Critical Heat Flux Ratio (DNBR)

C) The form of the correlation for axial uniform thermal flux

K.1/ Critical Heat Flux Ratio (DNBR)

Uncertainties treatment in the calculations Uncertainties treatment in the calculations

One utilizes a statistical approach to combine the uncertainties affecting One utilizes a statistical approach to combine the uncertainties affecting

the DNBR.

The uncertainties representing a random character and a well probability The uncertainties representing a random character and a well probability

law are treated with statistic methods, the others are treated with the law are treated with statistic methods, the others are treated with the

deterministic methods.

This approach is utilizes to guarantee the respect of the DNBR criteria for This approach is utilizes to guarantee the respect of the DNBR criteria for

all transients excepted the transient of uncontrolled-withdrawal of all transients excepted the transient of uncontrolled-withdrawal of control rod cluster in the case of reactor core un-critical or at low control rod cluster in the case of reactor core un-critical or at low

power and for transient of steamline break on which the uncertainties power and for transient of steamline break on which the uncertainties

are combined under deterministic manner.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

a) Statistical approach a) Statistical approach

To establish the relation between the uncertainties affecting the DNBR To establish the relation between the uncertainties affecting the DNBR

and the variation of DNBR, one utilizes a variable defined by the and the variation of DNBR, one utilizes a variable defined by the

following equation:

Y = DNBR(r) /DNBR(c) Y = DNBR(r) /DNBR(c)

Where DNBR(r) is the real DNBR and DNBR(c) is the calculated value, Where DNBR(r) is the real DNBR and DNBR(c) is the calculated value,

determined by taking into account all the parameters associated to the determined by taking into account all the parameters associated to the

calculation of the DNBR to their most probable value.

DNBR(c) is the calculated DNBR in operating by an algorithm set in place DNBR(c) is the calculated DNBR in operating by an algorithm set in place

on the I&C.

Prob (DNBR(r) > T) = 95% with a confidence level of 95% and is equivalent Prob (DNBR(r) > T) = 95% with a confidence level of 95% and is equivalent

to: Prob(DNBR(c) x Y > T) = 95% with a confidence level of 95%.

If m and σ are the mean value and standard deviation of the distribution If m and σ are the mean value and standard deviation of the distribution of probability for the random variable Y, the Prob(DNBR(c) x Y > T) = of probability for the random variable Y, the Prob(DNBR(c) x Y > T) =

95% with a confidence level of 95% is guaranteed if DNBR(c) > T/m(y)(1 95% with a confidence level of 95% is guaranteed if DNBR(c) > T/m(y)(1

– 1,645 V(y)).

– 1,645 V(y)). 159159

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

DNBR(r) is a random variable could be decomposed in random variable DNBR(r) is a random variable could be decomposed in random variable

products as following:

DNBR = Ф(rc)/Ф(cp) x Ф(cp)/Ф(LDC) x Ф(LDC)/Ф(rl) x P DNBR = Ф(rc)/Ф(cp) x Ф(cp)/Ф(LDC) x Ф(LDC)/Ф(rl) x P Where:

Where: Ф(rc) is the real CHF Ф(rc) is the real CHF

Ф(cp) is the predicted CHF determined by the CHF correlation Ф(cp) is the predicted CHF determined by the CHF correlation

Ф(LDC) is the local CHF calculated by the computer code Ф(LDC) is the local CHF calculated by the computer code

Ф(rl) is the local real CHF in the same thermal-hydraulic conditions Ф(rl) is the local real CHF in the same thermal-hydraulic conditions

P is the penalty factor P is the penalty factor

Ф(cp)/Ф(LDC) is DNBR(DC) is the DNBR caculated by the computer Ф(cp)/Ф(LDC) is DNBR(DC) is the DNBR caculated by the computer codecode

DNBR(DC) is a random variable which is function of variavles of the DNBR(DC) is a random variable which is function of variavles of the

system (temperature, local power, etc.).

DNBR(DC) could be decomposed as following:

DNBR(DC) = DNBR(DC)/ DNBR(DCO) x DNBR(DCO)/ DNBR(AO) x DNBR(DC) = DNBR(DC)/ DNBR(DCO) x DNBR(DCO)/ DNBR(AO) x

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Where:

Where: DNBR(DCO) is the calculated DNBR by the computer code to the DNBR(DCO) is the calculated DNBR by the computer code to the most probable values.

most probable values.

DNBR(AO) is the one-line calculated DNBR by the algoritm in the I&C DNBR(AO) is the one-line calculated DNBR by the algoritm in the I&C to the same most probable values.

to the same most probable values.

Consequently:

Y = DNBR(r)/DNBR(AO) = Ф(rc)/ Ф(cp) x DNBR(DC)/DNBR(DCO) Y = DNBR(r)/DNBR(AO) = Ф(rc)/ Ф(cp) x DNBR(DC)/DNBR(DCO)

xDNBR(DCO)/DNBR(AO)/ Ф(LDC)/ Ф(rl) x P xDNBR(DCO)/DNBR(AO)/ Ф(LDC)/ Ф(rl) x P

Y is the product of P factor with the following variables:

Ф(cp)/ Ф(LDC) : Distribution of probabilities of that variable, is provided Ф(cp)/ Ф(LDC) : Distribution of probabilities of that variable, is provided

by the correlation of CHF. It is a normal distribution characterized by a by the correlation of CHF. It is a normal distribution characterized by a

mean value m(c) and a standard deviation σ(c).

DNBR(DC)/DNBR(DCO): This random variable is function of independent DNBR(DC)/DNBR(DCO): This random variable is function of independent

random variables:

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• Operating parameters of NPP and measured on site (temperature, Operating parameters of NPP and measured on site (temperature, reactor pressure, local power, relative measured primary mass flow reactor pressure, local power, relative measured primary mass flow rate);

rate);

• Parameters which are not detected but influencing the DNBR Parameters which are not detected but influencing the DNBR

(uncertainties related to pellets enrichment, to diameter and to the (uncertainties related to pellets enrichment, to diameter and to the dishing)

dishing)

The distribution of probabilities of that variable representative of the The distribution of probabilities of that variable representative of the

global uncertainties of the system is characterized by a value m(s) and global uncertainties of the system is characterized by a value m(s) and

a standard deviation (σ(s)).

DNBR(DCO)/DNBR(CAO) : This random variable is taking into account the DNBR(DCO)/DNBR(CAO) : This random variable is taking into account the

uncertainty of the computer code. The distribution of probabilities is uncertainty of the computer code. The distribution of probabilities is

characterized by two parameters: m(DC) and σ(DC).

* A supplementary uncertainty must be taking into account: transient

* A supplementary uncertainty must be taking into account: transient uncertainty with regard to the steady state.

uncertainty with regard to the steady state.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

This uncertainty gives all the discordance introduced by the utilization of This uncertainty gives all the discordance introduced by the utilization of

the local properties of the fluid resulting analyses of accidental the local properties of the fluid resulting analyses of accidental

transients for the determination of the DNBR in steady state. It is transients for the determination of the DNBR in steady state. It is

independent of uncertainties mentioned above.

The parameters characterizing the distribution of probabilities are:

m(tss) and σ(tss).

The P factor corresponds to the all uncertainties which are treated under The P factor corresponds to the all uncertainties which are treated under

deterministic way:

• Absolute total mass flow rate;Absolute total mass flow rate;

• Core by-pass mass flow rate;Core by-pass mass flow rate;

• Fuel rods bowing effect ;Fuel rods bowing effect ;

• Neutron data;Neutron data;

• Set-point limit of automatic shutdown of the reactor, etc.Set-point limit of automatic shutdown of the reactor, etc.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR

As all variables mentioned above (different measurements, I&C algorithm, As all variables mentioned above (different measurements, I&C algorithm,

DNBR correlation, computer code, manufacturing uncertainties) are DNBR correlation, computer code, manufacturing uncertainties) are

independent and as the perturbations with regard to the mean values independent and as the perturbations with regard to the mean values

are small, the coefficient taken into account the distribution of are small, the coefficient taken into account the distribution of

uncertainty associated to the DNBR could be calculated as follow:

V(r)expo2 = (σ(Y)/m(Y))expo2 = (σ(c)/m(c))expo2 + (σ(s)/m(s))expo2 + V(r)expo2 = (σ(Y)/m(Y))expo2 = (σ(c)/m(c))expo2 + (σ(s)/m(s))expo2 +

(σ(a)/m(a))expo2 (σ(tss)/m(tss))expo2 +(σ(DC)/m(DC))expo2 (σ(a)/m(a))expo2 (σ(tss)/m(tss))expo2 +(σ(DC)/m(DC))expo2

All the terms of the above equation are determined separately excepted All the terms of the above equation are determined separately excepted

σ(s)/m(s) is determined by Monte Carlo method.

On the other hand, the probability distribution function of Y is close to On the other hand, the probability distribution function of Y is close to

the normal distribution with:

Mean value: m(Y) = m(c) x m(s) x m(a) x m(tss) x m(DC) x P Mean value: m(Y) = m(c) x m(s) x m(a) x m(tss) x m(DC) x P

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

In consequence, the probability that the DNBR is superior to the In consequence, the probability that the DNBR is superior to the

threshold T is 95% with the confidence of 95% if the DNBR is superior threshold T is 95% with the confidence of 95% if the DNBR is superior

to the threshold of theoretical DNBR(th) defined as following:

DNBR(th) = T/m(Y)(1-1,645V(Y) DNBR(th) = T/m(Y)(1-1,645V(Y) b) Deterministic approach

b) Deterministic approach

All the uncertainties mentioned above are treated by deterministic way.

As the simplified calculation of the DNBR on site is utilized to protect the As the simplified calculation of the DNBR on site is utilized to protect the reactor against the low DNBR for the concerned transients by reactor against the low DNBR for the concerned transients by deterministic approach, even for the action of the protection system or deterministic approach, even for the action of the protection system or by utilizing the surveillance of limit operating conditions (LCO) of the by utilizing the surveillance of limit operating conditions (LCO) of the DNBR, each parameter has effect on the DNBR must be controlled by a DNBR, each parameter has effect on the DNBR must be controlled by a specific limit operating condition (LCO) and, per consequence, its specific limit operating condition (LCO) and, per consequence, its

uncertainty must be taking into consideration.

Those parameters are:

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR

• Mean temperature of the primary circuit;Mean temperature of the primary circuit;

• The reactor pressure;The reactor pressure;

• The local power.The local power.

About the the power distribution, the analyses of the transients will be About the the power distribution, the analyses of the transients will be

effectuated by utilizing the one most unfavourable.

c) Uncertainties relative to computer code and mixing coefficient c) Uncertainties relative to computer code and mixing coefficient

The results obtained from a sensibility study with the computer code The results obtained from a sensibility study with the computer code

has shown that the minimum DNBR in the hot channel is relatively less has shown that the minimum DNBR in the hot channel is relatively less

sensible to the variations of axial power distribution for the hole sensible to the variations of axial power distribution for the hole

reactor core. (for the same value of FΔH(N).

Studies have been performed to determine the sensibility of minimum Studies have been performed to determine the sensibility of minimum DNBR in the hot channel at axial and radial meshes, to the inlet mass DNBR in the hot channel at axial and radial meshes, to the inlet mass

flow rate, to the loss-pressure, to the power distribution, to the mixing flow rate, to the loss-pressure, to the power distribution, to the mixing

coefficient and to the void model.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Results obtained have shown the minimum DNBR in the hot channel is Results obtained have shown the minimum DNBR in the hot channel is

sensible with the three of them: mixing coefficients, double-phase sensible with the three of them: mixing coefficients, double-phase

model and the radial distribution of press-drop coefficients of the grid model and the radial distribution of press-drop coefficients of the grid

spacers.

d) Justification of the statistic combination of the uncertainties d) Justification of the statistic combination of the uncertainties..

As explained above, one utilizes a statistic approach to combine the As explained above, one utilizes a statistic approach to combine the

following uncertainties which have effect of the DNBR:

• Uncertainty related to the CHF correlation (m(c), σ(c));Uncertainty related to the CHF correlation (m(c), σ(c));

• Uncertainty related to the complete system (m(s), σ(s));Uncertainty related to the complete system (m(s), σ(s));

• Uncertainty related to the algorithm (m(a), σ(a));Uncertainty related to the algorithm (m(a), σ(a));

• Uncertainty related to the computer code (m(DC), σ(DC));Uncertainty related to the computer code (m(DC), σ(DC));

• Uncertainty of transient regime in function of the steady state Uncertainty of transient regime in function of the steady state (m(tss), σ(tss).

(m(tss), σ(tss).

The independent parameters on which the uncertainty presents a The independent parameters on which the uncertainty presents a

random character and a well know probability law are treated by the random character and a well know probability law are treated by the

deterministic method.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR

f) Uncertainties relative to the CHF correlation f) Uncertainties relative to the CHF correlation

The evaluation of the characteristics of the CHF correlation on the basis The evaluation of the characteristics of the CHF correlation on the basis

of the comparison of the results obtained from the tests led to the of the comparison of the results obtained from the tests led to the

definition of the distribution of probabilities of the measured CHF to definition of the distribution of probabilities of the measured CHF to

the predicted CHF. The latter presents a normal distribution.

g) Uncertainty relative to the whole system g) Uncertainty relative to the whole system

Two principal uncertainties are defined that each of them could be Two principal uncertainties are defined that each of them could be

divided in several uncertainties:

• Uncertainties related to the physical measured parameters in Uncertainties related to the physical measured parameters in operation:

operation:

The following operating parameters of the reactor core are used to The following operating parameters of the reactor core are used to

calculate the DNBR: the inlet temperature, the pressure of the calculate the DNBR: the inlet temperature, the pressure of the

pressurizer, the relative measured primary mass flow and the local pressurizer, the relative measured primary mass flow and the local

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

The inlet temperature is obtained by the pyrometric gage at the cold leg, The inlet temperature is obtained by the pyrometric gage at the cold leg, the pressure of the pressurizer is obtained by the primary pressure the pressure of the pressurizer is obtained by the primary pressure gage, the primary mass flow obtained by the mass flow gage at the gage, the primary mass flow obtained by the mass flow gage at the primary pumps and the power distribution of the hot channel is primary pumps and the power distribution of the hot channel is obtained directly from the in-core measurement of self-powered obtained directly from the in-core measurement of self-powered detectors. Each measurement is independent to each others. An detectors. Each measurement is independent to each others. An uncertainty, for example, on the pyrometric due to the scaling does not uncertainty, for example, on the pyrometric due to the scaling does not have any relation with the pressure gauge of the pressurizer neither to have any relation with the pressure gauge of the pressurizer neither to

the mass flow rate gage of the primary pumps.

 Between the gage and the utilized signal in the protection system, Between the gage and the utilized signal in the protection system, certain dispositive are intercalated (for example for the temperature:

certain dispositive are intercalated (for example for the temperature:

convector ohms-amps, convector amps-volts, isolation module if convector ohms-amps, convector amps-volts, isolation module if necessary and convector analogue/numerical), each of these necessary and convector analogue/numerical), each of these dispositive has independent uncertainty and random, it is treated dispositive has independent uncertainty and random, it is treated

statistically.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR

The power distribution of the hot channel induces uncertainties on the The power distribution of the hot channel induces uncertainties on the accuracy of the aeroball measurement system (taking into account the accuracy of the aeroball measurement system (taking into account the accuracy of the activation rate, the reconstruction of the relative accuracy of the activation rate, the reconstruction of the relative density of power, the discretization of the burn-up and the number of density of power, the discretization of the burn-up and the number of instrumented fuel assembly) and on the accuracy of signals obtained instrumented fuel assembly) and on the accuracy of signals obtained from self-powered detectors (derivation, provision relative to the from self-powered detectors (derivation, provision relative to the

burnable poisons).

The total uncertainty could be divided in several distributions of The total uncertainty could be divided in several distributions of probabilities (uncertainties relative to the gage, to the scaling of gage- probabilities (uncertainties relative to the gage, to the scaling of gage- transmitters, etc.). The resulting distribution of probabilities of such a transmitters, etc.). The resulting distribution of probabilities of such a great number of random variables is a normal distribution, as great number of random variables is a normal distribution, as

generally observed at the measurement uncertainties.

e) Uncertainties relative to the manufacturing tolerances e) Uncertainties relative to the manufacturing tolerances

The FΔH(E) taking into account the manufacturing variables which affect The FΔH(E) taking into account the manufacturing variables which affect

the thermal power along the channel.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

The FΔH(E) taking into account the manufacturing variables which affect The FΔH(E) taking into account the manufacturing variables which affect

the thermal power along the channel.

Those variables are the diameter, the density and the enrichment rate of Those variables are the diameter, the density and the enrichment rate of U235 of the pellets. The uncertainties relatives to those variables are U235 of the pellets. The uncertainties relatives to those variables are

determined by sampling measurements on the fabrication. The determined by sampling measurements on the fabrication. The

resulted uncertainty is independent of uncertainties notified, and it is a resulted uncertainty is independent of uncertainties notified, and it is a

normal distribution.

* Uncertainty relative to the algorithm

This uncertainty taking into account the difference between the This uncertainty taking into account the difference between the

calculations obtained from the design computer code and the calculations obtained from the design computer code and the

calculations obtained from the algorithm of the DNBR implemented on calculations obtained from the algorithm of the DNBR implemented on

the site in the same thermal-hydraulic conditions. The algorithm is the site in the same thermal-hydraulic conditions. The algorithm is

adjusted to the calculations of the design computer code.

A statistical analyse allows to determine the distribution of probabilities A statistical analyse allows to determine the distribution of probabilities

differences between the algorithm and the computer code. It is a differences between the algorithm and the computer code. It is a

normal distribution.

normal distribution. 171171

Constraints of the thermal-hydraulic core design

Heat conduction in fuel assembly