ITTC Recommended Procedures

Transcription

1 Example for est Page of 7 00 ONENS PUPOSE O POEDUE EXAMPLE O ESISANE ES. est Design. Measurement Systems and Procedure. Uncertainty Analysis.. ias Limit... Hull Geometry (Model Length and Wetted Surface Area)... Speed emperature/density/viscosity...5 Skin rictional oefficient...6 orm actor...7 otal ias Limit- otal oefficient...8 otal ias Limit- esiduary oefficient.. Precision Limit.. otal Uncertainties. EEENES Edited by Specialist ommittee of rd I: for, Propulsion and Propeller Open Water ests Approved Date Date 00 rd I 00

2 Example for est Page of 7 00 PUPOSE O POEDUE Example for est he purpose of the procedure is to provide an example for the uncertainty analysis of a model scale towing tank resistance test following the I ev 00, Uncertainty Analysis in ED, Uncertainty Assessment Methodology and ev 00, Uncertainty Analysis in ED, Guidelines for owing ank ests.. est Design y measuring the resistance ( x ), speed (V) and water temperature (tº), and by measuring or using reference values for the wetted surface (S) and density (ρ) the total resistance coefficient ( ) can be calculated for a nominal temperature of 5 degrees, according to: EXAMPLE O ESISANE ES 5 deg m 5 deg m + ( )( + k) (-) his procedure provides an example showing an uncertainty assessment for a model scale towing tank resistance test. he bias and precision limits and total uncertainties for single and multiple runs have been estimated for the total resistance coefficient, and residuary resistance coefficient in model scale at one roude number. In order to achieve reliable precision limits, it is recommended that 5 sets of tests with speed measurements in each set are performed giving in total 5 test points. In this example the recommended sequence was followed. Extrapolation to full scale has not been considered in this example. Although it might lead to significant sources of error and uncertainty, it is not essential for the present purpose of demonstrating the methodology. When performing an uncertainty analysis for a real case, the details need to be adapted according to the equipment used and procedures followed in each respective facility. where m m x 0.5ρV S (-) he residuary resistance coefficient can further be calculated as + m m 5 deg 5 deg ( + k) ( k) (-) In Eq. (-) the conversion of the resistance coefficients from the measured model temperature (index m) to a nominal temperature of 5 degrees is made by the I-978 prediction method. in Eq. (-) is calculated according to the I-957 frictional correlation line (Log 0 e- ) (-4) where e is the eynolds Number for the respective temperatures.

3 Example for est Page of Measurement Systems and Procedure igure. shows a block diagram for the resistance test including the individual measurement systems, measurement of individual variables, data reduction and experimental results. In Section.. the bias limits contributing to the total uncertainty will be estimated for the individual measurement systems: hull geometry, speed, resistance and temperature/density/viscosity. he elementary bias limits are for each measurement system estimated for the categories: calibration, data acquisition, data reduction and conceptual bias. EXPEIMENAL EO SOUES HULL GEOMEY SPEED ESISANE EMPEAUE, DENSIY, VISOSIY Individual measurement systems X, Y, Z, S, L V, V x, x 0, ρ, ν 0, ρ, ν Measurement of individual variables 5deg m + ( 5deg - m )(+k) m x m / (0.5ρV S) - (+k) Data reduction equations 5deg,, m, P 5deg (M), P (M), P 5deg (S), P (S) U (M), U (M), U (S), U (S) Experimental results igure. lock diagram of test procedure. Using the data reduction Eqs. (-) and (- ) the bias limits are then reduced to m, and respectively. As the adjustments in model temperature from the measured temperature to 5 degrees are very small the bias limits associated with the Eq. (-) conversion have not been considered.

4 Example for est Page 4 of 7 00 he precision limits for the total resistance coefficient at a nominal temperature of 5 degrees P 5deg, and residuary resistance coefficient P are estimated by an end-to-end method for multiple tests (M) and a single run (S). able. Ship particulars. Definitions Symbol Value (unit) Length between perp. L PP (m) Length in waterline L WL 6.66 (m) Length overall submerged L OS 6.8 (m) readth.00 (m) Draught even keel 0.00 (m) Wetted surface incl. rudder S (m ) Area water plane A WP 4.86 (m ) Displacement. (m ) lock coefficient /L PP (-) Water plane coefficient WP A WP /L PP (-) Wetted surface coefficient S S/ ( L PP ).695 (-) able. onstants. Definitions Symbol Value (unit) Gravity g 9.80 (m/s ) Density, model basin ρ 000 (kg/m ) Water temperature (resistance test average) tº 5 (degrees) In ables. and. the ship particulars and constants used in the example are tabulated.. Uncertainty Analysis he uncertainty for the total resistance coefficient is given by the root sum square of the uncertainties of the total bias and precision limits ( U ) ( ) + ( P ) ( U ) ( ) + ( P ) -5) (-6) he bias limit associated with the temperature conversion of the measured data, Eq. (-), will not be considered in the present example and therefore (-7) 5 deg m he bias limit for can therefore be calculated as: ( ) x ( ) + S x S + V + ρ ρ V he bias limit for Eq. (-) is + k k + (-8) (-9) he precision limits will be determined for 5deg and for by an end-to-end method where all the precision errors for speed, resistance and temperature/density/viscosity are included. he precision limits for a single run (S) and for the mean value of multiple test (M) are determined. egardless as to whether the precision limit is to be determined for single or multiple runs the standard deviation must be

5 Example for est Page 5 of 7 00 determined from multiple tests in order to include random errors such as model misalignment, heel, trim etc. If it is not possible to perform repeat tests the experimenter must estimate a value for the precision error using the best information available at that time. he precision limit for multiple tests is calculated according to K SDev P ( M ) (-0) M where M number of runs for which the precision limit is to be established, SDev is the standard deviation established by multiple runs and K according to the methodology. he precision limit for a single run can be calculated according to P ( S) K SDev (-).. ias Limit Under each group of bias errors (geometry, speed, resistance and temperature/density/viscosity) the elementary error sources have been divided into the following categories: calibration; data acquisition; data reduction; and conceptual bias. he categories not applicable for each respective section have been left out.... Hull Geometry (Model Length and Wetted Surface Area) he model is manufactured to be geometrical similar to the drawings or mathematical model describing the hull form. Even though great effort is given to the task of building a model no model manufacturing process is perfect and therefore each model has an error in form and wetted surface. he influence of an error in hull form affects not only the wetted surface but also the measured values by an error in resistance. or example, two hull forms, with the same wetted surface and displacement, give different resistance when towed in water if the geometry is not identical. his error in hull form geometry is very difficult to estimate, and will not be considered here. Only the bias errors in model length and wetted surface area due to model manufacture error are taken into account. Model length Data acquisition: he bias limit in model length (on the waterline) due to manufacturing error in the model geometry can be adopted from the model accuracy of ± mm in all co-ordinates as given in I Procedure ev Ship Models. Hence the bias limit in model length will be L mm. Wetted surface Data acquisition: In this example, the error in wetted surface due to manufacturing error in model geometry is estimated using an ad hoc method. y assuming the model error to be ± mm in all coordinates, as given in I Procedure ev, Ship Models, the length will increase by mm, beam by mm and draught by mm. If the dimensions are changed while keeping the block coefficient constant, the displacement becomes

6 Example for est Page 6 of 7 00 ( ) kg which is an increase of kg. Assuming the wetted surface coefficient to be constant, the wetted surface for the larger model becomes S.696 ( L PP )7.6 m, which corresponds to an increase of S -S0.0 m or 0.9% of the nominal wetted surface S. otal weight displ. x50 kg ± 0.75 kg (0.75) ±.06 kg 6x0 kg ± 0.05 kg 6(0.05) ± 0. kg x kg ± kg (0.005) ± kg kg ±.67 kg he model is loaded on displacement and therefore an error in hull form with, for example, too large a model are somewhat compensated by the smaller model draught. he increased displacement of 6.7 kg gives, with a water plane area of A WP 4.86 m, a decreased draught of.8 mm. With a total waterline length of L WL.7 meters the smaller draught decreases the wetted surface by m. otally, the bias limit in wetted surface due to the assumed error in hull form will be S m. alibration: he model weight (including equipment) is measured with a balance and the model is loaded to the nominal weight displacement. he balance used when measuring the model weight is calibrated to ±.0 kg. he errors in model and ballast weights are seen in able.. he total uncertainty in weight is given by the root sum square of the accuracy of the group of weights,.67 kg. An increase in model weight of kg gives, with ρ000 and a water plane area of 4.86 m, an additional draught of / mm. With a waterline length of.7 m this results in an increased wetted surface of m per kg. or the deviation in displacement of ±.67 kg, the error in weight displacement equals.67/ 0.85%, the error in draught equals mm and the error in wetted surface equals S m. inally the error in wetted surface is obtained by the root sum square of the two bias components as S )0.007 m corresponding to 0.0 % of the nominal wetted surface area of 7.6 m. able. Error in displacement. Item Weights Weights Individual Group weights weights Ship model 60 kg ±.0 kg ±.00 kg allast weights x00 kg ±.0 kg (.0) ±.7 kg... Speed he carriage speed measurement system consists of individual measurement systems for pulse count (c), wheel diameter (D) and bit DA and AD card time base ( t). he speed is determined by tracking the rotations of one of the wheels with an optical encoder. he en-

7 Example for est Page 7 of 7 00 coder is perforated around its circumference with 8000 equally spaced and sized windows. As the wheel rotates, the windows are counted with a pulse counter. he speed circuit has a 00 ms time base which enables an update of the pulse every 0 th of a second. A -bit DA conversion in the pulse count limits the maximum number of pulses in 00 ms to he output of the speed circuit is 0-0 V so that 4096 counted in 00 ms corresponds to 0 V output. he output from the encoder is calculated with the equation cπd V 8000 t (-) where c is the number of counted pulses in t00 ms and D is the diameter of the carriage wheel (0.8 m). he bias limit from blockage effects has not been considered. Pulse count (c) alibration: he optical encoder is factory calibrated with a rated accuracy of ± pulse on every update. his value is a bias limit and represents the minimum resolution of the -bit AD data acquisition card. herefore, the bias limit associated with the calibration error will be c pulse (0V/ V). Data acquisition: In the given data acquisition cycle, the speed data is converted to the P by two -bit conversions. he resolution is resol0 V/ V / bit. he AD boards are accurate to.5 bits or pulses, which was determined by calibrating the boards against a precision voltage source. herefore, the bias associated with the two conversions is c c.5 pulses ( V). Data reduction: he final bias occurs when converting the analogue voltage to a frequency that represents the pulse count over 0 time bases or one second. his is enabled if correlating the given frequency to a corresponding voltage output. he bias limit results from approximating a calibration (set of data) with a linear regression curve fit. he statistic is called standard error estimate (SEE) and is written from oleman and Steele (999) as SEE N i ( Y -(ax + b) ) i i N (-) It is proposed by oleman and Steele (999) that a ±(SEE) band about the regression curve will contain approximately 95% of the data points and this band is a confidence interval on the curve fit. he curve fit bias limit is calculated to be.5 Hz corresponding to c4 0.5 pulse ( V). be he total bias limit for pulse count will then c ( c + c + c + c 4 ) ( ).58 pulse ( V ) (-4) Wheel diameter (D) One of the driving wheels of the carriage is used for the speed measurement. he wheel is measured with constant time intervals to ensure the right calibration constant is used.

8 Example for est Page 8 of 7 00 alibration: he wheel diameter is measured with a high quality Vernier calliper at three locations at the periphery of the wheel which are averaged for a final value of D. he wheel diameter is considered accurate to within D m. ime base ( t) he time base of the speed circuitry is related to the clock speed of its oscillator module. alibration: he oscillator module is factory calibrated and its rated accuracy is seconds on every update giving t seconds. he data reduction equation is derived from Eq. (-) and can be written V V c c V + D D V + t t (-5) Using the nominal values of c8.4, D0.8 m and t0. s for the mean speed of V.70 m/s the partial derivatives can be calculated as V πd (-6) c 8000 t V cπ (-7) D 8000 t V cπ D (-8) t 8000 t he total bias limit can then be calculated according to Eq. (-5) as V ( ) + ( ) 5 ( ) (-9) he total bias limit for the speed is V m/s corresponding to 0.% of the nominal speed of.70 m/s. he bias limit for the speed could alternatively be determined end-to-end, by calibrating against a known distance and a measured transit time.... he horizontal x-force is to be measured for the model when towed through the water. alibration: he resistance transducer is calibrated with weights. he weights are the standard for the load cell calibration and are a source of error, which depends on the quality of the standard. he weights have a certificate that certifies their calibration to a certain class. he tolerance for the individual weights used is certified to be ± 0.005%. he calibration is performed from 0 to 8 kg with an increment of 0.5 kg. he bias error arising from the tolerance of the calibration weights, x, is calculated as the accuracy of the weights, times the resistance measured according to Eq. (-0).

9 Example for est Page 9 of 7 00 x accuracy of weights x N (-0) Data acquisition: he data from the calibration tabulated in able.4 shows the mass/volt relation. rom these values the SEE can be calculated with Eq. (-) to SEE0.085 resulting in a bias for the curve fit to be x N. able.4 transducer calibration. Output (Volt) Mass (kg) orce (N) Volt.58 he third error is manifest in the load cell misalignment, i.e., difference in orientation between calibration and test condition. his bias limit is estimated to be ±0.5 degrees and will effect the measured resistance as x - x ( cos 0. 5 x) 479. ( cos 0. 5 o ) N (-) data is acquired by an AD converter, which normally has an error of bit out of AD accuracy of bits. AD conversion bias error in voltage shall be given by AD converter error in bit multiplied by AD range (-0 volts to 0 volts) divided by AD accuracy. his voltage can be translated into Newton by using the slope value of calibration. 0 x N (-) Data reduction: he transducer is fitted in the middle of a special rod, which connects the model to the carriage and tows the model. During the resistance tests the running trim and sinkage of the model result in an inclination of the towing force compared to the calibration which is expressed as a bias limit x5. he mean running trim fore and aft are measured to be f4. mm and a8.4 mm. If the towing force is applied in Lpp/ the sinkage + trim in the towing point tp can be calculated as tp( f+ a)/6.8 mm. he rod used for towing the model is 500 mm long and therefore the inclination of the towing force will be arcsin(6.8/500)0.7 degrees compared to the calm water level. he bias limit can then be computed as x5 x - o ( cos 0. 7 ) 479. ( cos 0. 7 o x ) N (-) his error can be corrected for during the measurements if the angle in the rod is measured. If the transducer is mounted directly to the carriage and is constructed to take loads only in the x-direction this error will be eliminated.

10 Example for est Page 0 of 7 00 he total bias limit in resistance is obtained by the root sum square of the four bias components considered x ( ) 0.84 N corresponding to 0.4 % of the mean resistance of 4.79 N....4 emperature/density/viscosity emperature alibration: he thermometer is calibrated by the manufacturer with a guaranteed accuracy of ±0.0 degrees within the interval -5 to +50 degrees. he bias error limit associated with temperature measurement is tº 0. degrees corresponding to % of the nominal temperature of 5 degrees. ρ o t o ρ t kg/m (-6) Data reduction: he error introduced when converting the temperature to a density (table lookup) can be calculated as two times the SEE of the curve fit to the density/temperature values for the whole temperature range. omparing the tabulated values with the calculated values (Eq. -4) the bias error ρ can be calculated as ρ kg/m. onceptual: he nominal density according to the I- 78 method is ρ 000. Using this method introduces a bias limit as the difference between ρ (5 degrees) and ρ 000 such as ρ kg/m corresponding to % of the density. Density alibration: he density-temperature relationship (table) according to the I Procedure ev 00 Density and Viscosity of Water for g9.8 can be expressed as: he bias for ρ can then be calculated according to: kg/m ( ) + ( ) + ( ) ( ) ρ tº tº tº he bias limit for density is thus (-4) ρ kg/m corresponding to % of ρ 000. If ρ o o t t (-5) using the density value determined by the temperature, the bias limit ρ will be eliminated. t Viscosity Using Eq. (-5) with tº5 degrees and tº 0. degrees the bias ρ can be calculated alibration: according to: he viscosity-temperature relationship for fresh water adopted by I Procedure ρ ρ ρ ρ (-7)

11 Example for est Page of Density and Viscosity of Water can be calculated as υ (( ( t.0) 0.06) ( t.0) +.50)0 ( t t 6 Partial derivative of Eq. (-8) is υ 6 (0.07t )0 o )0 (-8) (-9) Using Eq. (-9) with º5 degrees and º 0. degrees the bias ν can be calculated according to: ν ν t 0.00 o t m (-0) Data reduction: or a nominal temperature of 5.0 degrees this formula results in ν m /s. Meanwhile the fresh water kinematic viscosity according to the table in I Procedure for 5.0 degrees is equal to ν m /s. Using this method introduces a bias error due to the difference between ν(5.0) m /s and ν m /s such as ν m /s. With these results the total bias limit can be calculated as /s viscosity calculation method is thus ν m²/s corresponding to 0.79 % of the kinematic viscosity....5 Skin rictional oefficient he skin frictional resistance coefficient is calculated through the I-957 skin friction line (-) VL ( log0 ) υ ias errors in skin friction calculation may be traced back to errors in model length, speed and viscosity. ias limit associated with can be a found as ( ) V + L V L (-) + υ υ partial derivatives of Eq. (-) by model speed, model length and viscosity are V VL V ln0 ( Log ) υ (-4) υ ( υ) ( υ ) + (-) he bias limit associated with fresh water viscosity due to temperature measurement and L VL ( Log ) υ L ln0 (-5)

12 Example for est Page of υ VL ( Log ) υ υ ln0 (-6) y substituting V m/s, L 0.00 m, ν m /s, bias limits associated with in model scale is corresponding to 0.4 % of the nominal value of orm actor he recommended method for the experimental evaluation of the form-factor is that proposed by Prohaska. If the wave-resistance component in a low speed region (say 0. < r 4 <0.) is assumed to be a function of r, the 4 straight-line plot of / versus r / will intersect the ordinate (r 0) at (+k), enabling the form factor to be determined. hence ( + k ) at low roude numbers (-7) In the case of a bulbous bow near the water surface these assumptions may not be valid and care should be taken in the interpretation of the results. he bias limit (+k) can be determined from the data reduction Eq. (-7). he determination of the precision limit requires about 5 set of tests for several speeds. As there was no example data available, the uncertainty in form factor has for the time being and for indicative purposes been assumed to be 0.0, equal to 0% of k or.66% of +k....7 otal ias Limit- otal oefficient In order to calculate the total bias and precision limits the partial derivatives have to be calculated using input values of x4.79 N, g9.8 m/s, ρ000 kg/m, S7.60 m and V.70 m/s. S V x ρ x 0.5ρV S x 0.5ρS V ρV S (-8) (-9) (-40) x (-4) 0.5V S ρ he total bias limit can then be calculated according to Eq. (-8) as corresponding to 0.65% of the total resistance coefficient otal ias Limit- esiduary oefficient esiduary resistance can be obtained from Eq. (-) as

13 Example for est Page of 7 00 ( + k) (-4) he bias limit of residuary resistance coefficient can be calculated according to ( ) + + k partial derivatives of Eq.(-4): k ( + k). by using Eq. (-4): k (-4) (-44) (-45) (-46) 5 (. 0 ) + ( ) 6 ( ) 5 + (-47) he total bias limit associated with residuary resistance coefficient is corresponding to.7 % of the nominal value of Precision Limit In order to establish the precision limits, the standard deviation for a number of tests, with the model removed and reinstalled between each set of measurements, must be determined. In this example 5 sets of testing (A-E) with speed measurements in each set have been performed giving totally 5 test points. his is the best way to include random errors in the set-up such as model misalignment, trim, heel etc. As resistance is highly dependent on viscosity, the resistance values measured have to be corrected to the same temperature. or a single towing tank the resistance values can preferably be corrected to the mean temperature of the tests in order not to make too large a correction. If the results are to be compared to results from other facilities all the resistance values must be corrected to the same temperature. In the present case the total resistance coefficient for the measured resistance and speed are corrected to the temperature of 5 degrees centigrade, according to the I-78 method, by the following: he residual resistance, which is considered temperature independent, is calculated by m m - ( + k) (-48) where index m measured temperature (compare also Eq. (-)). for 5 degrees is then calculated from: 5deg 5deg + ( + k) (-49) y combining equation Eq. (-48) and Eq. (-49) can be calculated as in Eq. (-).

14 Example for est Page 4 of 7 00 able.5 Standard deviation of and. Series /run Measured values Nominal speed /temp Eq.(-) Eq.(-) x (N) V (m/s) emp (deg) A A A D D D E E E MEAN SDev In the above table the total resistance coefficient is calculated for each run, using the measured resistance and speed. his corrects the measured resistance to the nominal speed by the assumption that the resistance is proportional to V. or small deviations in speed this assumption is considered accurate. 5deg he mean value over 5 runs for (corrected to nominal speed and temperature) is calculated as.79 0 as shown in table.5. With Eq. (-), using the nominal values for speed, density and wetted surface, the corrected, mean resistance can be recalculated to x 4.79N. he precision limit for the mean value of 5 runs is calculated as P K SDev M (-50) according to Eq. (-0) and corresponding to 0.6% of. or a single run the precision limit is calculated as P K SDev (-5) according to Eq. (-) and corresponding to. % of.

15 Example for est Page 5 of 7 00 he residual resistance coefficient can also be calculated as shown in table.5. he precision limit for the mean value of 5 runs is calculated as P K SDev M (-5) according to Eq. (-0) and corresponding to 4.87% of. or a single run the precision limit is calculated as P K SDev (-5) according to Eq. (-) and corresponding to 8.88 % of... otal Uncertainties ombining the precision limits for multiple and single tests with the bias limits the total uncertainty can be calculated according to Eq. (-5) and Eq. (-6). he total uncertainty for for the mean value of 5 runs will then be ( U ) ( ) + ( ) ) P ( ) which is corresponding to 0.67% of. (-54) orrespondingly the total uncertainty for a single run can be calculated as ( U ) ( ) + ( ) ) P ( ) which is.8% of. (-55) he total uncertainty for for the mean value of 5 runs can similarly be calculated as ( U ) ( ) + ( ) ) P ( ) which is corresponding to.09% of. (-56) orrespondingly the total uncertainty for a single run can be calculated as ( U ) ( ) + ( ) ) P ( ) which is 6.9% of. (-57) As can be seen from the values above the uncertainty will decrease if it is calculated for the mean value of 5 tests compared to the single run value. his is also displayed in igure. where the bias is constant regardless of the number of tests while the precision and total uncertainty are decreasing with increasing number of repetitions.

16 Example for est Page 6 of 7 00 % O IAS LIMI OAL UNEAINY PEISION LIMI NUME O ESS igure. ias, precision and total uncertainty. Expressed in relative numbers the bias for represents only 7% percent of the total uncertainty for a single run but as much as 85% of the total uncertainty for the mean value of 5 tests. he bias for represents 74% of the total uncertainty for a single run and 98% of the total uncertainty for the mean value of 5 tests. y comparing the bias and precision limits and the uncertainties, the relative contribution of each term can be calculated. his makes it possible to determine where an upgrade in the measurement system has the largest effect. he bias and precision limits and the uncertainties for the total resistance coefficient are summarised in able.6 where the relative contribution of each term is calculated. his makes it possible to determine where an upgrade in the measurement system has the largest effect. If considering the total resistance coefficient in this example, the most effective would therefore be to improve the speed and resistance measurement systems as they respectively contribute too 47% and 50% of the total bias limit. he uncertainty in speed consists of 98% of the uncertainty in pulse count c. his uncertainty consists of over 80% of the bias limits c and c. he bias limit in resistance consists of almost 00% of the uncertainty in acquisition, x and x4. It is therefore most important to:. Upgrade the resistance measurement system by changing the resistance transducer to a transducer with better linearity (eduction of error x ).. Upgrade the data acquisition cycle in the speed measurement system (eduction of error c and c ). able.6 Error contributions to total uncertainty. erm Value Percentage values Model geometry (m ) S (m ).666E % of S S (m ) 6.89E % of S S (m ) 7.9E % of S Model speed (m/s).70 c (bit) % of c c (bit) % of c c (bit) % of c c4 (bit) % of c c (bit) % of c8 D (m).50e % of D0.8 t (s).05e % of t 0. s θ V c c (m/s).59e % of V θ V D D (m/s) 5.4E % of V θ V t t (m/s) -.746E % of V V (m/s).570e-0 0. % of V Model resistance (N) 4.79 x (N).090E-0 0. % of x x (N).706E % of x x (N).978E % of x x4 (N) -6.4E-0.47 % of x x5 (N).96E % of x x (N).84E- 0.4 % of x Model Density (kg/m )

17 Example for est Page 7 of 7 00 emperature (deg) (deg) % of 5 deg ρ (kg/m) E % of ρ ρ (kg/m) 7.00E-0. % of ρ ρ (kg/m) 6.55E % of ρ ρ (kg/m) 6.605E % of ρ otal oefficient.79e-0 θ S S -.588E-06.7 % of θ V V -.589E % of θ x x.646e % of θ ρ ρ -.504E-06.6 % of.9e % of P (S).89E-05. % of P (M) 9.886E % of U (S) 4.48E-05.8 % of U (M).50E % of esidual esist. oefficient.00e-04 θ.9e % of θ k k E % of θ -5.09E % of 6.48E-05.7 % of P (S).8E % of P (M) 9.895E % of U (S) 7.49E % of U (M) 6.5E % of r where θ i r i. EEENES oleman, H.W. and Steele, W.G., 999, Experimentation and Uncertainty Analysis for Engineers, nd Edition, John Wiley & Sons, Inc., New York, NY. I, 999a, Uncertainty Analysis in ED, Uncertainty Assessment Methodology, nd International owing ank onference, Seoul/Shanghai, I ecommended, Procedure I, 999b, Uncertainty Analysis in ED, Guidelines for owing ank ests, nd International owing ank onference, Seoul/Shanghai, I ecommended, Procedure I, 999c, Density and Viscosity of Water, nd International owing ank onference, Seoul/Shanghai, I ecommended, Procedure I, 999d, Example for est, nd International owing ank onference, Seoul/Shanghai, I ecommended, Procedure , ev 00. I, 00, Ship Models, rd International owing ank onference, Venice, I ecommended, Procedure , ev.