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# Burj Khalifa

The Burj khalifa, the tallest building in the world, took 5 years to build with over 22 million man hours. Standing at 828m it is almost twice as a tall as the Empire State Building.

# How do they work?

A water jet is industrial tool used to cut a variety of materials from copper to titanium by using extremely high pressure water. It is the preferred tool used to cut materials sensitive at high temperature generated by other machines.

There are two types of water jets, plain water jets that only use water and abrasive ones that use the velocity of the water coming out of the nozzle to carry the abrasives (usually garnet minerals,high density and high specific gravity) and erode the material.

# What if we used the Burj Khalifa as our Plain water jet? Could it cut through steel?

p(initial)=0

g=9.8m/s^2

P=p(initial)+pgh

P=F/A

radius of hole that water is coming out of:(.05mm)

Burj khalifa height : 828m

P=(828m)(9.8m/s^2)(1000kg/m^3)

P=8114400pa

Force needed to pierce metal sheet:

F=PxTxPSI

F=force required

P=Perimeter around area to be pierced

T=Thickness of sheet

PSi=Shear strength rating (newtons per square inch, in this case change to m)

Below is force required to pierce 10cm of Steel, Structural ASTM-A36

F=(.000314159m)x(.1m)x((4x10^8)N/m^2)

F=12566.36N

P=F/A

P=12566.36N/((.00005 m)^2)(pi)

P=1.60x10^12 pascals to pierce 10 cm of steel with water

nowhere near enough water pressure

But wait, all of the above is wrong,

Water is non compressible, it cannot puncture steel, it erodes it little by little. It may seem like it is cutting because the pressure is so high and the erosion occurs in seconds but it is still erosion.

# Empirical graph of depth of cut vs pressure with a plain water jet

the graph above can be found in reference [1] page 16

This data was collected using a .15mm nozzle diameter at pressures one to 10k bars.

The graph above shows X amount of pressure a water jet with no abrasive needs to cut through an Y amount of millimeters of a material.

It takes around 10K Bars to cut through around 2mm of mild steel.

1 bar=100,000 Pascals

10,000x 100,Pascals=1,000,000,000 Pascals

Would the building be able to cut steel: Not even close

# How tall would the building need to be?

P=pgh

1,000,000,000Pa=(1000kg/m^3)(9.8m/s^2)(Height)

Height:102040.8163m or 102.04km

that would mean the building would be in the mesosphere where temperatures are -120degrees fahrenheit.

The picture below would be our view if the building like that existed...

# How fast is the water coming out of the building at its actual height and at its hypothetical height?

We can derive from Bernoulli's equation that v^2=2gh

P1=P2

P+.5p(v^2)+pgh=P+.5p(v^2)+pgh

P+.5p(v^2)+pg(0)=P+.5p(0)+pgh

v^=2gh

Actual building:

V^2=2(9.8m/s^2)(828m)

V=127.39m/s

Hypothetical building:

V^2=2(9.8m/s^2)(102040.8163m)

V=1414.2135m/s

The water would need to going as fast a jet plane in order to cut through 2mm of steelâ€¦

almost 5 times faster then speed of sound

speed of sound=340m/s

# Empirical equation of abrasive water cut

The equation above is on page 38 in reference [2]

The equation above was empirically derived from a paper that "assess the influence of abrasive waterjet cutting process parameters on depth of cut of stainless steel." (last 2 line of page 34)

We can see from the equation that if the pressure increases the depth of the cut increases.

We can also see from the equation that  the average diameter of particle and density of particle has a significant impact to the depth of cut of stainless steel.  As the average diameter of particle and the density of particle increases , the depth of the cut increases.

# Other empirical abrasive water jetting models

To derive a velocity needed to cut through 10cm of steel would not be possible because there are constants that we do not have. Luckily in thesis he has his own experimental data below

graph above can be found in reference [1] page 22

The data was collected by the use of an abrasive water jet with an orifice(hole) .1mm diameter and a abrasive flow rate of 100g/min cutting  stainless steel and aluminium at differing pressures of 400 mpa,600 mpa and 800 mpa. We can see that with increasing pressure the depth of the cut increases as well.

We can also see from the graph that it take around 500mpa to cut through 10mm of stainless steel.

# How tall would a Burj Khalifa filled with abrasive water need to be to cut through 1 cm of steel?

P=pgh

P=500x10^6=500,000,000pa (pressure it takes to cut through 1 cm of steel)

g=9.8m/s^2

p=1000kg/m^3(same because in a water jet the abrasive is added to at the end after pressure has accumulated at end, go back to reference "how do they work picture" to get an idea)

P=51020.4m or 51km tall

and your view would be something be like

# What if we used liquid mercury to fill the building?

P=pgh

P=1,000,000,000pa(same pressure needed as water because also non compressible)

p=13,594kg/m^3

g=9.8m/s^2

h=7506.31m or 7.5km

and this, would be your view.

# References

[1] Mostafa Ahmed Kamel, Mohamed. Waterjet Cutting up to 900 MPa. Thesis. Hannover, Univ.,, 2004. Hannover: n.p., 2004. Print

[2]: Selvan C., Raju M., (2011). Assessment of process parameters in abrasive waterjet cutting of stainless steel, International Journal of Advances in Engineering & Technology, 1(3), 34-40