Storage Tank Dike Design

George HarriottComputational Modeling CenterAir Products

P2SAC Spring MeetingPurdue UniversityMay 3, 2017

Tank Rupture Consequences

• Molasses tank, Boston MA, 1919, 2.5x106 gallons, 21 fatalities

• Water tank, Juarez, MX, 1986, 7.5 x105 gallons, 4 fatalities

• Oil tank, Floreffe PA, 1988, 3.8x106 gallons

• Molasses tank, Loveland CO, 1990, 6.2 x105 gallons

• HCl tank, McDonald, PA, 2002,104 gallons

Floreffe Aftermath

Regulatory Requirements

• Secondary containment volume must be sufficient to hold the contents of the tank (EPA/OSHA).

• Fire code constraints

• Dynamics not considered!

Design Considerations

• Dike height

• Dike diameter

• Dike shape

• Complete tank rupture

• Small hole in tank wall

Rupture Flow Scales

• Height: H ∼ 10 m

• Distance: L ∼ 10 m

• Velocity: (gH)1/2 ∼ 10 m/s

• Time: L/(gH)1/2 ∼ 1 s

• Reynolds number: Re ∼ 108

Initial Condition

H

L

Shallow-Water Theory

• Depth-averaged inviscid equations of motion

• Exact mass balance

• Hydrostatic pressure

• Analogous to compressible gas dynamics

h + ∇•(hu) = 0

u + u•∇u + ∇h = 0

•

• h ⇒u ⇒ ⇒

Shock Conditions

• Multi-valued SWT solutions

• Supplemental jump equations

u+ − λ( )h+ − u− − λ( )h− = 0

u+ − λ( )2 h+ − u− − λ( )2 h− +12(h+ )2 − (h− )2( ) = 0 (-) (+)

⇐ λ

Numerical Pitfalls

• Diffusive Errors

• Dispersive Errors

Computational Methods

• Method of Characteristics

• Explicit Finite-Difference (Lax, FCT, WENO)

• Implicit Finite Element (DG, SUPG)

• Projection Methods (POD)

Method of Characteristics

φ =hu⎧⎨⎩

⎫⎬⎭

⇒ Ai∂φ∂t

+ Bi∂φ∂x

+ψ = 0

Eigenvalue: λ ξiA( ) = ξiB

ξiA( )i ∂φ∂t

+ λ ∂φ∂x

⎧⎨⎩

⎫⎬⎭+ ξiψ = 0

ξiA( )idφdt

+ ξiψ = 0 : dxdt

= λ

Simple Planar WaveWavespeed : c = h

dhdt

± cdudt

= 0 : dxdt

= u ± c

x

t

c = 0u = 0

c = 1u = 0

c =23−

13

x −αt

⎛⎝⎜

⎞⎠⎟

: u = 23+

23

x −αt

⎛⎝⎜

⎞⎠⎟

Splash

Lax FD Method

x

t

i + 1, ji, j i - 1, j

i, j + 1

∂h∂t

+∂q∂x

= 0 : q ≡ uh

⇓

hi, j+1 =12hi−1, j + hi+1, j( ) − Δt

2Δx⎛⎝⎜

⎞⎠⎟qi+1, j − qi−1, j( )

Planar Rupture Splash

Complete Rupture Symmetric Planar Flow

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Wall Position

DikeSplash

Complete Rupture: Planar Flow

a d0.80 0.990.60 0.960.40 0.900.30 0.840.20 0.760.15 0.690.10 0.590.05 0.41

Polar Rupture Splash

Complete Rupture Symmetric Polar Flow

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Tank Radius

DikeSplash

Complete Rupture: Polar Flow

a d0.90 0.910.80 0.820.70 0.730.60 0.640.50 0.550.40 0.450.30 0.350.20 0.25

Similarity Solution

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

r

h

Polar Paraboloid Wave

t=0.0

t=0.2

t=0.4

0 ≤ r ≤ R(t ) = α 2 + 4t 2 h(r,t ) = 2α2

R2 1− r2

R2

⎛⎝⎜

⎞⎠⎟

u(r,t ) = 4rtR2

Polar Parabaloid Splash

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Tank Radius

H

Polar Paraboloid Splash

Rupture Flow Experiments

• Greenspan & Young (1978)

(a) planar flow, validated SWT

(b) sloped dikes particularly ineffective

(c) initial splash exceeds tank height

• Greenspan & Johansson (1981)

(a) polar flow

(b) dike shape study

(c) proposed trip rings, deflectors

Summary

• Low (equilibrium) dikes do not contain rupture flow

• Shallow Water Theory can be applied to address:

(a) small hole splash

(b) dike overflow

(c) multiple dike interaction

• Small-scale experiments are representative