■ Statistical Methodoloy for Limit DNBR  -  ITDP

가) ITDP(Improved Thermal Design Procedure,  Westinghouse)

Determine norminal value and uncertainty of

parameters at 95% confidence level

↓

Determine variation in MDBNR associated

with variations in parameter values

↓

Determine limit DNBR value applicable to all

Condition I and II events

↓

Set reactor protection limits to insure that for

all Condition I and II events the MDNBR is

greater than or equal to the limit DNBR

나) Mathematical Relationships

To relate the variations in design parameter to DNBR variations, an                 uncertainty factor is defined as,

[eqn: y`=`{DNBR(variable)} over {DNBR(nominal)}]

To account for DNBR uncertainty factor change by changes in the design            parameters,

[eqn: {dy} over {y} = s_1 {dx_1} over {x_1} +s_2 {dx_2} over {x_2} +cdots cdots+s_n {dx_n} over {x_n}]                            (1)

where [eqn: s_i] is the sensitivity factor associated with the i-th parameter

[eqn: s_i = {partial y} over {y} /{partial x_i} over {x_i } RIGHT | _{all~ other~parameter~are~held~const.} ={partial (ln y)} over {partial {(ln x_i )}}]           (2)

Integrating Eq. (1),

[eqn: y=C`x_1^s_1` x_2^s_2` cdotscdots`x_n^s_n]                                           (3)

Now relate y to the uncertainties in the design parameters,

Let [eqn: mu_i] be a mean value of the distribution. of independent design parameter          [eqn: x_i] and expand y in an Taylor's serise about [eqn: mu_i]

[eqn: y-mu_y = {partial y} over {partial x_1}(x_1 -mu_1 )+{partial y} over {partial x_2}(x_2 -mu_2 )+cdotscdots+{partial y} over {partial x_n}(x_n -mu_n )+ H(O)]       (4)

From Eq. (3)

[eqn: mu_y =C `mu_1 ^s_1 `mu_2 ^s_2 cdotscdots`mu_n ^s_n]                                             (5)

And from Eq. (4), the variance of y is

[eqn: sigma_y ^2 =({ partial y} over {partial x_1})^2 sigma_1 ^2 +({ partial y} over {partial x_2})^2 sigma_2 ^2 +cdotscdots+({ partial y} over {partial x_n})^2 sigma_n ^2]                    (6)

Using Eqs (3), (5) and (6)

[eqn: ({partial sigma_y} over {mu_ y}) ^2 =s_1 ^2 ( sigma_1 over mu_1 )^2 +s_2 ^2 ( sigma_2 over mu_2 )^2 +cdotscdots+s_n ^2 ( sigma_n over mu_n )^2]                    (7)

Thus, if [eqn: s_i , ~mu_i , ~ sigma_i] are known, [eqn: {sigma} over {mu}](coefficient of variation) for DNBR                 uncertainty factor can be determined.

And from the central limit theorem of statistics, pdf for y has a nomonal             distribution having [eqn: mu_y] and [eqn: sigma_y] even through [eqn: x_i] are not normal.

The restriction of Eq. (7) is

☞ [eqn: x_i] are independent distributed

☞ the variation in [eqn: x_i] are small

☞ [eqn: s_i] is constant

다) Sample Calculation

A) Parameter variation and uncertainties

㉮ Plant Operating Parameters

㉠ coolant flow rate ; 그림 4.1

․ predicted by [eqn: TRIANGLE p] and pump performance

․ measured by measuring S/G power output and coolant temperature rise

㉡ core bypass flow

㉢ flow misdistribution at core inlet

㉣ outlet pressure distribution

㉤ core power

㉥ inlet temperature

㉦ system pressure

Ex) From Fig. 4,1

sample mean ; [eqn: bar x ={sumx_i} over n =`-1.1 %]

standard deviation ; [eqn: s= root{sum{(x_i -barx )^2} over {n-1}}=2.7%]

Population parameters

; If the population is normally distributed, [eqn: bar x , ~s] → estimates of population               mean [eqn: mu] and[eqn: sigma]

(1) [eqn: mu]

For normal distribution [eqn: {( bar x -x)root{n}} over s] possesses a Student's t distribution              with n-1 degree of freedom

∴ [eqn: mu =bar x -{ts} over {root n}]

To get t

[eqn: A=2sigma -1=0.9] (confidence level=0.95)

[eqn: nu=n-1=17]

∴ t = 1.740     and then,    [eqn: mu] = -1.1-1.74×[eqn: {0.027} over {root18}]=-1.11107

(2) [eqn: sigma]

For normal distribution, the [eqn: {nus^2} over {sigma^2}] possess [eqn: chi ^2] distribution with [eqn: nu] degree of            freedom

∴ [eqn: sigma=s root{{nu} over {chi^2}}]

To get [eqn: chi ^2]

[eqn: Q=gamma=0.95]    and   [eqn: nu=17]

→ [eqn: chi ^2] = 8.718    and then,  ∴ [eqn: sigma~=~0.027 root{17 over 8.718}] = 0.0377

Let total variation be composed of additive uncertainty and accuracy comp. as

[eqn: sigma_t = (sigma_u ^2 +sigma_a ^2 )^{1 over 2}]  [eqn: sigma_a] ; measurement accuracy

Put [eqn: sigma_a]=0.01

∴ [eqn: sigma_u =[0.0377^2 -0.01^2 ]^{1 over 2} =]0.0363

Typical value for [eqn: underline W] 4 loop 17 [eqn: TIMES] 17 core  ; TABLE 4-1

㉯ Nuclear and Thermal Parameter

㉠ [eqn: F_ {TRIANGLE H} ^N]

․measurement uncertainty

; INST drift, reproducibility error, integration and location error, burnup

→ 0.021 at 95% confidence level

․calaulation uncertainty ; calculation model

㉡ [eqn: F_Z ^N]

․ burnup

․ Xe

㉢ Thermal diffusion coeff. (TDC)

→ Thermal mixing between adjacent flow channel

㉰ Fabrication Parameters

㉠ Fuel rod pitch and bowing

㉡ Fuel stack height

㉢ surface roughness

B) Sensitivity factors

- THINC → calculate DNBR corresponding to local conditions using a DNB

correlation

- Reference case ; all input parameter to their nominal value.

- Next calculate DNBR by changing parameters one by one and plot the                resulting DNBR against individual parameter value → [eqn: s_i] from Eq. (3)

▶ 문제는 parameter의 dependence 이다. 즉 condition에 따라 sensitivity값이 심하          게 변하면 dependence를 무시못함. → largest [eqn: s_i]를 선택(즉 most conservative)

[eqn: mu_1 ,~mu_2 ,` cdotscdots `mu_n ```-> ```mu_y]  from Eq. (5)

Then from Eq. (7) → [eqn: sigma_y]  for both typical cell and thimble cell

C)  Uncertainty in Computer program

- must be tested with comparing the predicted result with measured one (or            operating plant data)

- include uncertainties in DNB correlation

D) DNBR Uncertainty factor for establishing limit DNBR

․DNBR uncertainty factor

․design value               → Table 4.5

★ For normal distribution,

[eqn: mu]=0.992

[eqn: sigma]=0.0904

[eqn: t={x-mu} over {sigma}] , t=-1.645

∴ the uncertainty factor =[eqn: mu]-1.645[eqn: sigma] = 0.843

∴ Limit DNBR = 1.3/0.843 = 1.54

E) Design Application

In Condition II event analysis, compute min.

DNB with COBRA-IV using norminal values

for all inputs → compare with the limit DNB

shown before

In fixed value method, use the fixed

conservative value and compare MDNBR to 1.3

→ used to set the core protection system.