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By Charles Rhodes, P.Eng., Ph.D.

Elsewhere on this website Fast Neutron Reactors (FNRs) have been identified as the primary source of energy for meeting mankind's future energy needs. This web page focuses on FNR features which prevent earthquake damage.

Earthquakes with peak horizontal accelerations of up to 3 g have been known to occur but are extremely rare. Peak ground accelerations of over 1.26 g seldom occur. If the primary sodium was completely confined then the primary sodium pool walls would have to be sufficiently strong to accelerate the entire primary sodium mass at the peak ground accelertion. That acceleration would impose an extreme hoop stress on the primary sodium pool steel walls.

To reduce earthquake induced hoop stress on the primary sodium pool walls the primary sodium is permitted to slosh around like water in a half full drinking glass. In essence the reactor structure moves with the ground but the inertia of the primary sodium reduces its movement, although the primary liquid sodium level at any point in the pool may rise or fall.

In order to allow thermal expansion and contraction of the secondary sodium pipes there is no rigid connection between the intermediate heat exchangers and the primary sodium pool liner. The enclosure walls above the primary sodium pool are not fixed to the pool deck. These walls are permitted to flex to maintain a gas seal while allowing for thermal expansion and contraction of the secondary sodium piping.. There is a small gap between the bottom of the enclosure side wall and the pool deck so that the wall can move with respect to the pool deck. This gap is gas sealed by a 26 m diameter flexible metal boot. The secondary sodium pipe jackets are sealed to this wall by ceramic insulation and bellows fittings.

The intermediate heat exchangers are located in the guard band of the primary sodium pool such that the intermediate heat exchangers can move up to +/- 0.2 m with respect to the pool wall without any of the secondary sodium pipes colliding with the pool wall.

Severe earthquakes can cause short term horizontal oscillating ground accelerations of up to 3 g, although 1.25 g is a more practical design limit. The design approach taken herein is to design the FNR fuel bundles to resist normal crane handling stresses and to mount the fuel bundles in a pool assembly such that if a severe earthquake occurs the liquid sodium stays almost stationary while the fuel assembly and intermediate heat exchangers move within the liquid sodium. Thus these assemblies can be subject to significant translational drag forces.

During an earthquake the primary liquid sodium will slosh around within the pool walls due to relative movement of the pool structure, fuel bundles and intermediate heat exchangers with respect to the primary liquid sodium.

The individual fuel tubes are protected from translational forces by the shroud plates and diagonal plates. However, on one side of the fuel assembly the shroud plates must exert force for displacing the liquid sodium ahead of the fuel assembly.

The horizontal ground displacement during an earthquake can be expressed in the form:
(X - Xo) = A sin(W t)
Xo = initial horizontal position of the primary sodium pool with respect to the ground
X = primary sodium pool horizontal position as a function of time
A = maximum value of (X - Xo)
W = angular frequency of earthquake vibrations in radians / s
t = time

The velocity V is given by:
V = d(X - Xo) / dt
= W A cos(W t)

Hence the peak velocity Vp is given by:
Vp = W A

The horizontal ground acceleration Ah is given by:
Ah = dV / dt
= - W^2 A sin(W t)
= - [(W A)^2 / A] sin(W t)

Hence the peak horizontal ground acceleration Ahp is given by:
Ahp = [(W A)^2 / A]

Ahp = Vp^2 / A
A = Vp^2 / Ahp

A violent earthquake is characterized by:
Ahp = 1.24 g
= 1.24 X 9.8 m / s^2
= 12.15 m / s^2
and by:
Vp = 1.16 m / s

Note that at a sustained 1.24 g horizontal acceleration the surface of the liquid sodium will be a more than 45 degrees to a horizontal reference.

A = Vp^2 / Ahp
= [1.16 m / s]^2 / [12.15 m / s^2]
= 0.1107 m
which is a typical horizontal ground displacement.

Recall that:
W = Vp / A
F = Vp / (2 Pi A)
= (1.16 m / s) / [6.28 (0.1107 m)]
= 1.67 Hz

The liquid drag force F is given by:
F = K V^2

Thus the peak drag force Fp is given by:
Fp = K (W A)^2

= K (Vp)^2
= K (1.16 m / s)^2

Thus the fuel bundles and the intermediate heat exchangers must both be sufficiently robust to withstand a transverse liquid sodium flow rate of 1.16 m / s.

Thus in terms of drag force on the fuel assembly and intermediate heat exchangers we are concerned about the maximum horizontal ground velocity. The intermediate heat exchangers may need perforated cylindrical shields to limit the transvese drag forces on their heat exchange tubes.

It is possible that there might be an issue with the natural surface wave resonant frequency of a 20 m diameter pool being excited by the earthquake frequency. The pool could be considered to be like a U tube where if the liquid rises on one side it falls on the other side. This system is in some respects like a pendulum. There is a kinetic enegy associated with the liquid sodium moving up and down. There is potential energy associated with one side of the pool being higher than the other side. A large slosh corresponds to a pendulum radius of 10 m. Smaller sloshes correspond to larger pendulum radii.

Rp = pendulum length

Theta = pendulum angular deviation from vertical.

Pendulum KE = M V^2 / 2
= (M / 2) (Rp d(Theta) / dt)^2

H = fuel assembly center of mass height above its height when the pendulum is upright.

For small angles:
Pendulum PE = M g H = M g Rp Theta^2

Total Energy
= TE = KE + PE
= (M / 2) [Rp d(Theta) / dt]^2 + M g Rp Theta^2
= (M Rp / 2) [Rp (d(Theta) / dt)^2) + 2 g |Theta|]

Theta = B sin(Wp t)
Theta^2 = B^2 sin^2(Wp t)
dTheta / dt = B Wp cos (Wp t)
(d(Theta) / dt)^2 = B^2 Wp^2 cos^2(Wp t)
TE = (M Rp / 2) [Rp (d(Theta) / dt)^2) + 2 g Theta^2]
= (M Rp / 2) [Rp B^2 Wp^2 cos^2(Wp t) + 2 g B^2 sin^2(Wp t)]
= (M Rp / 2) B^2 2 g [(Rp Wp^2 / 2 g) cos^2(Wp t) + sin^2(Wp t)
= constant

Recall identity that:
cos^2(Wp t) + sin^2(Wp t) = 1

Rp Wp^2 / 2 g = 1
Wp = [2 g / Rp]^0.5
Fp = (1 / 2 Pi)[2 g / Rp]^0.5

For Rp = 20 m:
Fp = (1 / 2 Pi)[2 g / Rp]^0.5
= (1 / 6.28)[2 (9.8 m /s^2) / 20 m]^0.5
= 0.1576 Hz

For Rp = 10 m:
Fp = (1 / 2 Pi)[2 g / Rp]^0.5
= (1 / 6.28)[2 (9.8 m /s^2) / 10 m]^0.5
= 0.223 Hz

For Rp = 5 m:
Fp = (1 / 2 Pi)[2 g / Rp]^0.5
= (1 / 6.28)[2 (9.8 m /s^2) / 5 m]^0.5
= 0.315 Hz

Note that for severe earthquakes the cavity resonant frequency is much less than the earthquarke frequency. At low intensity earthquakes, where the earthquake frequency may be lower and periodic, we rely on the fuel bundles and the intermediate heat exchangers to provide sufficient primary sodium flow damping to prevent earthquake excited surface waves in the liquid sodium from growing.

Note that the pond will not support large waves at Rp > 10 m.

The FNR design set out herein must safely withstand a maximum horizontal ground velocity of 1.16 m / s.

This web page last updated August 2, 2020.

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