Preliminary Assessment of Decay Heat Removal Systems in the ESFR-SMART Design: The Role of Natural Air Convection Around Steam Generators Outer Shells in Accidental Conditions

Bittan, Jeremy; Bore, Clement; Guidez, Joel

doi:10.1115/1.4048991

Abstract

In the frame of the ESFR-SMART European project, aiming at improving the safety level of the European sodium cooled fast reactor (ESFR), this paper presents the preliminary assessment of decay heat removal systems (DHRSs) in the ESFR-SMART design: the role of natural air convection around steam generators (SGs) outer shells in accidental conditions. Both theoretical and CATHARE code (thermal hydraulics reference code) calculations are presented. The impact of an additional chimney at the top of each casing as well as running primary and secondary pumps on the heat removal capacity is equally evaluated. This paper shows that the evacuation of decay heat thanks to completely passive air natural circulation alone, in case of Fukushima like accident, should lead to temperatures of sodium in the reactor vessel temporarily exceeding the safety criterion of 650 °C. The addition of chimneys increases the capacities but is not sufficient to evacuate the decay heat safely. If the primary and secondary side pumps are running, the safety criterion should be met.

1 Introduction

A new European project, entitled ESFR-SMART, was launched in 2017 for a 4-year period [1,2]. This project's goal is to improve the safety level of the European sodium cooled fast reactor (ESFR) by simplifying the design and using positive features of sodium cooled fast reactors. The ESFR concept embeds, as presented in Fig. 1, a sodium pool type vessel containing the reactor core and is associated with six intermediate heat exchangers (IHX). Each IHX is linked to six modular steam generators (SGs), which are located in casings. In case of an accidental transient with loss of electrical supply, it is possible to evacuate a part of the decay heat generated from the core via these casings, by cooling the SGs outer shells by natural convection of air thanks to openings located at the bottom and the top of each casing (system called DHRS-2). This paper describes the calculations performed to assess the capability of the SGs to evacuate the decay heat from the reactor vessel. Both theoretical and CATHARE (Code for Analysis of THermalhydraulics during an Accident of Reactor and safety Evaluation) code (Thermal Hydraulics reference code) [3] calculations are presented. The impact of both an additional chimney at the top of each casing and running primary and secondary pumps on the heat removal capacity are evaluated.

Fig. 1

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ESFR-SMART main view of the reactor without chimneys above the casings of the steam generators

2 Theoretical Evaluations

2.1 Methodology.

In order to evaluate the amount of decay heat generated from the core that can be extracted from the reactor vessel thanks to natural circulation via the SGs casings, different energy balance equations are established in steady-state in each system: the reactor vessel, the intermediate loops, and the SGs casings. This theoretical analysis is performed in addition to the CATHARE evaluation to make sure CATHARE results are in accordance with a simplified theoretical model. Modeling natural circulation around SGs casings with CATHARE is complex and providing theoretical solutions is relevant.

It has to be noted that all the temperature-dependent parameters presented in Sec. 2 have been considered as constant in the range of temperature for each part of the reactor (vessel, loops, SGs) as a sensitivity analysis has shown no significant impact on the decay heat removed by the SGs when considering linear variations of these parameters with temperature.

2.2 Energy Balance in the Reactor Vessel.

The energy balance between the core decay heat (Q), the total natural circulation flow

W_{Na Vessel}

in the vessel, the specific heat capacity of the sodium in the reactor vessel

{C p}_{Na Vessel}

⁠, and the temperature at the inlet and outlet of the core (respectively,

T_{Ves}^{Cold}

and

T_{Ves}^{Hot}

⁠) is presented in the following equation:

Q = W_{Na Vessel} {Cp}_{Na Vessel} (T_{Ves}^{Hot} - T_{Ves}^{Cold})

(1)

The thermal pump

Δ P

produced by the core and the heat exchangers can be evaluated as follows:

Δ P = Δ ρ g H

(2)

where $Δ ρ$ is the sodium density variation associated with the sodium heating and H is the height of heat exchangers and the core.

The head losses in the reactor vessel (⁠

Δ P'

⁠) are directly linked to the natural circulation flow

W_{Na Vessel}

in the vessel, as presented in Eq. . The k_vessel parameter is the global head loss coefficient in the reactor vessel

Δ P^{'} = k_{vessel} W_{Na Vessel}^{2}

(3)

The density variation of sodium (⁠

Δ ρ

⁠) is related to the core inlet and outlet temperatures thanks to the following equation:

Δ ρ = β_{Na} (T_{Ves}^{Hot} - T_{Ves}^{Cold})

(4)

In Eq. ,

β

_Na is the linear coefficient linking density variation to temperature variation. A polynomial equation linking sodium density variation to temperature could be used but would only increase the precision of the calculation of a few percent, which is not relevant for this preliminary theoretical analysis. In steady-state, the pressure increase caused by the core and the heat exchangers (2) exactly compensates the head losses in the vessel (3). Combining Eqs. –, Eq. can be written

Δ P = Δ ρ g H = β_{Na} (T_{Ves}^{Hot} - T_{Ves}^{Cold}) g H = Δ P^{'} = k_{vessel} W_{Na Vessel}^{2}

(5)

Hence

W_{Na Vessel} = γ_{Vessel} \sqrt{(T_{Ves}^{Hot} - T_{Ves}^{Cold})}

(6)

With:

γ_{Vessel} = \sqrt{\frac{β_{Na} g H}{k_{vessel}}}

(7)

Using the log mean temperature difference (LMTD) (SGs), it is possible to write for each IHX:

\frac{Q}{6} = K_{IHX} S_{IHX} [\frac{(T_{Ves}^{Hot} - T_{IHX}^{Hot}) - (T_{Ves}^{Cold} - T_{IHX}^{Cold})}{\ln (\frac{T_{Ves}^{Hot} - T_{IHX}^{Hot}}{T_{Ves}^{Cold} - T_{IHX}^{Cold}})}]

(8)

$K_{IHX}$ is the exchange coefficient for each IHX (⁠ $W / m^{2} K)$ and $S_{IHX}$ is the exchange surface area for each of the 6 IHX ${(m}^{2})$ ⁠. $T_{IHX}^{Hot}$ and $T_{IHX}^{Cold}$ are the temperatures of the secondary side sodium fluid just before and after the exchange with IHX.

2.3 Energy Balance in the Steam Generators Casings.

The same methodology as presented in Sec. 2.2 applies to the SGs casings and results in Eqs. –:

\frac{Q}{6} = W_{Air} {C p}_{air} (T_{Air}^{Hot} - T_{Air}^{Cold})

(9)

W_{air}

is the air flow in each SG casing,

{Cp}_{air}

is the specific heat capacity of the air, and

T_{Air}^{Cold} a nd T_{Air}^{Hot}

are, respectively, the temperatures of air entering and exiting each SGs casings

W_{Air} = γ_{SGs Casing} \sqrt{(T_{Air}^{Hot} - T_{Air}^{Cold})}

(10)

γ_{SGs Casing} = \sqrt{\frac{β_{Air} g H_{SG}}{k_{Casing}}}

(11)

In Eq. ,

β

_Air is the linear coefficient linking density variation to temperature variation. LMTD leads to

\frac{Q}{6} = K_{Cas} S_{Cas} [\frac{(T_{IHX}^{Hot} - T_{Air}^{Hot}) - (T_{IHX}^{Cold} - T_{Air}^{Cold})}{\ln (\frac{T_{IHX}^{Hot} - T_{Air}^{Hot}}{T_{IHX}^{Cold} - T_{Air}^{Cold}})}]

(12)

$K_{Cas}$ is the exchange coefficient for each casing (⁠ $W / m^{2} K)$ ⁠. This coefficient is evaluated in Sec. 2.5. $S_{Cas}$ is the exchange surface area for each SGs casing ${(m}^{2})$ ⁠. As mentioned above, each SGs casing contains 6 SGs. Hence, $S_{Cas}$ is the exchange surface area for 6 SGs. $T_{IHX}^{Hot}$ and $T_{IHX}^{Cold}$ are defined in Sec. 2.2. It is supposed that the heat losses in the secondary circuit are negligible.

2.4 Energy Balance in the Secondary Side Loops.

In the same way as for the reactor vessel and the SGs casings, it is possible to write two more Eqs. and for each secondary side loop

\frac{Q}{6} = W_{Na Sec} {Cp}_{Na Sec} (T_{IHX}^{Hot} - T_{IHX}^{Cold})

(13)

W_{Na Sec} = γ_{Na Sec} \sqrt{(T_{IHX}^{Hot} - T_{IHX}^{Cold})}

(14)

$W_{Na Sec}$ is the flow of sodium in each secondary side loop and ${Cp}_{Na Sec}$ is the specific heat capacity.

2.5 Exchange Coefficients Evaluation.

Both IHX and SGs exchange coefficients, respectively, $K_{IHX}$ and $K_{Cas}$ need to be evaluated. This is done using the Dittus–Boelter correlation [4] for the IHX (natural circulation in pipes) and Morgan empirical natural convection correlations for the SGs casings.

2.5.1 Intermediate Heat Exchanger Exchange Coefficient.

Since the exchange surface in each IHX is very large, the value of the coefficient retained in this evaluation should have a low impact on the results.

In order to evaluate the exchange coefficient

K_{IHX}

for each heat exchanger, the Dittus–Boelter correlation is taken into account; since the exchange can be considered as forced convection

Nu = 0.023 {{(Re)}^{0.8} (\Pr)}^{0.3}

(15)

With usual definitions for Reynolds (Re), Prandtl (Pr), and Nusselt (Nu)

Re = \frac{ρ_{Na} V_{Na} L_{exchange}}{μ_{Na}}

(16)

\Pr = \frac{μ_{Na} {Cp}_{Na}}{λ_{Na}}

(17)

Nu = \frac{K_{IHX} L_{exchange}}{λ_{Na}}

(18)

With

$ρ_{Na}$ ⁠: the density of sodium in the vessel IHX,
$V_{Na}$ ⁠: the speed of sodium in the vessel IHX,
$L_{exchange}$ ⁠: the exchange length in the IHX,
$μ_{Na}$ ⁠: the dynamic viscosity of sodium in the IHX,
${Cp}_{Na}$ ⁠: the heat capacity of sodium in the IHX,
$λ_{Na}$ ⁠: the thermal conductivity of sodium in the IHX.

Numerically, the values obtained for those coefficients are presented below:

$Re = 2.5 \times 10^{5}$ ⁠,
$\Pr = 5.5 {\times 10}^{- 3}$ ⁠.

The numerical value obtained for the coefficient $K_{IHX}$ is hence 874.5 $W / m^{2} K$ ⁠.

2.5.2 Steam Generators Exchange Coefficient.

In order to evaluate, in each casing, the value of the exchange coefficient for SGs, natural convection correlations are used. In literature, the recommended correlations when it comes to model natural convection, based on Nu, Grashof (Gr), and Pr coefficients are Morgan empirical natural convection correlations [5]

Nu = C {(Gr \Pr)}^{n}

(19)

With

\Pr = \frac{μ_{air} {Cp}_{air}}{λ_{air}}

(20)

Gr = \frac{g β Δ T L^{3} ρ_{air}}{μ_{air}^{2}}

(21)

Nu = \frac{K_{Cas} L}{λ_{air}}

(22)

And

•

$μ_{air}$ ⁠: the dynamic viscosity of air in the SGs casings,
$λ_{air}$ ⁠: the thermal conductivity of air in the SGs casings,
$ρ_{air}$ ⁠: the density of air in the SGs casings,
$β$ ⁠: the thermal expansion rate of air,
$Δ T :$ the temperature difference between hot air and cold air in the casings,
L: the length of exchange between air and the SGs.
According to the value of the $Gr \Pr$ product, the values to be considered for C and n coefficients are presented in Table 1.
Taking into account the characteristics of the SGs casings, the following values for Pr and Gr coefficients are calculated: $Gr = 3.1 \times 10^{14}$ and $\Pr = 6.5 {\times 10}^{- 1}$ ⁠.
The value obtained for $K_{Cas}$ is hence 15.9 $W / m^{2} K$ ⁠. This value enables to calculate the power exchanged with the cold outside air.

Table 1

C and n parameters depending on Gr and Pr dimensionless numbers for Morgan empirical correlations

	C	n
10⁴ < $Gr \Pr$ < 10⁹	0.59	0.25
10⁹ < $Gr \Pr$ < 10¹⁵	0.021	0.4

2.6 Numerical Scheme to Solve Equations

2.6.1 Known and Unknown Parameters.

In order to completely characterize the system, the following parameters need to be calculated:

$T_{Ves}^{Hot}$ and $T_{Ves}^{Cold}$ are the temperatures of primary side fluid just before and after the IHX exchanger,
$T_{IHX}^{Cold}$ and $T_{IHX}^{Hot}$ are the temperatures of the secondary side sodium fluid just before and after the IHX exchanger,
$T_{Air}^{Cold} and T_{Air}^{Hot}$ are the temperatures of air entering and exiting each SGs casing.

The temperature of cold air entering the SGs casings (⁠ $T_{Air}^{Cold}$ ⁠) is known as it is the environment temperature. This means that there are five unknown parameters in this study. Consequently at least five equations are needed to solve this problem.

In order to solve this system of equations and simplify things, three new parameters are introduced:

$X_{1} = T_{Ves}^{Hot} - T_{Ves}^{Cold},$
$X_{2} = T_{IHX}^{Hot} - T_{IHX}^{Cold}$ ⁠,
$X_{3} = T_{Air}^{Hot} - T_{Air}^{Cold} .$

The decay heat generated by the core $Q$ is considered as a known parameter. In addition, the different flows in the loops will be determined once temperatures are known.

2.6.2 Calculation of $T_{Air}^{Hot}$ ⁠.

Since

T_{Air}^{Cold}

is known, combining Eqs. and , it is possible to calculate

T_{Air}^{Hot}

(9) and (10) \leftrightarrow \frac{Q}{6} = γ_{SGs Casing} {Cp}_{air} \sqrt[3]{{(T_{Air}^{Hot} - T_{Air}^{Cold})}^{2}} {= γ}_{SGs Casing} {Cp}_{air} \sqrt[3]{{X_{3}}^{2}}

(23)

Hence

X_{3} = {(\frac{Q}{6 γ_{SGs Casing} {Cp}_{air}})}^{\frac{2}{3}}

(24)

{T_{Air}^{Hot} = T_{Air}^{Cold} + X}_{3} = {T_{Air}^{Cold} + (\frac{Q}{6 γ_{SGs Casing} {Cp}_{air}})}^{\frac{2}{3}}

(25)

2.6.3 Calculation of $T_{IHX}^{Hot}$ and $T_{IHX}^{Cold}$ ⁠.

Combining Eqs. –, it is possible to calculate

T_{IHX}^{Hot}

and

T_{IHX}^{Cold}

(12) \leftrightarrow \frac{Q}{6} = K_{Cas} S_{Cas} [\frac{X_{2} - X_{3}}{\ln (\frac{T_{IHX}^{Hot} - T_{Air}^{Hot}}{T_{IHX}^{Cold} - T_{Air}^{Cold}})}]

(26)

(13) and (14) \leftrightarrow \frac{Q}{6} = γ_{Na Sec} {Cp}_{Na Sec} \sqrt[3]{{(T_{IHX}^{Hot} - T_{IHX}^{Cold})}^{2}}

(27)

(13) and (14) \leftrightarrow \frac{Q}{6} = γ_{Na Sec} {Cp}_{Na Sec} \sqrt[3]{X_{2}^{2}}

(28)

Then

X_{2} = T_{IHX}^{Hot} - T_{IHX}^{Cold} = {(\frac{Q}{6 γ_{Na Sec} {Cp}_{Na Sec}})}^{\frac{2}{3}}

(29)

Thanks to Eq. ,

X_{2}

can be calculated. Using Eq. , Eq. can be written:

(26) \leftrightarrow \ln (\frac{T_{IHX}^{Hot} - T_{Air}^{Hot}}{T_{IHX}^{Cold} - T_{Air}^{Cold}}) = \frac{6}{Q} K_{Cas} S_{Cas} (X_{2} - X_{3})

(30)

The

α_{1}

parameter is defined as presented in the below equation:

α_{1} = e^{\frac{6}{Q} K_{Cas} S_{Cas} (X_{2} - X_{3})} = e^{\frac{6}{Q} K_{Cas} S_{Cas} [{(\frac{Q}{6 γ_{Na Sec} {Cp}_{Na Sec}})}^{\frac{2}{3}} - {(\frac{Q}{6 γ_{SGs Casing} {Cp}_{air}})}^{\frac{2}{3}}]}

(31)

Rewriting Eq. using

α_{1}

leads to the below equation:

(26) \leftrightarrow T_{IHX}^{Hot} - α_{1} T_{IHX}^{Cold} = T_{Air}^{Hot} - α_{1} T_{Air}^{Cold}

(32)

Combining Eqs. and , a linear system of two equations with two unknowns has to be solved, since

T_{Air}^{Hot}

and

T_{Air}^{Cold}

are known (cf. Sec. 2.6.2)

\{\begin{matrix} T_{IHX}^{Hot} - α_{1} T_{IHX}^{Cold} = T_{Air}^{Hot} - α_{1} T_{Air}^{Cold} \\ T_{IHX}^{Hot} - T_{IHX}^{Cold} = {(\frac{Q}{6 γ_{Na Sec} {Cp}_{Na Sec}})}^{\frac{2}{3}} \end{matrix}

(33)

The solutions of theses equations are presented in the below equation:

\{\begin{matrix} T_{IHX}^{Hot} = \frac{1}{1 - α_{1}} [T_{Air}^{Hot} - α_{1} T_{Air}^{Cold} - α_{1} {(\frac{Q}{6 γ_{Na Sec} {Cp}_{Na Sec}})}^{\frac{2}{3}}] \\ T_{IHX}^{Cold} = \frac{1}{1 - α_{1}} [T_{Air}^{Hot} - α_{1} T_{Air}^{Cold} - {(\frac{Q}{6 γ_{Na Sec} {Cp}_{Na Sec}})}^{\frac{2}{3}}] \end{matrix}

(34)

T_{IHX}^{Hot}

can be rewritten as presented in Eq. below using the expression of

T_{Air}^{Hot}

evaluated in Eq. so that it only depends on the known parameter

T_{Air}^{Cold}

⁠:

T_{IHX}^{Hot} = T_{Air}^{Cold} + \frac{1}{1 - α_{1}} [{(\frac{Q}{6 γ_{SGs Casing} {C p}_{air}})}^{\frac{2}{3}} - α_{1} {(\frac{Q}{6 γ_{Na Sec} {C p}_{Na Sec}})}^{\frac{2}{3}}]

(35)

In the same way,

T_{IHX}^{Cold}

can be rewritten as presented in the following equation:

T_{IHX}^{Cold} = T_{Air}^{Cold} + \frac{1}{1 - α_{1}} [{(\frac{Q}{6 γ_{SGs Casing} {C p}_{air}})}^{\frac{2}{3}} - {(\frac{Q}{6 γ_{Na Sec} {C p}_{Na Sec}})}^{\frac{2}{3}}]

(36)

2.6.4 Calculation of $T_{Ves}^{Hot}$ and $T_{Ves}^{Cold}$ ⁠.

The same methodology is applied to calculate

T_{Ves}^{Hot}

and

T_{Ves}^{Cold}

⁠. Combining Eqs. and , Eq. can be formulated

(1) and (6) \leftrightarrow Q = γ_{Vessel} {C p}_{Na Vessel} \sqrt[3]{{T_{Ves}^{Hot} - T_{Ves}^{Cold}}^{2}} = γ_{Vessel} {C p}_{Na Vessel} \sqrt[3]{{X_{1}}^{2}}

(37)

Hence,

X_{1}

is evaluated thanks to the following equation:

X_{1} = {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}}

(38)

Using Eq. again, it can be written as follows:

(8) \leftrightarrow \ln (\frac{T_{Ves}^{Hot} - T_{IHX}^{Hot}}{T_{Ves}^{Cold} - T_{IHX}^{Cold}}) \frac{Q}{6} = \frac{6}{Q} K_{IHX} S_{IHX} (X_{1} - X_{2})

(39)

The

α_{2}

parameter is then defined as presented in the below equation:

α_{2} = e^{\frac{6}{Q} K_{IHX} S_{IHX} (X_{1} - X_{2})} = e^{\frac{6}{Q} K_{IHX} S_{IHX} . [{(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}} - {(\frac{Q}{6 γ_{Na Sec} {C p}_{Na Sec}})}^{\frac{2}{3}}]}

(40)

Then, rewriting Eq. using

α_{2}

leads to new Eq. below:

(8) \leftrightarrow T_{Ves}^{Hot} - α_{2} T_{Ves}^{Cold} = T_{IHX}^{Hot} - α_{2} T_{IHX}^{Cold}

(41)

Combining Eqs. and , a linear system of two equations with two unknowns has to be solved (⁠

T_{IHX}^{Hot}

and

T_{IHX}^{Cold}

are evaluated in Sec. 2.6.3)

\{\begin{matrix} T_{Ves}^{Hot} - α_{2} T_{Ves}^{Cold} = T_{IHX}^{Hot} - α_{2} T_{IHX}^{Cold} \\ T_{Ves}^{Hot} - T_{Ves}^{Cold} = {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}} \end{matrix}

(42)

The solutions of theses equations are presented in the following equation:

\{\begin{matrix} T_{Ves}^{Hot} = \frac{1}{1 - α_{2}} [T_{IHX}^{Hot} - α_{2} T_{IHX}^{Cold} - α_{2} {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}}] \\ T_{Ves}^{Cold} = {T_{IHX}^{Hot} - X}_{1} = \frac{1}{1 - α_{2}} [T_{IHX}^{Hot} - α_{2} T_{IHX}^{Cold} - {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}}] \end{matrix}

(43)

Using Eqs. and ,

T_{Ves}^{Hot}

and

T_{Ves}^{Cold}

can be evaluated only using known

T_{Air}^{Cold}

temperature, as presented in Eqs. and below:

T_{Ves}^{Hot} = T_{Air}^{Cold} + \frac{1}{(1 - α_{1}) (1 - α_{2})} [(1 - α_{2}) {(\frac{Q}{6 γ_{SGs Casing} . {C p}_{air}})}^{\frac{2}{3}} - (α_{1} - α_{2}) {(\frac{Q}{6 γ_{Na Sec} {C p}_{Na Sec}})}^{\frac{2}{3}}] - \frac{α_{2}}{1 - α_{2}} {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}}

(44)

and

T_{Ves}^{Cold} = T_{Air}^{Cold} + \frac{1}{(1 - α_{1}) (1 - α_{2})} [(1 - α_{2}) {(\frac{Q}{6 γ_{SGs Casing} {C p}_{air}})}^{\frac{2}{3}} - (α_{1} - α_{2}) {(\frac{Q}{6 γ_{Na Sec} {C p}_{Na Sec}})}^{\frac{2}{3}}] - \frac{1}{1 - α_{2}} {(\frac{Q}{γ_{Vessel} {C p}_{Na Vessel}})}^{\frac{2}{3}}

(45)

The most relevant parameter is the sodium temperature getting out of the core. This temperature should not exceed 650 °C to protect the vessel mechanical integrity (decoupling value).

2.7 Results of Theoretical Analyses.

For nominal values of exchange coefficients, theoretical analysis predicts that around 17 to 18 MW_th can be evacuated by circulation of air around the SGs in the casings (for a $T_{Ves}^{Hot}$ equaled to 650 °C), corresponding to the decay heat generated by the core around 48 h after the reactor shutdown. Table 2 presents the power evacuated by the SGs casings for different exchange coefficients values (increased or decreased of 20% compared to nominal values) at a fixed cold air temperature of 20 °C. This shows that IHX exchange coefficient is not a sensitive parameter since the exchange surface is high (about 2000 m²) and exchanged power is relatively low compared to nominal power exchanged (3600 MW_th). The exchange coefficient for the SGs casings appears to play a significant role regarding the power evacuated.

Table 2

Power evacuated by steam generators for a cold air temperature of 20 °C

$T_{Air}^{Cold}$ (°C)	20	20	20	20	20
$T_{Air}^{Hot}$ (°C)	144.5	158.9	144.7	131.6	144.7
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	17.55	20.63	17.56	14.86	17.55

$T_{Air}^{Cold}$ (°C)	20	20	20	20	20
$T_{Air}^{Hot}$ (°C)	144.5	158.9	144.7	131.6	144.7
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	17.55	20.63	17.56	14.86	17.55

Note: The values in bold correspond to modified values compared to reference values.

Tables 3 and 4 present the power evacuated by the SGs at a fixed cold air temperature of respectively 30 °C and 40 °C. A 10 °C increase in the cold air temperature $(T_{Air}^{Cold})$ has a low impact on the power evacuated of around −1.8%. The power evacuated by the SGs is hence rather stable. The values in bold correspond to modified values compared to reference values.

Table 3

Power evacuated by Steam Generators for a cold air temperature of 30 °C

$T_{Air}^{Cold}$ (°C)	30	30	30	30	30
$T_{Air}^{Hot}$ (°C)	153.3	167.4	153.3	140.4	153.3
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	17.26	20.29	17.26	14.61	17.25

$T_{Air}^{Cold}$ (°C)	30	30	30	30	30
$T_{Air}^{Hot}$ (°C)	153.3	167.4	153.3	140.4	153.3
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	17.26	20.29	17.26	14.61	17.25

Note: The values in bold correspond to modified values compared to reference values.

Table 4

Power evacuated by steam generators for a cold air temperature of 40 °C

$T_{Air}^{Cold}$ (°C)	40	40	40	40	40
$T_{Air}^{Hot}$ (°C)	161.9	175.8	161.9	149.1	161.9
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	16.96	19.94	16.97	14.36	16.96

$T_{Air}^{Cold}$ (°C)	40	40	40	40	40
$T_{Air}^{Hot}$ (°C)	161.9	175.8	161.9	149.1	161.9
$K_{IHX} (W / m^{2} K)$	874.5	874.5	1050	874.5	729
$K_{Cas} (W / m^{2} K)$	15.9	19.1	15.9	13.2	15.9
$T_{Ves}^{Hot}$ (°C)	650.0	650.0	650.0	650.0	650.0
Power evacuated (MW_th)	16.96	19.94	16.97	14.36	16.96

Note: The values in bold correspond to modified values compared to reference values.

Figure 2 presents the results of Tables 2–4 graphically allowing to confirm that the most sensitive parameter related to the evaluation of the decay heat that can be extracted from the SGs is the $K_{Cas}$ coefficient.

Fig. 2

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Power extracted by the SGs (MW_th) depending on the cold air temperature entering the SG casings $(T_{Air}^{Cold})$ and on the exchange coefficients $K_{IHX}$ and $K_{Cas}$

3 CATHARE Calculation

The CATHARE code (Code for Analysis of THermalhydraulics during an Accident of Reactor and safety Evaluation) has been used to evaluate the power that can be evacuated by the circulation of air around the SGs outskirts in the casings during a loss of offsite power (LOOP) transient. It is assumed to happen at the initial time while the reactor is under nominal operating conditions (3600 MW_th produced in the core). The LOOP leads to the reactor scram as well as the loss of all active systems (all pumps): decay heat is only removed by the air circulation around the SGs after they have dried out (shortly after the beginning of the accident).

3.1 Methodology.

A CATHARE input deck has been created to model ESFR-SMART reactor design. In order to have the most reliable modeling, all three circuits have been considered in this deck: the reactor vessel, the secondary loops, and the SGs located inside the casings. The most challenging part of creating this input deck is related to the SGs and the modeling of air circulation around them. To model the heat exchange between sodium inside SGs and the air inside the casings in contact with the outer part, a thermal wall has been added to the SGs. This enables to model this exchange through a heat exchange coefficient (parameter HEXT) and the air temperature inside the casing (parameter TEXT).

The air temperature is considered to be constant due to the large volume inside each of the casings, which is a difference between theoretical and CATHARE calculations. In the theoretical evaluations, the temperature of air along the outer shell of the SGs is increasing. This is evaluated thanks to the LMTD calculation as presented in Eq. (11). This means that in the theoretical evaluations, the mean temperature of air along the SGs shells is higher than the cold air entering the casing, leading to a smaller exchange compared the CATHARE evaluation. CATHARE calculation is hence a little bit less conservative than theoretical evaluations since it has a higher exchange for the same given conditions in the secondary loops.

The airflow is not directly modeled by CATHARE, but depends on the heat exchange efficiency, so that the air flow rate may be modeled implicitly through the value of CATHARE parameter HEXT.

Table 5 presents the flows and temperatures in the primary and secondary circuit just before the initiating LOOP event.

Table 5

Flows and temperatures in the primary and secondary circuit in the CATHARE simulation just before the initiating LOOP event

$Primary circuit$
Vessel $mass flow rate$ (kg/s)	18,688
$Core inlet temperature$ (°C)	368.5
$Core outlet temeprature$ (°C)	515.3
$Secondary circuit$
$Individual loop mass flow rate (kg / s)$	2496
$IHX inlet temperature$ (°C)	325.6
$IHX outlet temperature$ (°C)	510.8

$Primary circuit$
Vessel $mass flow rate$ (kg/s)	18,688
$Core inlet temperature$ (°C)	368.5
$Core outlet temeprature$ (°C)	515.3
$Secondary circuit$
$Individual loop mass flow rate (kg / s)$	2496
$IHX inlet temperature$ (°C)	325.6
$IHX outlet temperature$ (°C)	510.8

3.2 Results of CATHARE Simulation.

Figure 3 shows the evolution of the core inlet and outlet temperatures inside the reactor vessel over 24 h, for a fixed value of HEXT equaled to 20 W/m² K and a temperature of cold air of 20 °C. As it can be seen on Fig. 3, showing the evolution of core inlet and outlet temperatures during the LOOP transient, a few quick drops of temperature at the core outlet temperature can be noted. In terms of temperature, the decreases are of around 30 °C and last less than 200 s (plot interval). The core inlet temperature is not affected by these temperature drops.

Fig. 3

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Core inlet and outlet temperatures (°C) during the LOOP transient (s)

The variations of core outlet temperature that can be observed in Fig. 3 cannot be seen on Fig. 4 related to the evacuated power by the SGs with air: the decay heat that can be removed from the core is smooth without any quick perturbations.

Fig. 4

View large Download slide

Core decay heat and power evacuated by SGs (MW_th) versus time (s)

This is explained by the core natural circulation fluctuations (driven by density differences between core inlet and outlet). Due to numerical convergence difficulties, for some limited time steps, CATHARE estimates that the natural circulation in the reactor vessel has a fast increase. This creates a drop in the core outlet temperature. Once the temperature at the core outlet is decreased, for a rather constant core inlet temperature, then the natural circulation flow is decreased (because the density differences are reduced) and progressively gets back to its initial value.

The exchanged power between the sodium in the vessel and the core is not affected. This phenomenon propagates to the secondary circuit (six loops) with some natural circulation flow fluctuations induced by the primary circuit. Since the power exchanged between the core and sodium in the vessel and IHX (between vessel and secondary loops) is not affected by those quick phenomena, this is not observed in Fig. 4 related to the evacuated power by the SGs with air.

Those quick phenomena are limited to a few points and do not call into question the rest of the CATHARE results.

CATHARE calculation is stopped after around 24 h of transient because of numerical instabilities. As it can be seen, there is a drop in the core outlet temperature after around 86,000 s that is related to a decrease in the time-step and does not appear to be physical. After this drop, CATHARE code crashed due to numerical nonconvergence.

Figure 3 shows that after around 7 h of transient, the temperature of the sodium out of the core exceeds the 650 °C criterion.

The power evacuated by the SGs, presented in Fig. 4, is not sufficient to prevent high sodium temperatures in the vessel. One way to increase the power evacuated is to add chimneys (cf. Sec. 4.1).

Figure 4 shows that even if the core decay heat (evaluated by CATHARE using eight energy groups) is lower than the power evacuated by the SGs, the temperature in the reactor vessel is increasing. This is explained by the enormous inertia of the reactor vessel and secondary circuits and the small flows of sodium in the vessel and different loops during natural circulation. After one day of transient, the decay heat is around 20 MW_th [1]. Figures 3 and 4 also show that, for example, for a temperature at the outlet of the core of around 800 °C, the power evacuated by the SGs is around 29 MW_th. It is possible to compare the CATHARE results to the theoretical calculation when considering that when the temperature at the outlet of the core in the CATHARE calculation is around 800 °C, then the reactor is in a quasi-steady-state. When applying those data to the theoretical calculations, roughly the same power is extracted from the SGs (28.7 MW_th). Table 6 presents the results for theoretical evaluations. The difference of power evacuated by the theoretical method and CATHARE is low: 0.3 MW_th. As expected, CATHARE evaluation remains conservative.

Table 6

Power evacuated by steam generators for a cold air temperature of 20 °C using CATHARE conditions

$T_{Air}^{Cold}$ (°C)	20
$T_{Air}^{Hot}$ (°C)	192.3
$K_{IHX} (W / m^{2} K)$	874.5
$K_{Cas} (W / m^{2} K)$	20
$T_{Ves}^{Hot}$ (°C)	800
Power evacuated (MW_th)	28.7

This validates that the theoretical correlations enable to estimate correctly the reactor configuration (temperatures in the vessel, power extracted by the SGs…) compared to CATHARE reference code.

4 Sensitivity Analyses

4.1 Impact of Adding Chimneys.

Since the natural convection of air in the SGs casings does not prevent from reaching high temperatures in the reactor vessel, the impact of adding one chimney (Fig. 5) at the top of each casing to increase the airflow and the evacuated power is evaluated.

Fig. 5

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ESFR-SMART main view of the reactor with chimney addition

The depressurization induced by the chimney,

Δ P_{chimney}

depends on inlet conditions (hot air density

ρ_{c}

⁠) and outlet conditions (ambient air density

ρ_{0}

⁠) of the chimney and on its height h_chimney.

Δ P_{chimney}

is calculated through the following theoretical formula:

Δ P_{chimney} = (ρ_{0} - ρ_{c}) g h_{chimney}

(46)

The entrainment term $Δ P_{chimney}$ is initiated by the difference of densities between cold and hot air, that is to say that the hot air is going high through the casing thanks this ΔP created.

Regular head losses

Δ P_{loss}

inside the chimney can be written as presented in Eq. below:

Δ P_{loss} = \frac{Λ h_{chimney}}{{2 ρ_{c} d}_{chimney} {S_{chimney}}^{2}} W_{air}^{2}

(47)

$Λ$ is the head loss coefficient inside the chimney and $W_{air}$ is the air flowrate going through the chimney. $S_{chimney}$ is the chimney area and $d_{chimney}$ the chimney diameter.

Λ

can be estimated thanks to iterative calculations using the classical Prandtl–Von Karman correlation [6], as shown in the following equation:

\frac{1}{\sqrt{Λ}} = 2 \log (Re \sqrt{Λ}) - 0.8

(48)

The entrainment term

Δ P_{SG exchange}

created by the SG heat exchange can be calculated as

Δ P_{SG exchange} = (ρ_{0} - ρ_{c}) g h_{SG exchange}

(49)

Where:

$ρ_{0}$ is the cold air density, which enters into the casing,
$ρ_{c}$ is the air density at the top of SG after heat exchange along the shell height $h_{SG echange}$ to cool sodium.

The total entrainment pressure difference term (⁠

Δ P_{Total ent}

⁠) can be written as a sum of three terms, as presented in the below equation:

Δ P_{Total ent} = Δ P_{SG exchange} + Δ P_{chimney}

(50)

Thanks to the previous evaluation with no chimney (Sec. 2), the air flow rate

W_{air}^{without chimney}

is estimated as shown in the following equation:

Δ P_{SG exchange} = (ρ_{0} - ρ_{c}) g h_{SG exchange} = k_{0} {W_{air}^{without chimney}}^{2}

(51)

The value of coefficient

k_{0}

in the casing is calculated in the following equation:

k_{0} = \frac{Δ P_{SG exchange}}{{W_{air}^{without chimney}}^{2}} = \frac{(ρ_{0} - ρ_{c}) g h_{SG exchange}}{{W_{air}^{without chimney}}^{2}}

(52)

It is then possible to evaluate the new air flow rate (⁠

W_{air}

⁠) considering both the chimney effect and the SG heat exchange by solving the following equation:

\begin{matrix} Δ P_{Total ent} = Δ P_{SG exchange} + Δ P_{chimney} = (ρ_{0} - ρ_{c}) g h_{SG exchange} + (ρ_{0} - ρ_{c}) g h_{chimney} \\ = k_{0} W_{air}^{2} + Δ P_{loss} = (k_{0} + \frac{Λ h_{chimney}}{{2 ρ_{c} d}_{chimney} S_{chimney}^{2}}) W_{air}^{2} \end{matrix}

(53)

The air flow rate can hence be evaluated as presented in the following equation:

W_{air} = \sqrt{\frac{(ρ_{0} - ρ_{c}) g {(h}_{SG exchange} + h_{chimney})}{\frac{Λ h_{chimney}}{{2 ρ_{c} d}_{chimney} S_{chimney}^{2}} + k_{0}}}

(54)

Numerically, the effect of taking into account chimneys of a height of 45 m leads to an increased airflow of about 45%. Once the new airflow considering chimneys is evaluated, the updated value of the exchange coefficient (K_Cas) between air and SGs casings is obtained (using Gr and Pr adimensional numbers, Sec. 2.5). This new evaluation shows that the exchange coefficient is increased of 54% from 15.9 W/m² K to 24.5 W/m² K.

By adjusting the reactor inertia (I) to fit to CATHARE sodium temperature increase in the reactor vessel and applying Eq. , it is possible to show (Fig. 6) that for a K_Cas of 24.5 W/m² K, the maximum temperature in the reactor vessel should still exceed 650 °C, which is not sufficient to evacuate the decay heat safely. In Eq. ,

Q_{Decay}

is the core decay heat and

Q_{Exchanged SGs}

is the power exchanged by the different SGs in the casings with air.

I \frac{d T}{d t} = Q_{Decay} (t) - Q_{Exchanged SGs} (t)

(55)

Fig. 6

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Core outlet temperature (°C) for different values of K_Cas versus time (s)

4.2 Primary and Secondary Side Pumps Running.

An additional evaluation is performed to determine if assuming the primary and secondary side pumps are running, the decay heat can be removed safely by the SGs via natural circulation of air. One chimney is considered to be at the top of the each casing. The same methodology based on the CATHARE result extrapolation is applied (Sec. 4.1). For this evaluation, the flow rates considered are based on constant volume flow rates both in the vessel and in the secondary loops. The nominal volume flow considered is based to the one presented in Table 1 for the CATHARE calculation.

Figure 7 shows that in this configuration, the maximum core outlet temperature during the LOOP transient should be of 620 °C and meet the safety criterion.

Fig. 7

View large Download slide

Core outlet temperature (°C) for running pumps versus time (s)

5 Conclusions

This paper shows that it is possible to evaluate the power evacuated thanks to the natural circulation of air around SGs using theoretical equations that match well compared to CATHARE results. In a “Fukushima situation” (with no water in the SGs and no power supply), the natural circulation of sodium cannot safely remove alone the decay heat generated by the core since the temperature of the sodium in the reactor vessel temporarily exceeds the safety criterion of 650 °C. The addition of chimneys increases the capacities but is not sufficient to evacuate the decay heat safely.

Other safety systems embedded in the ESFR-SMART design, DHRS-1 (decay heat removal system-1) and DHRS-3 (decay heat removal system-3) [1] would be relevant to remove the decay heat safely in addition to the DHRS-2 (decay heat removal system-2) evaluated in this paper. DHRS-1 is a passive system that would enable to remove a greater part of the decay heat in case of a Fukushima like accident scenario and prevent the core from being damaged.

If the primary and secondary side pumps are running, the safety criterion should be met.

Funding Data

Euratom research and training program 2014–2018 (Grant Agreement No. 754501).

Nomenclature

$Cp$ =: specific heat at constant pressure, J/kg K
d =: diameter, m
$g$ =: gravitational acceleration, m/s²
H =: height of heat exchangers and the core, m
I =: inertia, J/K
k =: global head loss coefficient
K =: exchange coefficient, W/m² K
L =: length, m
P =: pressure, Pa
P' =: pressure, Pa
Q =: core decay heat, W
S =: area, m²
t =: time, s
$T$ =: temperature, °C
V =: speed, m/s
$W$ =: flow, kg/s
$X_{1}$ =: temperature difference in the vessel, °C
$X_{2}$ =: temperature difference in the IHX, °C
$X_{3}$ =: temperature difference in the SG casing, °C

$α_{1}$ =: exchange parameter
$α_{2}$ =: exchange parameter
$Λ$ =: head loss coefficient
$β$ =: volumetric thermal expansion coefficient, 1/K
$γ$ =: proportionality coefficient, kg/s K^0.5
$Δ$ =: difference
$λ$ =: thermal conductivity, W/m K
$μ$ =: dynamic viscosity, Pa s
$ρ$ =: density, kg/m³

$Gr$ =: Grashof number; $(\frac{g β Δ T L^{3} ρ}{μ 2})$
$Nu$ =: Nusselt number; $(\frac{K L}{λ})$
$\Pr$ =: Prandtl number $; (\frac{μ Cp}{λ})$
Re =: Reynolds number; $(\frac{ρ V L}{μ})$

Air =: air
air =: air
c =: hot
Cas =: casing
casing =: casing
chimney =: chimney
Cold =: cold
Decay =: decay heat
ent =: entrainment
exchange =: exchanged
Exchanged =: exchange
Hot =: hot
IHX =: intermediate heat exchanger
loss =: loss
Na =: sodium
sec =: secondary side
Sec =: secondary side
SG =: steam generator
th =: thermal
Total =: total
ves =: vessel
vessel =: vessel
$without$ =: without
0 =: constant properties, scale, reference, characteristic, initial, or axial value

CATHARE =: Code for Analysis of THermalhydraulics during an Accident of Reactor and safety Evaluation
DHRS-1 =: decay heat removal system-1
DHRS-2 =: decay heat removal system-2
DHRS-3 =: decay heat removal system-3
ESFR =: European sodium fast reactor
ESFR-SMART =: European Sodium Fast Reactor Safety Measures Assessment and Research Tools
HEXT =: heat exchange coefficient
IHX =: intermediate heat exchangers
LMTD =: log mean temperature difference
LOOP =: loss of offsite power
SFR =: sodium cooled Fast Reactors
SGs =: steam generators
TEXT =: temperature of air in the casing

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Preliminary Assessment of Decay Heat Removal Systems in the ESFR-SMART Design: The Role of Natural Air Convection Around Steam Generators Outer Shells in Accidental Conditions

Abstract

1 Introduction

2 Theoretical Evaluations

2.1 Methodology.

2.2 Energy Balance in the Reactor Vessel.

2.3 Energy Balance in the Steam Generators Casings.

2.4 Energy Balance in the Secondary Side Loops.

2.5 Exchange Coefficients Evaluation.

2.5.1 Intermediate Heat Exchanger Exchange Coefficient.

2.5.2 Steam Generators Exchange Coefficient.

2.6 Numerical Scheme to Solve Equations

2.6.1 Known and Unknown Parameters.

2.6.2 Calculation of $T_{Air}^{Hot}$ ⁠.

2.6.3 Calculation of $T_{IHX}^{Hot}$ and $T_{IHX}^{Cold}$ ⁠.

2.6.4 Calculation of $T_{Ves}^{Hot}$ and $T_{Ves}^{Cold}$ ⁠.

2.7 Results of Theoretical Analyses.

3 CATHARE Calculation

3.1 Methodology.

3.2 Results of CATHARE Simulation.

4 Sensitivity Analyses

4.1 Impact of Adding Chimneys.

4.2 Primary and Secondary Side Pumps Running.

5 Conclusions

Funding Data

Nomenclature

References

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Preliminary Assessment of Decay Heat Removal Systems in the ESFR-SMART Design: The Role of Natural Air Convection Around Steam Generators Outer Shells in Accidental Conditions

Abstract

1 Introduction

2 Theoretical Evaluations

2.1 Methodology.

2.2 Energy Balance in the Reactor Vessel.

2.3 Energy Balance in the Steam Generators Casings.

2.4 Energy Balance in the Secondary Side Loops.

2.5 Exchange Coefficients Evaluation.

2.5.1 Intermediate Heat Exchanger Exchange Coefficient.

2.5.2 Steam Generators Exchange Coefficient.

2.6 Numerical Scheme to Solve Equations

2.6.1 Known and Unknown Parameters.

2.6.2 Calculation of TAirHot⁠.

2.6.3 Calculation of TIHXHotandTIHXCold⁠.

2.6.4 Calculation of TVesHot and TVesCold⁠.

2.7 Results of Theoretical Analyses.

3 CATHARE Calculation

3.1 Methodology.

3.2 Results of CATHARE Simulation.

4 Sensitivity Analyses

4.1 Impact of Adding Chimneys.

4.2 Primary and Secondary Side Pumps Running.

5 Conclusions

Funding Data

Nomenclature

References

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2.6.2 Calculation of $T_{Air}^{Hot}$ ⁠.

2.6.3 Calculation of $T_{IHX}^{Hot}$ and $T_{IHX}^{Cold}$ ⁠.

2.6.4 Calculation of $T_{Ves}^{Hot}$ and $T_{Ves}^{Cold}$ ⁠.