, and to fit the thermodynamic properties of the reference process as a data source and test the relative accuracy of simplified thermal properties . Saturated thermodynamic properties of the simplified model used in [3] proposed fitting function form: <P style=thermodynamic properties, [4] did not give the complete reunification of the form fitting function. overheated zone of a simplified model of a unified form, in order to reduce the model fit and complexity, the specific form: <P style = C to ensure accuracy in the temperature range of 20 ° C extrapolated trends within the guarantee to ensure the correct simulation carried out. 2 implicit fitting bifurcation problems and solutions fitting implicit The biggest problem is the bifurcation problem. In the regression equation implicit equations for calculating the conversion of explicit when it comes to the root of the discrimination problem. cubic equation involving three roots, representing a different root path. but in the end may be needed is one of the one or more root-locus combination. However, the interface between the root locus will appear at a small fracture, which is the bifurcation phenomenon. breaks in called the bifurcation point, which caused the curve is not continuous (bifurcation point error is large) and the curve is not smooth (bifurcation point first derivative is not continuous). Although the bifurcation point is small, in most cases, a simulation model of the refrigeration system does not have a big impact, but it is a huge the risks (which may lead to abnormal calculated values simulation model). through a series of attempts, without changing the premise of the model proposed a solution: by changing the process of fitting the data to adjust the fit, fitting out of the scope of the bifurcation point, while ensuring that the fitting precision. mathematical principle of this method is to fit the data points by changing the fitting function of the curvature can change. Therefore, as long as the fitting function to fit within the curvature reduced, would enable the bifurcation point (ie, the greatest change in curvature points) relocation. By this means the scope of the model there is no bifurcation in the fit issue, pushing out a few models within the range of bifurcation, but this Common conditions within the refrigeration and air conditioning system simulation has no effect. specific methods: 1 change fitting range, the range is usually fitted extended the range to ensure the accuracy of fit; 2 change the fit of the points, in most cases is to reduce the number of points ; 3 with the fitting of non-uniform data points (an increase of the number of data within a region). In contrast, the position 1 on the bifurcation point of maximum 3 minimum. from the current situation, fitting the range of data, points and distribution of the choice of different thermal parameters are different, depends largely on the experience.
3 Comparison of fitting results with the calculation speed fitting results shown in Table 1 to 3 below. in Table 1 and Table 3, e1 and e * 1, respectively, in the fitting and extrapolation within the range of known temperature T, using the function f (x, T) Find the relative maximum of x absolute error; e2 and e * 2, respectively, in the fitting and extrapolation within the scope of the known properties of x, using the function f (x, T) find the temperature T, the absolute value of the maximum relative error; table T0 and x0 is the left edge of the scope of the actual fitting; fitting coefficients for x in order to prevent too large or too small and difficult to calculate a result of this paper is not directly using the basic international units, but to a certain scale, a scale in the unit column shows, for example, in Table 1 ρL of units (× 103kg/m3), which shows the table multiplied by the coefficient is ρL 10-3 in the post-fitting results. subscript v said saturated steam, L said saturated liquid . in Table 2, e1 and e * 1, respectively, in the fitting and extrapolation within the range, known pressure p and temperature T, find hot gas specific volume v, enthalpy or entropy h value s, the maximum relative error of the absolute value; e2 and e * 2, respectively, in the fitting and extrapolation within the range of known thermal gas pressure p and specific volume v, enthalpy h, or entropy s, find hot gas temperature T, the absolute value of the maximum relative error. Table 4 for the simplified model and the results of the calculation speed of the reference model comparison. In order to highlight the superiority of the simplified model, so there is no accurate model for the iterative calculation of the function does not to be compared, but only some of the more typical function selected to compare. In order to accurately measure the calculation speed,You are not allowed to view links. Register or Login, each function is called thousands of times, came to the conclusion of each call to the average time required by the table shows, the simplified model advantages in terms of speed is obvious. 4 Conclusion established R410A and thermal properties of R407C simplified model and reference materials to compare the relative error in the saturated zone is the maximum absolute value of 0.19%, the average error was 0.07%; in the hot zone is the maximum absolute relative error of 0.61%, the average error 0.18%. As a result of implicit fitting method, while the range in the fitting to avoid the bifurcation point, so the simplified model performed well in every respect, in particular, to avoid the iterative calculations because of greatly increased computing speed, while also ensuring the stability of the calculation. References [1] Thermodynam ic properties of Suva 9100 refrigerant (R410A) [R]. DuPont Technical Inform ation, 1996. [2] Thermodynamic properties of Suva 9000refrigerant (R407C) [R]. DuPont Technicalnform ation, 1995. [3] Zhang Road, Ding Guoliang, Li Hao. refrigerant thermodynamic properties of saturated implicit fitting methods [ ,],[. the Chinese Society of Engineering Thermophysics of thermodynamics and energy utilization Annual Conference Proceedings [C]. Zhenjiang, 1999.91 ~ 95.
Abstract: implicit polynomial fitting of the three main alternative refrigerant R22 R410A and R407C of the thermodynamic properties, given the form of a unified thermodynamic properties of refrigerant simplified model to analyze the implicit fitting process appears bifurcation problem and proposed solutions to further improve the model of consistency and stability. Compared with the reference model, the model in the saturation region the maximum absolute relative error is 0.19, the average error is 0.07, overheated zone the maximum absolute relative error of 0.61, the average error is 0.18, calculated the average speed an order of magnitude for product design based on computer-aided design and optimization. Keywords: thermodynamic properties of refrigerant R410A R407C a variety of refrigeration and air conditioning industry, exploration and research shows that mixture in the substitution of refrigerant has great potential, which R410A and R407C as a substitute for R22 is much attention. In order to study their impact on the existing refrigeration system, computer simulation is a good means. and thermal properties of the foundation program as part of the simulation program, the simulation efficiency and results have a considerable impact. but the domestic little research in this area, the general equation of state directly to the use of complex iterative calculations, so a certain extent, reduce the simulation speed and stability. To compensate for this shortfall , we use the implicit explicit calculation of the fitting method, refer to DuPont's data, the thermodynamic properties of R410A and R407C re-fitting the results to avoid iteration, a significant increase in calculation speed while ensuring the required accuracy . 1 In this paper, based on fitting model literature [1,2] prepared to simplify the thermodynamic nature of procedures, and to fit the thermodynamic properties of the reference process as a data source and test the relative accuracy of simplified thermal properties . Saturated thermodynamic properties of the simplified model used in [3] proposed fitting function form: <P style=thermodynamic properties, [4] did not give the complete reunification of the form fitting function. overheated zone of a simplified model of a unified form, in order to reduce the model fit and complexity, the specific form: <P style = C to ensure accuracy in the temperature range of 20 ° C extrapolated trends within the guarantee to ensure the correct simulation carried out. 2 implicit fitting bifurcation problems and solutions fitting implicit The biggest problem is the bifurcation problem. In the regression equation implicit equations for calculating the conversion of explicit when it comes to the root of the discrimination problem. cubic equation involving three roots, representing a different root path. but in the end may be needed is one of the one or more root-locus combination. However, the interface between the root locus will appear at a small fracture, which is the bifurcation phenomenon. breaks in called the bifurcation point, which caused the curve is not continuous (bifurcation point error is large) and the curve is not smooth (bifurcation point first derivative is not continuous). Although the bifurcation point is small, in most cases, a simulation model of the refrigeration system does not have a big impact, but it is a huge the risks (which may lead to abnormal calculated values simulation model). through a series of attempts, without changing the premise of the model proposed a solution: by changing the process of fitting the data to adjust the fit, fitting out of the scope of the bifurcation point, while ensuring that the fitting precision. mathematical principle of this method is to fit the data points by changing the fitting function of the curvature can change. Therefore, as long as the fitting function to fit within the curvature reduced, would enable the bifurcation point (ie, the greatest change in curvature points) relocation. By this means the scope of the model there is no bifurcation in the fit issue, pushing out a few models within the range of bifurcation, but this Common conditions within the refrigeration and air conditioning system simulation has no effect. specific methods: 1 change fitting range, the range is usually fitted extended the range to ensure the accuracy of fit; 2 change the fit of the points, in most cases is to reduce the number of points ; 3 with the fitting of non-uniform data points (an increase of the number of data within a region). In contrast, the position 1 on the bifurcation point of maximum 3 minimum. from the current situation, fitting the range of data, points and distribution of the choice of different thermal parameters are different, depends largely on the experience.
3 Comparison of fitting results with the calculation speed fitting results shown in Table 1 to 3 below. in Table 1 and Table 3, e1 and e * 1, respectively, in the fitting and extrapolation within the range of known temperature T, using the function f (x, T) Find the relative maximum of x absolute error; e2 and e * 2, respectively, in the fitting and extrapolation within the scope of the known properties of x, using the function f (x, T) find the temperature T, the absolute value of the maximum relative error; table T0 and x0 is the left edge of the scope of the actual fitting; fitting coefficients for x in order to prevent too large or too small and difficult to calculate a result of this paper is not directly using the basic international units, but to a certain scale, a scale in the unit column shows, for example, in Table 1 ρL of units (× 103kg/m3), which shows the table multiplied by the coefficient is ρL 10-3 in the post-fitting results. subscript v said saturated steam, L said saturated liquid . in Table 2, e1 and e * 1, respectively, in the fitting and extrapolation within the range, known pressure p and temperature T, find hot gas specific volume v, enthalpy or entropy h value s, the maximum relative error of the absolute value; e2 and e * 2, respectively, in the fitting and extrapolation within the range of known thermal gas pressure p and specific volume v, enthalpy h, or entropy s, find hot gas temperature T, the absolute value of the maximum relative error. Table 4 for the simplified model and the results of the calculation speed of the reference model comparison. In order to highlight the superiority of the simplified model, so there is no accurate model for the iterative calculation of the function does not to be compared, but only some of the more typical function selected to compare. In order to accurately measure the calculation speed,You are not allowed to view links. Register or Login, each function is called thousands of times, came to the conclusion of each call to the average time required by the table shows, the simplified model advantages in terms of speed is obvious. 4 Conclusion established R410A and thermal properties of R407C simplified model and reference materials to compare the relative error in the saturated zone is the maximum absolute value of 0.19%, the average error was 0.07%; in the hot zone is the maximum absolute relative error of 0.61%, the average error 0.18%. As a result of implicit fitting method, while the range in the fitting to avoid the bifurcation point, so the simplified model performed well in every respect, in particular, to avoid the iterative calculations because of greatly increased computing speed, while also ensuring the stability of the calculation. References [1] Thermodynam ic properties of Suva 9100 refrigerant (R410A) [R]. DuPont Technical Inform ation, 1996. [2] Thermodynamic properties of Suva 9000refrigerant (R407C) [R]. DuPont Technicalnform ation, 1995. [3] Zhang Road, Ding Guoliang, Li Hao. refrigerant thermodynamic properties of saturated implicit fitting methods [ ,],[. the Chinese Society of Engineering Thermophysics of thermodynamics and energy utilization Annual Conference Proceedings [C]. Zhenjiang, 1999.91 ~ 95.