(06xx2012) Computational Model: Pressure Change

Purpose: 

Due to cohesion and adhesion properties of water, a container of water does not level off at the breaking point between gas and liquid. Instead, it forms a parabolic shape around the breaking point known as meniscus. The small change in height can cause a change in pressure, therefore it is necessary to account for these behaviors.

Figure 1: A view of cohesion properties in water and Mercury

Figure 2: Reading of Meniscus

Solution:

One way of accounting the pressure change is to make a triangle approximation of the change in height. The volume of the parabola would be accounted by using a triangle rotating around z - axis. This will create a larger pressure change, as x*tan15 > x²  for values around zero, and we justify this by arguing that the integrity of a structure should account for at least five percent change around threshold for force and pressure.

Below is a visual representation of the triangle approximation.

from __future__ import division>
from visual import *
from time import clock







tri = Polygon( [(-20,0), (0,0.267949192*20), (0,0)] )




"""

p = paths.circle( pos=(10,4,2), radius=10 )
#tri=shapes.triangle(pos=(0,2), length=4)


extrusion(pos=p, shape=tri, color=color.yellow)


"""

tri = Polygon( [(-3,0), (0,0.267949192*3), (0,0)] )




p = paths.circle( pos=(0,0,-10), radius=10 )
#tri=shapes.triangle(pos=(0,2), length=4)


extrusion(pos=p, shape=tri, color=color.yellow)

Figure 3: Simulated meniscus using triangle rotating around z - axis

Analysis and Uncertainty:

The triangle approximation can be mathematically expressed by :

y = x * tan(θ)

x = the radius of the circle

θ = to the angle of elevation from the lowest point of meniscus to the highest point of meniscus

The theoretical change of height is approximated by a parabola :

y = x⁴

or

y = x²

The uncertainty for this approximation would therefore be expressed by:

Δy = x * tan(θ) -  x²

% error = {(x * tan(θ) -  x²) / x²} * 100

The uncertainty remain small for small values of x, and our approximation is justified.