Uncertainty of measurements of podded propulsor performance characteristics

Ocean Engineering 81 (2014) 130–138

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Uncertainty of measurements of podded propulsor performance characteristics Mohammed Islam a,n, Brian Veitch b, Pengfei Liu c a

Oceanic Consulting Corporation, 95 Bonaventure Avenue, Suite 401, St. John's, NL, Canada A1B 2X5 Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John's, NL, Canada c Oceanic, Coastal and River Engineering, National Research Council, Canada b

art ic l e i nf o

a b s t r a c t

Article history: Received 5 March 2013 Accepted 17 February 2014 Available online 14 March 2014

One of the fundamental aspects of any physical experiment is the uncertainty or error limits of the measurement. Unfortunately, majority of published model experimental results do not come with the much needed error analysis. It is imperative that a fundamental and rigorous uncertainty assessment is carried out for all measurements, especially for measurements using a newly designed apparatus. This paper presents the uncertainty analysis methodology and results for a newly designed fully functional podded propulsor performance measurement apparatus. The measurements include propeller thrust, torque, rotation rate and advance speed as well as global forces and moments of a pod unit. The facility and measurement systems are briefly described, and detailed uncertainty assessment methodologies with examples for each measurement are provided with descriptions of bias and precision limits and total uncertainties. The generalized methodology can also be used for other relevant measurements. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Uncertainty analysis Error limits Bias error Precision error Podded propulsors Measurement uncertainty

1. Introduction Experimental uncertainty analysis (UA) has evolved over the last 50 years from formalization (Kline and Mcclintock, 1953; Abernethy et al., 1985; Coleman and Steele, 1999) standard procedures (ASME, 1998; AIAA, 1999; ITTC, 2002a) with emphasis on simplification and practical application. However, consistent and uniform use both in research and design at university, industry, and government laboratories continues to be a problem. As with fields of mechanical and aerospace engineering, ship hydrodynamics has also struggled to rectify this situation. Experimental uncertainty estimates are imperative for risk assessments in design when using data directly or in calibrating and/or validating simulation methods (Longo and Stern, 2005). The work by Bose and Luznik (1996) offered the first detailed methodology to assess the uncertainty in the classic open water propulsive characteristics measurements. An in-depth discussion on each of the sources of error in the measurements and their calculation is provided. Taylor (2006) used the methodology presented by Bose and Luznik (1996) to assess the uncertainties in the measurements using an apparatus to measure podded propulsors performance characteristics. In the analysis, the author

n

Corresponding author. Tel.: þ 1 709 722 9060x13; fax: þ1 709 722 9064. E-mail addresses: [email protected], mfi[email protected] (M. Islam). http://dx.doi.org/10.1016/j.oceaneng.2014.02.009 0029-8018 & 2014 Elsevier Ltd. All rights reserved.

presented the uncertainty in propeller performance and pod unit thrust without any discussion on the uncertainty in the unit forces and moments measurements. Islam et al. (2007) presented results of an uncertainly analysis for the measurements of a podded propulsor's forces and moments but did not present detail aspects of the analysis. An objective of the present paper is to provide detailed examples of UA for typical open water performance coefficients of podded propulsors following the most recent standard procedures and thereby also providing benchmark values for uncertainty estimates. Work is done at Ocean Engineering and Research Centre of Memorial University of Newfoundland in collaboration with National Research Council, Oceanic Consulting Corporation and Thordon Bearing, Inc. Measurements of open water propulsive performance characteristics of podded propulsors are made with varied geometry, configurations, loading conditions and azimuthing conditions. An overview is provided of the UA methodology. The podded propulsor's open water experiments are described, including datareduction equations (DREs) and definitions and measurement systems (instrumentation and data acquisition and reduction). Detailed examples of UA are provided with discussion of bias and precision limits and total uncertainties along with possible unaccounted error sources and methods for uncertainty reduction. Lastly, conclusions and recommendations are given. Hopefully, provision of such detailed analysis and benchmark values will stimulate and facilitate broader use of UA.

M. Islam et al. / Ocean Engineering 81 (2014) 130–138

131

2. Models, apparatus and experiments In the current experimental investigation, a model pod unit, named Pod 01 is used, see Fig. 1. Two propellers with the same geometry but different hub taper angle are used for the pusher and puller configurations; see Liu (2006) for details. The geometric particulars of the pod-strut model can be found in Islam et al. (2009a). The values for the model propulsor are selected to provide an average representation of in-service, full-scale single screw podded propulsors. Open water tests of a model pod unit in straight course and at static azimuth angles are performed in accordance with the ITTC recommended procedure, Podded Propulsor Tests and Extrapolation, 7.5-02-03-01.3, and the description provided by Mewis (2001). A custom-designed dynamometer system (Macneill et al., 2004) is used to measure propeller thrust, torque, and unit forces in the three orthogonal coordinate directions. With the exception of torque, all forces are measured with off the shelf load cells. Torque is measured using strain gauges. Water temperature, carriage speed, VA, and the rotational speed of the propeller shaft, n, are also measured. The definition of the forces, moments and coordinates that was used to analyze the data and present the results is shown in Fig. 2. There are two coordinate systems – one in carriage fixed and one is pod unit fixed. The carriage fixed (inertia frame) coordinate system measured the three forces FX, FY and FZ and the three moments MX, MY and MZ. In the pod fixed coordinates, propeller thrust, TProp and propeller torque Q were measured. The pod unit coordinate system rotates with the pod with its origin located at the center of the propeller shaft and center of the propeller hub. A right handed coordinate system with positive Z downward is used for the carriage fixed system. The coordinate center is located 1.68 m vertically above the pod center, which is at the intersection of the horizontal axis through the propeller shaft center and the vertical axis through the strut shaft center. The propeller thrust and torque are measured at the propeller end of the shaft. The propeller forces, pod unit forces and moments are presented in the form of traditional non-dimensional coefficients as defined in Table 1.

3. Methodology for uncertainty analysis The general recommendations and guidelines provided by the International Towing Tank Conference (ITTC, 2002b,c) for

Fig. 1. Physical model of the podded propulsor unit – Pod 01.

Fig. 2. Definitions of forces, moments, coordinates of the puller azimuthing podded propulsors. Note that the unit forces and moments FX, MZ, etc. are measured at the global dynamometer which is 1.68 m above the center of the pod.

uncertainty analysis for resistance and propulsion tests are most closely aligned with the testing techniques used to study the uncertainly in present measurement system. The methodology used in the analysis follows the recommended guidelines set out by the ITTC in combination with approaches described by Bose and Luznik (1996) as well as Coleman and Steele (1999). As described in Coleman and Steele (1999), uncertainty in a measurement consists of two major components: bias error and precision error. Bias error is a constant, systematic error in the system or process, which may be reduced through calibration, whereas, precision error is the random contribution often referred to as repeatability error, which can be reduced through the use of multiple readings. The bias errors are consisted of many elemental sources of error, which basically depended on the approaches followed to measure the variables. However, for the precision error estimates of most variables, only one source of error (repeatability) is considered significant. The coefficients used to present the performance characteristics of podded propulsor and the associated measurement quantities are presented in Table 1. The overall uncertainly in the nondimensional performance coefficients of the podded propulsors require proper identification of all the variables contained within the data reduction expressions. Thus, the variables of interest in the podded propulsors uncertainty analysis are propeller thrust, pod unit thrust, propeller torque, forces and moments on the propulsor in the three orthogonal directions, propeller shaft rotational speed, carriage advance speed, azimuthing angle, water density (function of water temperature) and propeller diameter. Fig. 3 shows a block diagram for the podded propulsor open water tests including the individual measurement systems, measurement of individual variables, data reduction and experimental results. The uncertainties in all of the variables of interests but the forces and moments mentioned above are obtained using the method provided in Bose and Luznik (1996) and Taylor (2006). A methodology to derive the uncertainties in the forces and moments is described in Section 3.1.

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M. Islam et al. / Ocean Engineering 81 (2014) 130–138

Table 1 List of performance coefficients for the podded propulsor unit. Performance characteristics

Data reduction equation

K T Prop – propeller thrust coefficient

T Prop =ρn2 D4

K T Unit – unit thrust coefficient, K T X or longitudinal force coefficient, K F X

T Unit =ρn2 D4 or F X =ρn2 D4

10KQ – propeller torque coefficient

10Q =ρn2 D5 V A =nD J=2π  ðK T Prop =K Q Þ J=2π  ðK T Unit =K Q Þ

J – propeller advance coefficient ηProp – propeller efficiency ηUnit – unit efficiency K F Y – transverse force coefficient

F Y =ρn2 D4

K F Z – vertical force coefficient

F Z =ρn2 D4

K MX – moment coefficient around x axis

10M X =ρn2 D5

K MY – moment coefficient around y axis

10M Y =ρn2 D5

K MZ – moment coefficient around z axis (Steering moment)

10M Z =ρn2 D5

Where, TProp – propeller thrust TUnit – unit thrust Q – propeller torque VA – propeller advance speed, in the direction of carriage motion

The experimental approaches used to obtain the data for each of the variables of interest are influenced by a variety of elemental sources of error. These elemental sources are estimated, and combined using a root mean square method to give the bias and precision limits for each of the variables. While it is possible to identify many elemental error sources for bias limits, there is primarily only one source of error identified for precision limits for most variables. The error estimates used in this study are considered to be a 95% coverage estimates. The elemental sources of error for each of the variables that comprises the corresponding total bias errors of the corresponding variables are provided in tabular form, see Table 2. The errors associated with the propeller thrust measurements are conveniently divided into three primary categories: Calibration errors, curve fit errors, and testing errors. The data acquisition errors influence the uncertainty levels of the calibration data, in the way that the data is used to generate the curves to convert the test data voltage signals into loading values. The data acquisition error actually influences the overall uncertainty in two ways, in that it first introduces uncertainty into the calibration data used to get the curve fits, and secondly the actual test data collected is subject to these uncertainties. Therefore, the errors associated with the data acquisition would be best treated as contributors to the error of both to the calibration and test data sets. The precision limit for the propeller thrust are estimated for several advance speeds and the data from one run is subdivided into 10 segments of sufficient length (at least 10 seconds of data acquisition time) and then applying the standard deviation theory as suggested in Bose and Luznik (1996). A similar approach is followed for propeller torque, shaft speed, and advance speed. All of the components of the bias errors of the global forces (FX, FY and FZ) in the three coordinate directions are calculated in a similar fashion as done for the propeller thrust. The only component of the bias error that is calculated differently is the least square curve fit error. Similarly, all of the components of the bias errors of the global moments (MX, MY and MZ) in the three coordinate directions are calculated in a similar fashion as done for the propeller torque except the least square curve fit error. The theoretical background to calculate the curve fit error for the sixcomponent global dynamometer is described in Section 3.1. The bias uncertainty and the precision uncertainty are combined using the root-sum-square (RSS) method shown in the following equation to provide estimates of overall uncertainty levels in these variables. The overall uncertainly is thus considered

ρ – water density n – propeller rotational speed D – propeller diameter FX,Y,Z – components of the hydrodynamic force on the pod MX,Y,Z – components of the hydrodynamic moment on the pod

to be a 95% coverage estimate: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U v ¼ B2v þ P 2v

ð1Þ

The final step in the methodology of uncertainly analysis is to determine how uncertainties in each of the variables propagate through the data reduction equations, see Table 1. Using the approaches described by Bose and Luznik (1996) and Coleman and Steele (1999) the uncertainty expressions for each set of experiments are developed as shown in the following equations: !  2  2 U K T Prop 2 U T Prop 2 U ρ 2 Un UD ¼ þ þ4 þ 16 ð2Þ K T Prop T Prop ρ n D  2  2  2  2   UKQ 2 UQ Uρ Un UD ¼ þ þ4 þ25 KQ Q ρ n D UKT

!2 Unit

K T Unit

 ¼

U T Unit T Unit



2 þ

Uρ ρ

2 þ4

 2  2 Un UD þ 16 n D

ð3Þ

ð4Þ

 2  2      2 UFY 2 UFY 2 Uρ Un UD ¼ þ þ4 þ 16 K FY FY ρ n D

ð5Þ

 2  2  2     UFZ 2 UFZ 2 Uρ Un UD ¼ þ þ4 þ 16 K FZ FZ ρ n D

ð6Þ

 2  2    2 U MX 2 Uρ Un UD þ þ4 þ 25 MX ρ n D

ð7Þ

 2  2  2     U K MY 2 U MY 2 Uρ Un UD ¼ þ þ4 þ 25 K MY MY ρ n D

ð8Þ

 2  2  2   U MZ 2 Uρ Un UD þ þ4 þ 25 MZ ρ n D

ð9Þ

  U K MX 2 K MX

  U K MZ 2 K MZ

¼

¼

 2   2  2  UJ UVA 2 Un UD ¼ þ þ J VA n D

ð10Þ

In the expressions for the podded propulsors’ tests, it should be noted that for both thrust and torque coefficient uncertainties, the tare thrust and frictional torque are imbedded in the corresponding measurements, hence it have not been treated as an independent contributor of error to the corresponding coefficients, but

M. Islam et al. / Ocean Engineering 81 (2014) 130–138

Fig. 3. Block diagram for podded propulsor open water tests and uncertainty analysis.

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M. Islam et al. / Ocean Engineering 81 (2014) 130–138

Table 2 Bias error sources for each of the variables of interest. Bias error sources

Calibration error, loading angle error, load cell alignment error, A/D card error, curve fit error, test condition errors, equipment positioning error, static zero error, A/D card error

Equation for total bias error

BT Prop ¼ ðB2T Prop _Cal þ B2T Prop _LA þ B2T Prop _LCA þ B2T Prop _ADC þ B2T Prop _CF þ B2T Prop _TC þ B2T Prop _EPA þ þ B2T Prop _SZT þ B2T Prop _ADT Þ1=2

Propeller torque

Bias error sources

Calibration error, static zero error during calibration, A/D card error during calibration, curve fit error, test condition errors, equipment positioning error, static zero error, A/D card error BQ ¼(BQ_Cal2 þ B2Q_SZC þ B2Q_ADC þ B2Q_CF þ B2Q_TC þ B2Q_EPA þ B2Q_SZT þ B2Q_ADT)1/2

Advance speed

Bias error sources

Propeller thrust

Equation for total bias error

Equation for total bias error

Calibration error, static zero error during calibration, A/D card error during calibration, curve fit error, test condition errors, static zero error, A/D card error BV ¼ (BV_Cal2 þ B2V_ADC þB2V_Tide þ BV_CF2 þ B2V_TC þ B2V_SZC þ B2V_SZT þ B2V_ADT)1/2

Shaft speed

Bias error sources Equation for total bias error

Calibration error, A/D card error during calibration, curve fit error, test condition errors, static zero error Bn ¼ (Bn_Cal2 þB2n_ADC þB2n_CF þ Bn_TC2 þ B2n_SZC þ B2n_SZT þ B2n_ADT)1/2

Azimuthing angle

Bias error sources Equation for total bias error

CNC machining errorþ polishing errorþ tolerance errorþ equipment position error BAA ¼ (B2AA_CNC þ B2AA_Polish þ B2AA_TE þ B2AA_EP)1/2

Propeller diameter

Bias error sources Equation for total bias error

CNC machining errorþ polishing error BD ¼ (B2D_CNC þ B2D_Polish)1/2

Temperature

Bias error sources Equation for total bias error

Calibration errorþ scale errorþ temperature range error BTemp ¼ (B2Temp_Cal þ B2Temp_Scale þ B2Temp_Range)1/2

Density

Bias error sources Equation for total bias error

Calibration errorþ data reduction errorþ conceptual errorþtemperature related density error BDen ¼ (B2Den_Cal þB2Den_DataR þ B2Den_Conc þ B2Den_Temp)1/2

Viscosity

Bias error sources Equation for total bias error

Calibration errorþ data reduction error BVis ¼(B2Vis_Cal þ B2Vis_DataR)1/2

rather has been treated as a bias error on the static-zero value of the thrust and the torque measurements.

 þ

∂m Uy ∂y1 1

2

 þ

∂m Uy ∂y2 2

2

 þ

∂m Uy ∂y3 3

2

 þ⋯þ

∂m Uy ∂yN N

2 #1=2

ð13Þ 3.1. Curve fit error for six component dynamometer In order to calculate the curve fit error in a six-component dynamometer measurement, one must determine how the uncertainties in the calibration data propagates into each element of the interaction matrix and into future measured forces and moments. The calibration of the six-degree-of-freedom global dynamometer yields an N  M array of applied force and moment components, F, and an N  M array of corresponding output voltages, V (Islam, 2009b). To each element of the F and V arrays there is a corresponding bias and precision uncertainty that is determined. As described in Hess et al. (2000), the solution for interaction matrix is given in the following equation: C ¼ ðVT VÞ  1 VT F

ð11Þ

If we consider one dynamometer axis at a time (one column of C and F), then the equation can be rewritten as in the following equation: T

T

V V Ci ¼ V Fi

ð12Þ

Eq. (12) is a classic form of the normal equations for a least squares fit problem. To determine the bias and precision uncertainty propagated into each column of the interaction matrix, one must determine how uncertainty in input data propagates into the coefficients of a least squares fit. Similarly, to determine the uncertainty present in future measured forces and moments from the fit (F¼ VA), one must understand how uncertainty propagates through a least squares fit into the output. As given in Hess et al. (2000), the uncertainty propagated into the slope, m, for a linear least square fit of the form, y¼ mxþb is of the form as given in the following equation: " 2  2  2  2 ∂m ∂m ∂m ∂m U x1 þ U x2 þ U x3 þ ⋯ þ U xN Um ¼ ∂x1 ∂x2 ∂x3 ∂xN

Thus, the uncertainty in the slope depends upon the uncertainties in each of the abscissas and ordinates of the raw data used to construct the fit. Thus, one must determine the partial derivatives, ∂m=∂xi and ∂m=∂yi , which are found to be of the form given in the following equations:   ∂m Nyi  ∑yi  2m Nxi ∑xi ¼ ð14Þ   2 ∂xi N∑x2  ∑xi i

∂m Nxi  ∑xi ¼   ∂yi N∑x2  ∑xi 2

ð15Þ

i

where m¼

N N N∑N i ¼ 1 xi yi ∑i ¼ 1 xi ∑i ¼ 1 yi  N 2 N 2 N∑i ¼ 1 xi  ∑i ¼ 1 xi

For a six-degree-of-freedom dynamometer system, the uncertainty in each of the element of the interaction matrix is obtained using Eq. (13) where U x1 and U y1 are the uncertainties in the voltage and applied load measurements for each of the N loading conditions. The applied loads to the calibration frame designed specifically for the dynamometer system are converted to forces and moments in the three coordinate directions (Islam, 2009b). In the matrix form, the forces and moments and the corresponding voltage output from the six-component dynamometer are expressed as in the following equation: 2 3 F1; 1     F1; 6 6       7 6 7 6 7 7 and       F¼6 6 7 6       7 4 5 F195; 1     F195; 6

M. Islam et al. / Ocean Engineering 81 (2014) 130–138

2 6 6 6 V¼6 6 6 4

V 1; 1









V 1; 6

 

 

 

 

 

 













V195; 1









V195; 6

3

which reduces to the following equation:

7 7 7 7 7 7 5

" UFX ¼

ð16Þ

"

∂R UX ∂X 1 1

2

 þ

∂R UX ∂X 2 2

2

 þ

∂R UX ∂X 3 3

2

 þ⋯þ

∂R UX ∂X n n

ð18Þ

F Y ¼ C 21 V 1 þ C 22 V 2 þ C 23 V 3 þ C 24 V 4 þ C 25 V 5 þ C 26 V 6

ð19Þ

F Z ¼ C 31 V 1 þ C 32 V 2 þ C 33 V 3 þ C 34 V 4 þC 35 V 5 þ C 36 V 6

ð20Þ

M X ¼ C 41 V 1 þ C 42 V 2 þ C 43 V 3 þ C 44 V 4 þC 45 V 5 þ C 46 V 6

ð21Þ

M Y ¼ C 51 V 1 þC 52 V 2 þ C 53 V 3 þ C 54 V 4 þ C 55 V 5 þ C 56 V 6

ð22Þ

M Z ¼ C 61 V 1 þ C 62 V 2 þ C 63 V 3 þ C 64 V 4 þ C 65 V 5 þC 66 V 6

ð23Þ

UFX UFY UF Z U MX UM Y U MZ

∂F X ∂C 12 U C 12

2

þ



∂F X ∂C 13 U C 13

#1=2

2

ðV 1 U C 21 Þ2 þ ðV 2 U C 22 Þ2 þ ðV 3 U C 23 Þ2 þ ðV 4 U C 24 Þ2 þ ðV 5 U C 25 Þ2 þ ðV 6 U C 26 Þ2 þ

#1=2

ðC 21 U V 1 Þ2 þ ðC 22 U V 2 Þ2 þ ðC 23 U V 3 Þ2 þ ðC 24 U V 4 Þ2 þ ðC 25 U V 5 Þ2 þ ðC 26 U V 6 Þ2

U C 11 U C 21  ¼   U C 61

V1 V 2 V3 V 4 V5 V6  UV1 UV2 UV 3 UV4 UV 5 UV6

U C 12



U C 13

U C 14

U C 15

U C 16

C 11

C 12

C 13

C 14

C 15

U C 22







U C 26

C 21





























 

 

 

 

 

 

 

 

 

 

U C 62







U C 66

C 61









C 16 C 26    C 66

ð27Þ

The details of the uncertainty calculation for each component of the dynamometer are provided in Islam (2009b). The biases and precision limits are combined using RSS to determine the overall uncertainty estimates for each of the variables of interests as shown in Table 3. Substitution of the uncertainty values from Table 3 into the appropriate uncertainty equations (Eqs. (2)–(10)) yields the overall uncertainty levels for the propulsive performance coefficients

 2  2  2 3 ∂F X ∂F X ∂F X þ ∂C U C 14 þ ∂C U C 15 þ ∂C U C 16 7 14 15 16 71=2  2  2  2  2  2  2 5 ∂F X ∂F X ∂F X ∂F X ∂F X ∂F X þ ∂V U V 1 þ ∂V U V 2 þ ∂V U V 3 þ ∂V U V 4 þ ∂V U V 5 þ ∂V UV6 5 1 2 3 4 6 

2

4. Results on uncertainties

UFX ¼ þ

2

These equations and the rest four components when put in the matrix form yields, which is the final curve fit error for each of the forces and moments components as given in the following equation.

Now applying Eq. (13) to Eq. (18) yielded the following equation:

2

2

ð26Þ

2 #1=2

F X ¼ C 11 V 1 þ C 12 V 2 þ C 13 V 3 þC 14 V 4 þ C 15 V 5 þ C 16 V 6

∂F X ∂C 11 U C 11

2

ðC 11 U V 1 Þ þ ðC 12 U V 2 Þ þ ðC 13 U V 3 Þ þ ðC 14 U V 4 Þ þ ðC 15 U V 5 Þ þ ðC 16 U V 6 Þ

" UFY ¼

In the present case of the six component dynamometer, the defining equation, F¼VA, where F¼ F(FX, FY, FZ, MX, MY, MZ) gives us the following equations:

6 6 4

2

In a similar fashion, applying Eqs. (13)–(19) we get the following equation:

ð17Þ

2

ðV 1 U C 11 Þ2 þ ðV 2 U C 12 Þ2 þ ðV 3 U C 13 Þ2 þðV 4 U C 14 Þ2 þ ðV 5 U C 15 Þ2 þ ðV 6 U C 16 Þ2 þ

ð25Þ

The uncertainties in the interaction matrix are then obtained using Eq. (13). Each of the 36 elements of the interaction matrix has corresponding uncertainties that are calculated using Eq. (13). It is to be noted that the uncertainties in each elements in the F and V matrices is assumed to be equal for simplicity. Next thing to do in the uncertainty analysis of the dynamometer is to consider how the uncertainties in the calibration matrix propagate into a future calculation. A general formula for the uncertainty, UR, which propagates into a results, R, from uncertainties in M different variables, Xi; i¼1, 2, …, M, where R¼ R(X1, X2, …, XM) is given by the following equation (Hess et al., 2000): UR ¼

135

2

ð24Þ

Table 3 Overall uncertainty estimates for podded propulsor variables. J

Uρ

UD

UAA

Un

UVA

U T Prop

UQ

U T Unit

U FY

UFY

U MX

U MY

U MZ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

0.0940 0.0001 0.0508 0.0154 2.2789 0.0662 1.0823 1.2523 1.9170 1.9880 2.5301 1.3076 0.0940

0.0940 0.0001 0.0525 0.0154 2.2760 0.0769 0.7920 1.2528 1.9307 1.3465 2.2517 0.8556 0.0940

0.0940 0.0001 0.0522 0.0154 2.2536 0.0749 0.8550 1.4277 1.9085 1.1181 2.2774 0.9633 0.0940

0.0940 0.0001 0.0528 0.0154 2.2733 0.0682 0.9309 1.0634 1.8383 1.1284 2.3626 0.9827 0.0940

0.0940 0.0001 0.0521 0.0154 2.2479 0.0664 1.0273 1.3137 1.8736 0.9637 2.3299 1.2638 0.0940

0.0940 0.0001 0.0530 0.0154 2.3497 0.0666 0.9319 1.2702 1.8569 0.8465 2.1401 0.8930 0.0940

0.0940 0.0001 0.0530 0.0154 2.2369 0.0803 1.0607 1.1249 1.8593 0.9178 2.1009 0.6993 0.0940

0.0940 0.0001 0.0521 0.0154 2.2485 0.0674 1.1925 1.2815 1.8510 1.0239 2.2418 0.7820 0.0940

0.0940 0.0001 0.0530 0.0154 2.3954 0.0794 3.7040 1.2639 1.7746 0.7071 2.2393 0.6724 0.0940

0.0940 0.0001 0.0522 0.0154 2.3153 0.0702 1.1706 1.2136 1.7178 0.5921 2.1941 1.0635 0.0940

0.0940 0.0001 0.0528 0.0154 2.3699 0.0852 0.8736 1.0864 1.7100  0.4653 2.1608 0.9901 0.0940

0.0940 0.0001 0.0521 0.0154 2.3569 0.0714 0.6411 1.0182 1.6135  0.6530 2.1106 0.9568 0.0940

0.0940 0.0001 0.0531 0.0154 2.4384 0.0662  0.7897 1.1486 1.5827  0.6875  2.1477 0.9520 0.0940

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M. Islam et al. / Ocean Engineering 81 (2014) 130–138

Table 4 Overall uncertainties in advance coefficients, propeller thrust and torque coefficients and unit thrust coefficients. J

J (7)

% error of J ( 7 )

K T Prop ( 7)

% error of K T Prop ( 7 )

KQ (7 )

% error of KQ ( 7 )

K T Unit ( 7 )

% error of K T Unit ( 7 )

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

– 5.20E  03 5.27E  03 5.37E-03 5.52E  03 5.70E 03 5.90E  03 6.15E  03 6.40E  03 6.71E  03 7.02E  03 7.32E  03 7.67E 03

– 5.20 2.63 1.79 1.38 1.14 0.98 0.88 0.80 0.75 0.70 0.67 0.64

5.78E  03 5.59E  03 5.28E  03 5.02E  03 4.74E  03 4.62E  03 4.22E  03 4.02E  03 3.33E 03 2.89E  03 2.43E  03 1.43E  03  2.26E  03

1.21 1.24 1.28 1.34 1.42 1.60 1.71 1.96 2.03 2.40 3.21 4.85 12.58

7.61E  03 7.57E  03 7.25E  03 6.68E  03 6.27E  03 5.90E  03 5.92E  03 5.18E  03 5.30E  03 4.66E  03 4.94E  03 4.15E  03 3.82E  03

1.11 1.17 1.20 1.19 1.23 1.28 1.45 1.47 1.79 2.00 2.94 4.38 45.91

4.75E  03 4.39E  03 4.05E  03 3.74E  03 3.40E  03 3.07E 03 2.66E  03 2.44E  03 5.22E  03 2.21E  03 1.43E  03 9.99E  04  1.34E  03

1.01 1.00 1.02 1.05 1.08 1.13 1.17 1.33 3.71 2.30 2.86 2.80 2.70

Table 5 Overall uncertainties in global forces and moments in the three orthogonal directions for the podded propulsors. J

K FX ( 7 )

0.00 4.76E  03 0.10 4.86E  03 0.20 4.44E  03 0.30 4.08E  03 0.40 3.67E 03 0.50 3.20E  03 0.60 2.93E  03 0.70 2.60E  03 0.80 4.47E  03 0.90 2.18E  03 1.00 1.56E  03 1.10 9.99E  04 1.20  1.36E  03

% error of K F X ( 7)

KFY ( 7 )

1.01 1.11 1.11 1.15 1.16 1.18 1.29 1.42 3.18 2.27 3.13 2.80 2.75

1.37E  03 4.33 1.47E  03 4.75 1.46E  03 4.77 1.35E 03 4.95 1.49E  03 5.64 1.44E  03 6.21 1.45E  03 7.33 1.37E  03 7.92 1.39E  03 8.93 1.39E  03 9.95 1.40E  03 11.64 1.34E  03 11.90 1.33E 03 16.78

% error of K FY ( 7 )

K FZ ( 7 )

% error of K F Z ( 7)

K MX ( 7 )

% error of K MX ( 7)

K MY ( 7 )

3.01E  03 3.02E  03 2.94E  03 2.89E  03 2.90E  03 2.92E  03 2.86E  03 2.80E  03 2.81E  03 2.72E 03 2.70E 03 2.56E  03 2.45E  03

17.02 15.71 15.14 14.92 15.98 15.59 16.63 16.51 17.53 16.70 16.68 15.15 14.54

1.83E  02 1.12E  02 9.88E  03 9.91E  03 8.06E  03 6.39E  03 6.51E  03 6.41E  03 4.23E  03 3.75E  03  2.66E  03  3.98E  03  3.77E 03

2.08 1.41 1.40 1.66 1.56 1.59 2.04 2.71 2.63 4.50 4.76 5.07 2.43

1.44E  02 5.23 1.34E  02 4.93 1.34E  02 4.87 1.40E  02 4.89 1.36E  02 5.75 1.26E  02 4.79 1.24E 02 5.09 1.31E  02 5.98 1.31E  02 7.25 1.30E  02 9.81 1.25E  02 14.46 1.22E  02 69.82  1.24E 02 21.12

% error of K MY ( 7)

K MZ ( 7 )

% error of K MZ ( 7 )

1.89E  02 1.96E  02 1.82E  02 1.72E 02 1.60E  02 1.44E  02 1.26E  02 1.16E  02 1.04E  02 9.88E  03 8.65E  03 7.47E  03 7.41E  03

1.04 1.13 1.13 1.15 1.16 1.17 1.15 1.19 1.22 1.33 1.36 1.42 1.82

Fig. 4. Performance curves for average pod 01 in puller configuration with uncertainty (error) bars.

Fig. 5. Performance curves for average pod 01 in pusher configuration with uncertainty (error) bars.

of the podded propulsors as summarized in Tables 4 and 5. The uncertainty estimates are based on the test and calibration data presented in Islam (2009b). From Tables 4 and 5, it can be seen that the uncertainty levels of the propeller thrust coefficient are higher than those of the unit thrust coefficient. However, the uncertainty in torque coefficient is comparable with that of the bare propeller test uncertainty presented in Bose and Luznik (1996). Applying the uncertainty limits to the performance curves of pod 01 in pusher and puller configurations in the form of error bars results in a plot as shown in Figs. 4 and 5, respectively. It is observed in the figures that the

curves fitted to the data lie inside the error bars. Therefore, the fitted curves provide a good representation of the trends indicated by the results. A further discussion on the podded propulsors performance in different configurations and operating conditions are presented in Islam (2009b).

5. Discussions on uncertainty The podded propulsor tests are conducted using a custom designed pod dynamometer system. The uncertainty analysis

M. Islam et al. / Ocean Engineering 81 (2014) 130–138

results of this system are compared to that of a very high quality, well-established equipment (tests done in the OCRE-NRC towing tank) used to measure the performance of some bare podded propellers and performance of podded propulsor measured by the same dynamometer similar to the current experiments (Taylor, 2006). The uncertainties of the bare propeller tests results are used as a benchmark to compare the uncertainty levels of the new dynamometer system. When the results provided in Table 4 are compared with the corresponding results given in Taylor (2006) for the bare propellers and the pod test results, it is seen that the podded propulsor tests using the new pod instrumentation provides the level of accuracy comparable with the established equipment. The uncertainty levels observed in the propeller thrust in the podded propulsor tests are found to be slightly higher than the thrust uncertainty for the baseline tests, but less than the corresponding unit thrust uncertainty in the pod tests as given in Taylor (2006). The less error in the unit thrust is primarily due to the application of proposed calibration approach which was not used in Taylor (2006). Note, the proposed calibration method is applicable for a multi-component dynamometer system and not for single component dynamometer such as propeller thrust. The slight increase in the measurement uncertainty of propeller thrust in the current work as compared to the baseline results may be due to the repeatability error (precision error). It is observed in the detailed uncertainty analysis presented in Islam (2009b) that for majority of the cases, the primary element of the uncertainty of the performance coefficients is the bias error (90% or more on the total uncertainty). To reduce the overall uncertainty in the final results, the primary focuses should be to reduce the bias error in the equipment. Each individual variable in Table 2 should be examined for possible ways to improve the bias errors. Given the high degree of accuracy found in the temperature, density, propeller diameter, azimuthing angle, shaft speed and advance speed, these variables have not been given any further consideration for improvement. As presented in Islam (2009b), the major component influencing the bias limits of propeller thrust, torque and forces and moments for the podded propulsor tests is the curve-fit error. The current analysis incorporated a SEE analysis into the calibration procedure with reduced the error substantially. The calibration of the propeller thrust and torque measurement gauges are repeated 5–8 times and SEE analysis to the results determined whether or not a curve-fit is acceptable and ascertained the functionality of the equipment. The uncertainty levels of the propeller torque are less than the baseline propellers or the pod tests. This is primarily because of less weight and curve-fit errors. The uncertainty level of the unit thrust is less than the pod test results as given in Taylor (2006). This is primarily because of the different calibration approaches. One possible approach to further improve accuracy of the uncertainties in propeller thrust and torque of the podded propulsor tests is to run experiments at higher shaft speeds. As identified in Islam (2009b), for the torque readings of the podded propeller experiments, there is no one dominant factor influencing the overall uncertainty. Despite low error levels in the variables in the torque uncertainty expression, the magnitudes of the actual test measurements are small which results in larger overall error. At higher shaft speeds, higher advance speeds will be required to achieve the desired advance coefficients. Under these conditions the magnitudes of the thrust and torque will be larger relative to the uncertainty levels. Correspondingly, the percent error for each of these measured variables would be reduced, which results in less overall uncertainty in the thrust and torque coefficients.

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6. Concluding remarks A methodology to assess the uncertainty for the measurement of performance characteristics of podded propulsors by a newly developed apparatus is presented. The towing tank experiments are described, including data-reduction equations (DREs) and definitions and measurement systems (instrumentation and data acquisition and reduction). Detailed examples of UA are provided with discussion of bias and precision limits and total uncertainties along with possible unaccounted error sources and methods for uncertainty reduction. For majority of the cases, the primary element of the uncertainty of the performance coefficients is the bias error (90% or more on the total uncertainty). To reduce the overall uncertainty in the final results, the primary focuses should be to reduce the bias error in the equipment. The major component influencing the bias limits of propeller thrust, torque and unit thrust for the podded propulsor tests is the curve-fit error. The standard error estimate (SEE) analysis is incorporated into the calibration procedure and the error is reduced substantially. High degree of accuracy found in the temperature, density, propeller diameter, azimuthing angle, shaft speed and advance speed. The custom-made podded propulsor dynamometer system demonstrated the capability of achieving uncertainty levels close to those of the established equipment. The uncertainty analysis results provided strong evidence that the experimental data obtained using the new dynamometer system presented the true performance characteristics of the model scale podded propulsors.

Acknowledgments The authors would like to express their gratitude to the Natural Sciences and Engineering Research Council (NSERC), the National Research Council (NRC), Oceanic Consulting Corp., and Thordon Bearings Inc. and Memorial University for their financial and other support. The authors are indebted to Oceanic Consulting Corporation to facilitate writing this paper and allow access to all relevant information. References Abernethy, R.B., Benedict, R.P., Downdell, R.B., 1985. ASME measurement uncertainty. J. Fluids Eng. 107, 161–164. AIAA Standard, 1999. Assessment of Experimental Uncertainty with Application to Wind Tunnel Testing, AIAA S-071A-1999, Washington, DC. ASME, 1998. Test Uncertainty: Instruments and Apparatus, PTC 19.1-1998. Bose, N., Luznik, L., 1996. Uncertainty analysis in propeller open water tests. Int. Shipbuild. Prog. 43 (435), 237–246. Coleman, H.W., Steele, W.G., 1999. Experimentation and Uncertainty Analysis for Engineers. Wiley Interscience. Hess, D.E., Nigon, R.T., Bedel, J.W., 2000. Dynamometer Calibration and Usage, Research and Development Report No. NSWCCD-50-TR-2000/040, Hydromechanics Directorate, Carderrock Division, Naval Surface Warfare Center, West Bethesda, Maryland, 31p. Islam, M.F., Veitch, B., Molloy, S., Bose, N., Liu, P., 2007. Effect of geometry variations on the performance of podded propulsors. Trans. Soc. Nav. Archit. Mar. Eng. 115, 140–162. Islam, M.F., Veitch, B., Akinturk, A., Liu, P., 2009a. Study of podded propulsors with varied hub angles and configurations. Int. J. Marit. Eng. A1, 16p Islam, M.F., 2009b. Performance Study of Podded Propulsors with Varied Geometry and Azimuthing Conditions (Doctoral Dissertation), Memorial University of Newfoundland, St. John's, NL, Canada, 245p. ITTC, 2002a. Report of the resistance and flow committee. In: Proceedings of the 23rd International Towing Tank Conference, Venice, Italy. ITTC Quality Manual – Recommended Procedures, 2002b. Propulsion, Performance – Podded Propeller Tests and Extrapolation, 7.5-02-03-01.3, Revision 00. ITTC Quality Manual – Recommended Procedures, 2002c. Propulsion, Propulsor Uncertainty Analysis, Example for Open Water Test, 7.5-02-03-02.2, Revision 00. Joe, Longo, Fred, Stern, 2005. Uncertainty assessment for towing tank tests with example for surface combatant DTMB model 5415. J. Ship Res. 49 (March (1)), 55–68.

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