4. THE SOURCES OF UNCERTAINTIES

(1) Classification of error sources

by ① how the source behaves with time

② the sampling frequency of the observations

∙fixed error

∙random error

∙variable error but deterministic

(2) MEASURING SYSTEM ERRORS

The errors is introduced by every component of the measuring system.

★ Four options of "true" temperature in Handout

T(1) : the temperature of the thermocouple junction (the achieved value).

T(2) : the temperature of the gas at the junction location (the available value)

T(3) : the temperature the gas at the junction location would have had if the                   instrumentation system had not disturbed the distribution (the undisturbed                value)

T(4) : the mass-flow-weighted average temperature the gas in the duct would                 have had at the axial location of the thermocouple prove, had the                       instrumentation system not disturbed either the temperature of the flow                 distribution (the conceptual value)

Estimation Method

① experimentally

"end to end" calibration : by providing a known and constant input of each            channel of the measuring system and observing that channel's output as a              function of time : fixed errors are evidenced by an offset of the mean value, and        variable errors by variations in the output.

② analytical

: requires an estimate of the fixed and random components of error from each           component of the system, usually based on the manufacturer's specifications

(3) SYSTEM-SENSORS INTERACTION ERRORS

T(2) : the temperature of the gas at the junction location (the available value)

If T(2) is used as the true value, the system-sensor interaction errors (radiation,        velocity, conduction errors) must be added to the list of possible errrors.

Example)

Consider the radiation error encountered when measuring the temperature of a hot      gas in a duct with cold walls. The radiation error can be estimated by

[eqn: T_gas ~-~ T_probe ~=~ {sigma epsilon (T^{4 probe} ~-~ T^{4 wall} )} over h]

The above equation estimates the gas temperature(the available value) from the probe    temperature (the achieved value), using estimates of the heat transfer coefficient, the     emissivity of the wire, and the wall temp.

(4) SYSTEM DISTURBANCE ERRORS

T(3) : the temperature the gas at the junction location would have had if the                    instrumentation system had not disturbed the distribution (the undisturbed               value)

[eqn: {T_avail ~-~ T_undist } over { T_gas ~-~ T_undist } ~=~ 1+PHI over PHI]

[eqn: where ~~~~~ PHI ~=~ {1 ~+~ (hDK_10 )^1/2 / 2K_s }over {(hDk_w )^1/2 / 2k_s }]

Typical case of system disturbance effect a thermocouple attached to a hot metal        part but exposed to a cooler gas flow. The thermo-couple may act as a fin,             transferring heat to the gas and locally cooling the metal

(5) CONCEPTUAL ERRORS

T(4) : the mass-flow-weighted average temperature the gas in the duct would                  have had at the axial location of the thermocouple prove, had the                       instrumentation system not disturbed either the temperature of the flow                 distribution (the conceptual value)

In this time, the effects of velocity and temperature maldistribution muse be added       to the list of sources of errors.

(6) Identifying the True Value

Consider the end-use equations

[eqn: Q~=~ MC_p (T_1 ~-~ T_2 )]               (9)

[eqn: Q~=~ hA ( T_s ~-~ T_g )]                 (10)

For Eq (9), energy balance equation,

√[eqn: T_1 ~, ~T_2] ; the mass-flow-weighted temperature of the fluid flowing

√ [eqn: C_p ~] ; average value over the temperature range [eqn: rm T_1 ~t o ~ T_2]

For Eq (10),  heat transfer rate equation,

√ [eqn: rm T_s] : the average temperature of surface over the area A

√ [eqn: rm T_g] : the mass-flow-weighted gas temperature, or gas temperature far from

the  wall, according to the definition of h for use

따라서 어느 한 measurement 라도 여러 가지 error 가 정의될 수 있으니 이를 특히 주의해야        한다.

5. SINGLE SAMPLE UNCERTAINTY ANALYSIS

; where each test point is run only once or at most a very few times

o 따라서 이들 실험의 credibility를 확인하기 위해 많은 노력이 필요(shakedown 혹은          debugging)

o 3 concerns during experiments

1) acceptability of proposed instrumentation system during planning

2) scatter of data by repeated trials during debugging of experiment

3) overall uncertainty during reporting the results

➡ zeroth, first and Nth-order uncertainty estimate

□ multiple-sample uncertainty analysis 가 실제 반복실험을 수행하는 경우인데 비해         single-sample uncertainty analysis는 반복실험을 했다고 상상하는 것이다.

o 3 baseline levels of replication

1) Repeated trials with absolutely constant process ; zeroth-order replication

; zero variability in the process, but consider all fixed and random errors

− overall uncertainty of fixed and random errors

□ assess the suitability of the proposed instrumentation system

2) Repeated trials with the process running, but no fixed errors ; first-order

replication

; all random errors + process unsteadiness (new error source)

− estrimate the scatter of replication using same equipment, procedures and                 instrumentation each time

□ used during the debugging phase to assess the significance of scatter in output

3) Repeated trials with the process runing and fixed error ; Nth-order replication

− This is the value that must be reported in publication.

(1) Describing Fixed and Variable Errors in Measurements

① Fixed error

− evaluated in the same way for single-sample and multiple-sample analysis; as           RSS combination

② Variable Errors in Measurements

single-sample experiment	multiple-sample experiment
require an auxiliary experiment	determined from the variance of the set itself
the standard deviation of the population	the mean of set

③ Description

For a single observation of parameter, Xi ,

[eqn: X_i ~=~ X_i (measured)~+-~ 2 sigma]                  at (20:1) odds

Under the assumption that if an infinite number of observation were made with the      specified system, the mean value would be the true value(no fixed error) and the        random errors would be normally distributed around the true value, with a standard      deviation of [eqn: sigma],  We can estimate, with 95% confidence, that the mean must lie          within [eqn: +-2sigma] of the observation

□ The relation between the standard deviation of a sample and that of its parent           population can be described by the chi-squared statistics.

➡ estimate of [eqn: sigma] from auxiliary data of a single-sample uncertainty analysis

(2) Describing the Single Sample Uncertainty in Measurements

[例] Determine the heating value of e small portion of the deposit of lignite.

[eqn: rm H~=~ {(T_2 - T_1 )C - e} over M]                                         (11)

where C ; calibrated energy equivalent of the calorimeter(cal/R)

Three uncertainty values for each measurement ;

① The zeroth order uncertainty

; RSS combination of all the fixed and random uncertainty components introduced         by the measuring system.

[eqn: rm deltaX_i,0 ~=~ left {(deltaX_i,fixed )^2 ~+~ (2sigma_i.0 )^2 right}^1/2]

where [eqn: rm deltaX_i,fixed] ; overall fixed error uncertainty of the emasuring system

[eqn: rm sigma_i.0]     ; standard deviation of the population of individual measurement

from the measuring system when its input is stationary

여기에서 [eqn: rm deltaX_i,0] 가 단지 측정계통 error 만 생각할 경우, multi-sample uncertainty의        95% uncertainty와 같지만 random measurement error도 포함하므로 multi-sample         analysis의 overall bias limit와는 다르다.

[Ans]

− The calibration "constant" has been previously determined by using a high-            purity benzonic acid as the fuel. From multiple tests, mean of C is 1385 cal/R.

The precision and bias limits of this mean are ±8 and ±2 cal/R , respectively.

Variable	Bias limit	Precision limit
[eqn: T_1]	±2℉	±0.052℉
[eqn: T_2]	±2℉	±0.052℉
[eqn: C]	±8.2 cal/R	0
[eqn: e]	0	±0.7 cal
[eqn: M]	0	±0.00028g

Then, the uncertainty of the calibration

constant is

[eqn: rm U_C ~=~ [(8)^2 + (2)^2 ]^1/2 ~=~ +- 8.2 ~ cal/R]

□ the precision limit of thermometer

= [eqn: +-] 0.052 oF

여기에서 bias limit는 의미가 없음. (T1 - T2)

□ the precision limit of e = [eqn: +-] 0.7 cal

□ the precision limit of m from a digital

balance  = [eqn: +-] 0.00028 g

표 1  Zeroth-Order Estimates of the                                                      Bias and Precision Limits for Variables

From Eq. (11),  the partial derivatives needed for the uncertainty analysis are

[eqn: & { PARTIAL H} over {partial T_2 } = C over M ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(12)## & { PARTIAL H} over {partial T_1 } = ~ - C over M ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(13) ## & { PARTIAL H} over {partial C } = { T_2 - T_1 } over M ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(14) ## & { PARTIAL H} over {partial e } = ~ - 1 over M ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(15)## & { PARTIAL H} over { partial M } = { - [(T_2 - T_1 ) C~-~ e ]} over M^2 = ~-~ H over M ~~~~~~~~~~~~~~~~~~~(16)##]

Thus the bias limit will be

[eqn: B_H^2 = & left ({partial H } over {partial T_2 } B_T^2 right )^2 ~+~ left ({partial H } over {partial T_1 } B_T_1 right )^2 ~+~ left({partial H } over {partial C} B_C right )^2 ~+~ left({partial H } over {partial e } B_e right )^2 ### & + left({partial H } over {partial M } B_M right )^2 ~+~ 2 {partial H } over {partial T_2 } ~{partial H } over {partial T_1 } (1) B prime_T_2 B prime_T_1 ~~~~~~~~~~~~~~~~~~~~~~~~(17)]

Substituting for the partial derivatives and noting that [eqn: B_T_1 ~=~ B_T_2 ~=~ B prime_T_1 ~=~]

[eqn: B prime_T_2 ~=~ B_T], we get

[eqn: rm B_H^2 ~=~&left(C over M right)^2 B_T^2 + left(C over M right )^2 B_T^2 + left({T_2 - T_1} over M right )B_C^2 + left(1 over M right )^2 B_e^2 + left(H over M right)^2 B_M^2 ~-~ 2 left(C over M right )^2 B_T^2]

From the nominal values and the bias limit estimates in Table 1

(C)    (e)  (M)

[eqn: B_H^2 = 726 ~+~ 0 ~+~ 0]    or   [eqn: rm B_H = ±27 ~cal/g]               (18)

And the precision limit of the result is given as,

[eqn: P_H^2 ~=~ left ({partial H} over {partial T_2} P_T_2 right )^2 + left ({partial H} over {partial T_1 } P_T_1 right )^2 + left ({partial H} over {partial C } P_C right )^2 + left ({partial H } over {partial e } P_e right )^2 + left ({partial H } over {partial M } P_m right )^2]

Substituting for the partial derivatives,

[eqn: rm P_H^2 ~=~ left(C over M right ){P^2}_T_2 + left(C over M right )^2 {P^2}_T_1 + left ({T_2 ~-~ T_1 } over M right ) ^2 P_C^2][eqn: + left (1 over M right )^2 P_e^2 + left (H over M right )^2 P_M^2]

Using the values given in Tables 1 , the precision limit becomes

[eqn: &(T_2) ~~~~(T_1) ~~~(C)~~~(e)~~~(M) # P_H^2 = &5143 ~+~ 5143 ~+~ 0 ~+~ 0.5 ~+~ 1.6]      or   [eqn: rm P_H = ±101 ~rm cal/g]

Finally,  the uncertainty (with a 95% coverage) is found as,

[eqn: rm U_H ~=~ [ B_H^2 ~+~ P_H^2 ] ^1/2]     =   105 cal/g

This uncertainty, which represents the zeroth-order replication level estimate for         this experiment, is about 2% of the lignite heating value.

② The first order uncertainty

− describe the scatter which would be expected in a set of observations while             observed system is running.

□ includes all effects of process unsteadiness and all variable error effects, except          fixed error effects

□ take uncertainty interval for each measurement type with an auxiliary experi-           ment (air-water calibration before boiling experiment in SNU study)

□ [eqn: sigma] of a set of 30 or more readings using chi-squared statistics to estimate its            maximum and minimum probable values

□ Then the most likely value for [eqn: sigma_1] might lie between the max. and min. values

[Ans]

The mining company delivered 26 sealed samples to the laboratory as summarized       in Table 2,

TABLE 2  Heating Value Results for 26 Lignite Samples

Sample number	Heating value (cal/g)	Sample number	Heating value (cal/g)
1	4572	14	4381
2	4568	15	4642
3	4547	16	4481
4	4309	17	4528
5	4383	18	4547
6	4354	19	4541
7	4533	20	4612
8	4528	21	4358
9	4383	22	4386
10	4546	23	4446
11	4539	24	4539
12	4501	25	4478
13	4462	26	4470

[eqn: bar H = 4486] cal/g

[eqn: S_H = 87.2] cal/g                                        (19)

For the 26 data points, the value of [eqn: t] from Table ([eqn: nu] = 25) is 2.060, and so

[eqn: rm P_H = ±t``S_H ~=~ ±180] cal/g                                 (20)

This precision limit is greater than that (101 cal/g) found for a single sample,           which seems reasonable as it includes contributions to the precision both from the       measurement system and from the material composition variation from sample to        sample.

If we divide the contributors to the first-order precision limit into two categories -       zeroth-order and sample-to-sample material variation - we can write

[eqn: rm (P_H )_1st^2 ~=~ (P_H )_zeroth^2 ~+~ (P_H )_mat^2]

Then, the precision limit due to sample-to-sample variation in the lignite                composition can be found as,

[eqn: rm (P_H )_mat & ~=~ [(P_H )_1st^2 ~-~ (P_H )_zeroth^2 ] ^1/2 ## rm & ~=~ [ (180)^2 ~-~ (101)^2 ]^1/2 ~=~ ±149 ~ rm cal/g]

③ The N-th order uncertainty

; measure of its overall uncertainty, accounting for all source of fixed and variable         error in the experiment as it was run.

[eqn: deltaX_i.N ~=~ left{ (deltaX_i.1 )^2 ~+~ (RSS ~deltaX_i.fixed )^2 right }1/2]

where   [eqn: deltaX_i.1]         : the frist order uncertainty

[eqn: RSS ~ deltaX_i.fixed]   : RSS of all fixed error contribution

여기에서 [eqn: rm deltaX_i,N] multi-sample uncertainty의 95% uncertainty와 같다. 그러나 Nth-        measure는 process unsteadiness를 포함하고 multi-sample analysis는 이를 평균하여       처리하기 때문에 달라질 수도 있다.

[Ans]

Now determine what the probable range of heating values will be for the particular       portion of th lignite deposit being considered.

[eqn: rm bar H (4486 cal/g) ~±~ U_H]

where [eqn: U_H] is the Nth-order estimate of the uncertainty as given by

[eqn: U_H ~=~ [B_H^2 ~+~ P_H^2 ] ^1/2]

Here, [eqn: P_H] is the first-order estimate of the precision limit found in Eq. (20) and        [eqn: B_H] is the bias limit found at the zeroth-order replication level in Eq. (18).

Thus,

[eqn: U_H ~=~ [ (27)^2 ~+~ (180)^2 ]^1/2 ~=~ ±182] cal/g                     (21)

We find that the probable range of heating values will be 4304 to 4668 cal/g.

Now with the 26 samples, the uncertainty in the mean(4486 cal/g) is

[eqn: U_barH ~=~ [B_H^2 ~+~ P_H^2 ]^1/2]

where

[eqn: rm S_barH ~=~ S_H over SQRT {M }] [eqn: ~=~ {87.2 cal/g} over sqrt 26 ~=~ 17.1] cal/g  from Eq. (19)

[eqn: rm P_barH ~=~ t``S_barH]  =  (2.060)(17.1 cal/g) = 35.2 cal/g

Then,    [eqn: rm U_barH ~=~ [(27)^2 ~+~ (35.2)^2 ]^1/2 ~=~ ±44] cal/g

∴ the average lignite heating value is, with a 95% coverage, 4486±44 cal/g.

★ Note that often biases arise due to installation, sensor/environment interaction,           and other sources not considered at the zeroth level. Thus, in general,

[eqn: rm (B_r )_Nth ~>~ (B_r )_zeroth].

(3) Describing the Single-Sample Uncertainty in a Result

− The separate components of fixed and variable error are not described.

[eqn: rm R ~=~ R (X_1. ~ X_2. ~ X_3. ~ X_4. ~ … ~ X_N )]

Then, from   [eqn: rm delta``R_X_i ~=~ {PARTIAL R }over partialX_i ``delta``X_i] ,

[eqn: rm delta``R ~=~ left{ sum_i=1^N ~ left(partialR over partialX_i deltaX_i right )^2 right }^1/2]

(4) Interpreting Single Sample Uncertainty Statements

① The zeroth order uncertainty interval

; the contribution to its overall uncertainty arising from the instrumentation system        itsel, for experiment planning to assess the suitability of a proposed instrumen-          tation system.

② The first order uncertainty interval

; estimate the scatter observed in the results repeated trials using the same instru         -mentation each time, and can be used to assess the significance of scatter in           the result of repeated trial with the same instruments and equipment.

③ The N-th order uncertainty interval

; estimate the overall uncertainty in the result and acknowledges all recognized           sources fixed and variable error in the experiment

5. MULTIPLE SAMPLE UNCERTAINTY ANALYSIS

; where enough data are taken at each test point to support a sound statistical            interpretation of random error

☞ dominantly estimate fixed errors

o calibration

o traceability hierarchy

o elaboration of statistical method

☞ reported result는 각 test point에서의 개별측정치가 아닌 측정 set의 평균치로 계산

∴ uncertainty interval associated with mean result

☞ Individual measurements have "fixed" and "random" errors, not uncertainties.

즉 어떤 값(Tg)이 측정 최종결과가 아니고 다른 값을 계산하기 위한 값이면, 이것은        단지 “fixed" 및 ”random" error들로 처리

(1) Describing Fixed and Random Errors in Measurement

The residual fixed and random errors of each measurement are described       by,

∙ The mean value of a set of N observation of the measurement. [eqn: X_i]

∙ The precision index of the mean, [eqn: S_{X_i}]  an estimate of the standard deviation of         the mean of the set N observations

☞ precision index of individual measurement, Xi  : [eqn: S_{X i } ~=~ left{sum_i=1^N ~{(X_i - X_i ^prime ) }^2 over {N ~-~ 1} right }]

☞ precision index of mean  ;                      [eqn: S_bar X^prime ~=~ S_Xr over SQRT { N}]

∙ The number of degrees of freedom of the precision index of the mean, [eqn: nu]

☞ accompany with [eqn: S_{X_i}] in estimating uncertainty, so that an appropriate value of           the Student's t multiplier can be used to describe the 95% confidence level

[eqn: nu] = N - 1

∙ The bias limit of the measurement [eqn: B_{X i}]

[eqn: rm B_{Xi} ~=~ left{(B_cal )^2 ~+~ (B_acq )^2 ~+~ (B_{i n t} ) right }^1/2]

☞ This is combined with the 95% confidence [eqn: t``S_X_i] estimate of the random error in          calculating the overall uncertainty

(2) Describing the Overall Uncertainty in a Single Measurement

Three informations provided in a single measurement

∙The bias limit, [eqn: rm B_X_i]

∙The precision index of the mean, [eqn: S_{X i}]

∙The number of degrees of freedom.  [eqn: nu]

The uncertainty for a single measurement as an end in itself

[eqn: rm U_0.95 &~=~ left{ (B_{X i })^2 ~ + ~ (tS_{X i})^2 right }^1/2 ## & X_i ~=~ barX_i ~+- ~U_0.95]

(3) Describing the Overall Uncertainty in a Result

[eqn: R ~=~ R ~ (X_1. ~ X_2. ~ X_3. ~ …, ~ X_N )]

With

[eqn: rm & B_R ~=~ left{ sum_i=0^N ~ left(partialR over partialX B_i right )^2 right }## rm & S_R ~=~ left{ sum_i=0^N ~left(partialR over partialX B_i right)^2 right }## rm & nu ~=~ N-1 ~~~~~~~~~~~ N~ : ~ rm the ~n umber ~ of ~obserbations~ i n ~the ~ data~ set]

Then the uncertainty in the result will be

[eqn: (U_R )_0.95 ~=~ left{(B_R )^2 ~+~ (tS_R )^2 right }^1/2]

6. DATA ANALYSIS AND REPORTING OF RESULTS

In a experimental report, the most useful uncertainty estimate should have following    meanings;

∙ If the present experiment were repeated using the same apparatus and techniques,       and similar (but not the same) instruments, 19 of 20 repeated trials would produce       results within ± XX (or PP% ) of the persent value.

∙ The reported value is the best estimate for the result, and, with 95% confidence,        the true value is believed to lie within ± XX of that value.

Thus, minimum tabular informations in a report presenting experimental data

o the value (single reading or average)

o the bias limit

o the precision limit

o the uncertainty with the confidence level

☞ graph or correlation

(1) Regression (Curve-fitting)

☞ best equation or best line through the data

Assume the general linear equation,

[eqn: rm Y_o ~=~ a``X ~+~ b]

We wish to find the slope, a, and the intercept, b, that minimize

[eqn: rm eta ~=~ SUM from { {i }=0} to N (Y_i - Y_o )^2]   [eqn: rm =~ SUM from { {i }=0} to N (Y_i - aX -b )^2]

For [eqn: eta] to be minimum,

[eqn: partialeta over partiala ~=~ 0]      and     [eqn: partialeta over partialb ~=~ 0]

Then,     [eqn: rm a ~=~ {N SUM X_i Y_i - SUM X_i SUM Y_i } over {N SUM (X_i^2 ) - (SUMX_i )^2 }]    and  [eqn: rm b ~=~ {SUM (X_i^2 ) SUM Y_i - SUM X_i SUM (X_i Y_i ) } over {N SUM (X_i^2 ) - (SUMX_i )^2 }]

(2) Regression Uncertainty

The standard error of estimate for the regression equation

[eqn: rm SEE ~=~ left( { SUM from { { i}=0} to N [Y_i - (aX_i + b ) ]^2 } over (N-2) right)^1/2]

➡ the general form of a precision index which is a measure of the scatter of data

points about the linear curvefit

− By Schnek, [eqn: +-]2(SEE) band contains about 95% of data points when the bias            contribution to the uncertainty is negligible.

□ 일반적으로는 이 curvefit 의 uncertainty에 data 자체의 uncertainty를 포함해야 함

(3) Example of Presentation of Experimental Results

Consider an experiment for free convection heat transfer from horizontal and             electrically heated cylinders of finite length

o 1/8 in. D and 6 in. long (L/D = 48)

o 1 in. D and 6 in. long   (L/D = 6)

측정변수 ; wall 표면온도 및 air 온도

The h values are plotted versus [eqn: TRIANGLE]T in Fig 7.1 for L/D = 48.  The 95% condidence

uncertainty band is shown for each data point.

The curve in the figure represent the classic results for large L/D (L/D →∞)

[eqn: rm N u ~=~ C (Gr``Pr)^n]                                     (22)

GrPr	C	n
10-2 - 102	1.02	0.148
102 - 104	0.850	0.188
104 - 107	0.480	0.250

where

When, after reviewing the data used in the correlation, we conclude that assuming       an uncertainty of [eqn: +-]15% about the nominal curve seemed appropriate,

➡ the comparision at the Nth-order replication between the experimental data and          the classic results for L/D →∞ is excellent.

For the cylinder with L/D = 6, the data is distorted by the classic result of Eq.(22).

Thus, by a linear least-square curvefit,

log(Nu) = 0.192 log(GrPr) + 0.1232

or    [eqn: rm N u ~=~ 1.328 (Gr``Pr)^0.192]

[eqn: +-]2(SEE) band is plotted around the linear regression line in Fig. 7.4

※ 그러나 보는바와 같이 이 band 가 data의 uncertainty를 적절히 cover하지는 못한다.

When reporting the results of this reegression, a statement similar to the following      should be made;

For horizontal cylinders with L/D and GrPr between 2,600 and 34,000, Nu can be        expressed as,

[eqn: rm N u ~=~ 1.328 (Gr``Pr)^0.192]

with an uncertainty in Nu of

29 - 14%         2,600 < GrPr <  5,300

14 -  6%         5,300 < GrPr < 12,000

6  -  3%        12,000 < GrPr < 34,000