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**INTRODUCTION:**

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.

**EARTHQUAKE THREAT:**

Earthquakes with horizontal accelerations of up to 3 g have been known to occur. If the primary sodium pool was firmly attached to the ground then the primary sodium pool walls would have to be sufficiently strong to accelerate the entire primary sodium mass at 3 g. That acceleration would impose an extreme hoop stress on the primary sodium pool steel walls.

**EARTHQUAKE SOLUTION:**

To reduce earthquake induced hoop stress in the primary sodium pool walls the primary sodium pool rests on a layer of one inch diameter ball bearings. These bearings provide a low friction connection between the primary sodium pool structure and the ground. Then if the ground horizontally shakes the primary liquid sodium pool remains stationary and the ground shakes underneath it.

There is no rigid connection between the primary sodium pool and the primary sodium piping. The walls above the primary sodium pool do not rely on the primary sodium pool for support. There is a small gap between the wall and the pool deck so that the pool can move with respect to the wall. This gap is gassealed by a 26 m diameter rubber boot which has sufficient ply to allow the primary sodium pool to move horizontally in any direction 1 m with respect to the pool enclosure wall. The primary sodium pipes are sealed to this wall by ceramic insulation. This wall can flex enough to accommodate thermal expansion of the primary sodium pipes. The primary sodium pipes are located in the guard band of the primary sodium pool such that the pool can move up to 1 m with respect to the wall without the pipes colliding with either to pool wall or the fuel bundles.

**EARTHQUAKE OVERVIEW:**

Severe earthquakes can cause short term horizontal oscillating ground accelerations of up to 3 g. In principle a FNR fuel assembly could be fabricated to be sufficiently robust to tolerate that acceleration. However, the amount of steel required to achieve that level of robustness would seriously reduce the FNR's fuel breeding performance. Instead the design approach taken herein is to design the FNR fuel bundles to resist normal handling stresses and to mount the fuel bundles in an inertial pool assembly such that if a severe earthquake occurs the liquid sodium pool stays almost stationary while the ground shakes underneath it.

During an earthquake the liquid sodium between the pool walls and the fuel assembly will still slightly slosh around due to relative movement of the primary sodium pipes with respect to the primary sodium pool.

The horizontal ground displacement during an earthquake can be expressed in the form:

(X - Xo) = A sin(W t)

Where;

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 primary sodium pool and contents has a total mass of about 6000 tonnes. It rests on a layer of one inch diameter ball bearings that cover a disc under the primary sodium pool that is 26 m in diameter. That disc is made slightly bowled. The bottom plate of the primary sodium pool has a matching bowl shape. That bottom plate must have sufficient flexibility to conform to the ball bearing surface contour when under load.

If 1 inch diameter ball bearings are used the required number of bearings is:

Pi (13 m)^2 / (.0254 m / inch)^2 = 822,942 inch^2

implying that about 822,942 one inch diameter steel ball bearings would be required. The best grease for long term lubrication of these bearings remains to be determined.

During a severe earthquake the primary sodium pool will slide off the enclosure center. The purpose of slightly bowling the bottom of the sodium pool is to cause gravity to naturally return the pool assembly to the enclosure center after the earthquake subsides.

**BEARING FORCE**

The bottom of the primary liquid sodium pool is a slightly bowled disk 24 m in diameter. The area of this disk is:

Pi (12 m)^2 / (0.0254 m / inch)^2 = 701,205 inch^2.

The weight of the primary sodium pool is about:

6000 tonnes X 2200 lb / tonne = 13,200,000 lbs.

Hence on average the force born by each one inch diameter ball is:

13,200,000 lbs / 701,205 balls = **18.82 lb / ball**

Thus the bottom plate of the primary sodium pool should be edge welded to slightly flex to distribute the load and to match the ball bearing contour. The welds must be ground flush with the sheet steel surface. Around the bottom disc there must be a one inch high upward projecting rim to contain the ball bearings.

The approximate cost of the required ball bearings is:

822,942 balls X $2.00 / ball = $1,645,884

**FIND SYSTEM NATURAL FREQUENCY:**

Let (X - Xo) denote the horizontal displacement.

Assume an earthquake vibration of the form:

(X - Xo) = A sin(W t)

where W is the angular frequency of an earthquake horizontal vibration and A is the peak amplitudeof the horizontal displacement..

dX / dt = W A cos(wt)

d^2(X) / dt^2 = W^2 A [- sin(wt)]

where:

W^2 A < 3 g

< 3 (9.8 m / s^2)

< 29.4 m / s^2

The viscous damping by the primary liquid sodium flowing past the fuel bundles should be sufficient for critical damping. The earthquake excited oscillation of surface waves in the liquid sodium must not be permitted to grow.

Experimental measurements of the maximum horizontal ground velocity during earthquakes suggest that:

W A = 1.2 m / s

Thus:

W ~ W^2 A / W A

= (29.4 m / s^2) / (1.2 m / s)

= 24.5 rad / s

Then the horizontal vibration frequency is:

f < W / 2 Pi

< (24.5 rad / s) / 6.28

< 3.90 Hz

The maximum horizontal displacement amplitude A is given by:

A = (1.2 m / s) / W

= (1.2 m / s) / (24.5 rad / s)

= 0.049 m

The FNR design set out herein can withstand a maximum horizontal displacement of 1 m. Thus, there is a factor of 20 safety margin in the horizontal displacement, which can safely accommodate ground shake frequencies as low as:

3.9 Hz / 20 = 0.2 Hz.

The primary sodium pool moving on the ball bearings will have the same natural oscillation frequency fp as does a pendulum that describes the same arc as does the pool CM motion. The radius of arc curvature Rp is very long (~ 100 m) which makes the pendulum frequency low.

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 minium height.

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^2]

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)

giving:

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

Hence:

Rp Wp^2 / 2 g = 1

or

**Wp = [2 g / Rp]^0.5**

Let R = (X - Xo)

H / (Rp Theta) = sin(Theta)

R / (Rp Theta) = cos(Theta)

[H / (Rp Theta)]^2 + [R / (Rp Theta)]^2 = 1

or

H^2 + R^2 = [Rp Theta]^2

(Theta) = (H / R)

At small Theta angles:

tan(Theta) ~ Theta

giving:

H^2 + R^2 = [Rp Theta]^2

= [Rp H / R]^2

or

Rp^2 = R^2 + (R^4 / H^2)

or for R >> H:

**Rp = R^2 / H**

Hence:

Wp = [2 g / Rp]^0.5

= [2 g H / R^2]^0.5

or

fp = (1 / 2 Pi R)[2 g H]^0.5

where typically:

R = 1 m

H ~ 1 cm

giving the pendulum frequency:

fp = (1 / 2 Pi R)[2 g H]^0.5

= [1 / (2 Pi 1 m)] [(2 9.8 m 0.01 m)/ s^2]^0.5
=[1 / 2 Pi m][0.196 m / s]

= **0.031 Hz**

**Rp** = R^2 / H

= 1 m^2 / 0.01 m

= **100 m**

Thus with these parameters after an earthquake impulse it will take several minutes for the primary sodium pool to precisely re-center itself. The returning velocity generates relatively little viscous drag as compared to the earthquake. The natural frequency fp of the fuel assembly suspension is:

f / fp = 3.9 Hz / 0.031 Hz

= 125.8

fold less than the typical earthquake wave frequency f.

This reduction reduces horizontal acceleration forces on the primary sodium pool sidewalls which reduction is essential for earthquake protection.

Note that there must be a 1 m clearance between the primary sodium pipes and the nearest primary sodium pool wall to accommodate pipe and pool thermal expansion and earthquake vibrations.

This web page last updated May 24, 2020.

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