Michael Schmiechen. The quasi-steady 'model' propulsion test: results of its further advanced evaluation

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1 mod_7_effs_mcd_15_exz.xmcd / 1 Michael Schmiechen The quasi-steady 'model' propulsion test: results of its further advanced evaluation 7. Identify parameters of partial efficiencies Stand-alone test-programme demonstrating the usage and the fundamental defect of the Minerr function in PTC Mathcad Identify basic parameters Propulsive efficiency identified in the preceding programme p RS p RS1 j HC h RS p RS j HC p RS Advance ratios at the stationary conditions j last

2 mod_7_effs_mcd_15_exz.xmcd / 2 Values of the propulsive efficiency at the stationary conditions to be approximated η RSj η RS h RS p RS j The approximation function h RS.V p V j HC 1 p V1 p V2 p V3 j HC p V p V3 j HC p V4 p V5 j HC 2 1 p V p V2 p V3 j HC Raw start values of the physical parameters to be identified crudely estimated as under Mathcad 8.3 p raw of nominal wake fraction of nominal thrust deduction fraction of parameters of linear approximation of propeller jet efficiency of parameters of linear approximation of pump hydraulic efficiency Given = h RS.V p raw j HC Minerrp raw h RS p RS j HC H P j HC As has been repeatedly observed and communicated, the built-in function ERR available in the Numerical Recipes does no longer exist

3 mod_7_effs_mcd_15_exz.xmcd / 3 The parameters identified at the stationary conditions p 15.i j H P j i last p 15.i p 15.j.T p 15.i T Mean values of parameters displayed to only five digits for ready comparison p 15i mean p i 15.j.T manually copied from pdf file p p Conclusions This result confirms my repeated observation and communication, that the Minerr function in Mathcad 15 does not work properly, does not arrive at the global minimum, as did the Minerr function in Mathcad 8.3 starting from the same rather crude values guessed! The reason for this failure maybe due to 'lack' of the ERR function mentioned before. The consequence of the failure of the crucial routine is, that the following and further results are useless! Corrected and comments revised and completed January 29, 216.

4 mod_7_effs_mcd_15_exz.xmcd / 4 Comparison of approximation errors at the stationary conditions for the versions of Minerr under Mathcad 15 and 8.3 The values of the approximate functions at the stationary conditions i last η RS.15.rawi h RS.V p 15 i η RS.8.3i h RS.V p 8.3 i The differences of the approximate values from the original values approximated Δη RS.15.raw η RS.15.raw η RS Δη RS.8.3 η RS.8.3 η RS Joint plot of the differences Δη RS.8.3 Δη RS.15.raw 'Global error 15.raw' ERR 15.rawi ERR 15.raw.m ERR i mean ERR 15.raw ERR 15.raw.m Differences of differences ΔΔη RS.raw Δη RS.15.raw Δη RS ΔΔη RS.raw Evidently the results obtained using the function Minerr in PTC Mathcad 15 still deviate systematically from the results obtained using Mathcad 8.3, care having been taken to compare values arrived in exactly the same way.

5 mod_7_effs_mcd_15_exz.xmcd / 5 Relative systematic errors Δη RS.15.raw.reli Δη RS.15.rawi η RSi.2.1 Δη RS.15.raw.rel %.1 These very small systematic errors of less than.1 % cannot be avoided, but 'look' acceptable..2 Final check after further in-depth pondering during a walk Initial values of parameters to be identified, now the results of the solution under Mathcad 8.3 Given = h RS.V p 8.3 j HC Minerrp 8.3 h RS p RS j HC H P j HC The parameters identified at the mean advance ratio p 15.i j H P j i last p 15.i p 15.j.T p 15.i T Mean values of parameters displayed to only five digits for ready comparison p 15i mean p i 15.j.T manually copied from pdf file p p

6 mod_7_effs_mcd_15_exz.xmcd / 6 This result clearly demonstrates, that the Minerr function of PTC Mathcad 15 does not return to its former 'optimal' solution, and thus is inacceptable The values of approximate functions at the stationary conditions i last η RS i h RS.V p 15 i The differences of the approximate values from the original values approximated Δη RS η RS η RS Joint plot of the differences 41 4 Δη RS.8.3 Δη RS 'Global error ' ERR i ERR i ERR m mean ERR ERR m The 'global' errors according to PTC Mathcad 15 ERR 15.raw.m ERR m clearly show that the initial solution has not been optimal. Starting from the correct initial values reduces the error by two orders of magnitude. The 'global' error provided by the built-in function in Mathcad 8.3 had even been two orders of magnitude less ERR The differences of the final results may be due to different definitions.

7 mod_7_effs_mcd_15_exz.xmcd / 7 Differences of differences ΔΔη RS Δη RS Δη RS mean ΔΔη RS These remaining differences may be due to rounding of the parameters originally identified. Evidently the results obtained using the function Minerr in PTC Mathcad 15, starting from the correct initial values provided by Mathcad 8.3 (!), no longer deviate substantially from the results obtained using Mathcad 8.3, after care has been taken to compare only values arrived at in 'exactly' the same way as far as possible now. Relative systematic errors Δη RS reli Δη RS i η RSi Δη RS rel % Δη RS.15.raw.rel % These systematic errors of magnitudes less than.5 %, i. e. five times larger than at the local minimum, cannot be avoided, but still 'look' acceptable. At a later stage they may possibly serve to correct the results, if found considerably to affect the results of the subsequent analysis.

8 mod_7_effs_mcd_15_exz.xmcd / 8 Conclusions This concludes my work of five months to convince many experts of PTC, that the Minerr function in PTC Mathcad 15 is inherently faulty, does not work properly in case of my admittedly extremely delicate, ill-conditioned problem and thus urgently needs to be replaced by the original function, which evidently coped with the problem and provided meaningful physical values of the parameters, in question before PTC Case C can be closed. An important observation is, that in the example under investigation the relative systematic errors of the approximate solution at the global minimum are considerably larger than at the local minimum. The lesson learned is thus, that the relative systematic errors of an approximate solution provide no valid optimum criteria. For the time being I shall up-date the warning to my colleagues on my website according to these findings. Further, I wonder who is going to pay me for my work. END 7. Identify paramet ers of partial efficiencies etc