Pressure Vessel Burst Example

I have been doing some research lately into pressure vessel bursts in order to get an idea of the lethal distance in the event of a burst/explosion. I thought I would share an example as it might be useful to others.

In a pressure vessel burst there are two things that can hurt you, the pressure/shock wave and fragments of the vessel. In this post I will go through calculating the "overpressure" - the pressure above atmospheric and the impulse - The force due to the overpressure/ fast moving air.

What does that mean? Well, in a explosion, the pressure time relationship at a point some distance away from the blast will look like this:

The overpressure is the highest pressure experienced at the point and the impulse being the integral of the overpressure experienced as a force or wind. What does it mean? Well 10PSI or 0.069MPa will result in death of most people.

Internal Pressure:$P_1=3.5MPa$
Ambient Pressure:$P_0=0.1MPa$
Tank Volume: $V=9L$
Height of tank (mid point) $H_s = 1m$
Tank Diameter: $D=0.24m$
Tank Length: $L=0.5m$
Gamma: $\gamma=1.4$ (nitrogen)
Distance at which pressure damage is calculated: $D=5m$
Ambient speed of sound $a_0 = 340m/s$

Assumptions: Cylindrical pressure vessel at ground level with vertical orientation. All energy gets released from the vessel. In reality as much as %30-%40 would get transferred into fragments depending on the material and failure conditions.

Energy stored in vessel:

$E=\frac{(P_1-P_0)*2*V_1}{\gamma-1}$
$=\frac{(3.5-0.1)*2*9e-3}{\1.4-1}$
$=0.1125MJ$

Burst Pressure Ratio:

$=P_1/P_0$

$=3.5/0.1$

$=35$

Scaled Standoff Distance:

$\bar{D}=D(\frac{P_0}{E})^\frac{1}{3}$

$=D(\frac{0.1e6}{0.1125e6})^\frac{1}{3}$

$=4.8$

Scaled Side-On peak overpressure:

The above plot shows scales standoff distance vs scaled side-on peak overpressure for a variety of burst pressure ratios. So our scaled side-on peak overpressure is around 0.04

$\bar{P_s}=0.04$

Scaled Side-On Impulse:

Above is scaled standoff distance vs scaled side-on impulse. Scaled side-on impulse is around 0.03

$\bar{i_s}=0.03$

Correct for tank geometry:

The above plots are only true for a spherical pressure vessel in free air. In reality the ground reflects the shock wave generated when the vessel bursts and increases the pressure. Also a cylindrical pressure vessel can result in a higher overpressure depending on its orientation.

It is generally accepted that the ground doubles the effective length of the vessel for a upright cylinder

$L'/D = 2*L/D$

$=4$

Interestingly, for a horizontal vessel:

$L'/D = L/D^\frac{1}{2}$

The ratio of vessel height to diameter:

$H/R = H/D/2$

$=8$

The above are plots of Scaled standoff radius vs overpressure and impulse ratios (correction factors) for L/D and also H/R we can see we need that we need to multiply our scaled side on peak overpressure by 1.6 and 1.3. Also we need to multiply the impulse by 1.2 twice to account for the height and geometry.

$\bar{P_s}=0.04*1.3*1.6$

$=0.0768$

$\bar{i_s}=0.03*1.2*1.2$

$=0.0468$

Side-on peak overpressure:

So the side on pressure is simply the scales value multiplied by the atmospheric pressure.

$P_s=P_0*\bar{P_s}$

$=.0768MPa$ 

And the side on impulse is given by:

$I_s=\frac{\bar{i_s}*P_0^\frac{2}{3}*E^\frac{1}{3}}{a_0}$

$I_s=14.31Pa-s$

In the next post I will go into calculating the distance fragments from the explosion could travel. Fragments are what you would really want to worry about for a small, thin walled pressure vessel. They are much more dificult to account for because the fragmentation depends much more on the
vessel and failure conditions.