The Aluminum pressure vessel stress can most likely be simplified to Hoop Stress for Thin-Walled pressure vessels. If that is the case, the condition which the objective function (gross lift off weight) would be subject to would be; $$ \frac{P_{i}r}{t}\left (FS \right )\leq S_{y} $$ where $$P_{i}$$ is the internal pressure of the vessel,$$r$$ is the radius, $$t$$ is the thickness, $$FC$$ is the factor of safety and $$S_{y}$$ is the yield strength.
More specifically, the condition will become; $$ \frac{P_{i}r}{t}\left (FS \right )\leq 255 MPa $$
The weight of the vessel will be; $$ \rho _{Al} g\left ( \pi l\left ( r_{o}^{2}-r_{i}^{2} \right )+2\pi r_{i}^{2} \right )=W_{Al} $$
where $$\rho _{Al}$$ is the density, $$g$$ is the acceleration due to gravity, $$r_{o}$$ is the outer radius,and $$r_{i}$$ is the inner radius.
And, $$\rho _{Al}=2700\frac{kg}{m^{2}}$$
In [3]:
import math as m
from sympy import solveset
# Aluminum Pressure Vessel Dimensions
Pi = 10 # Internal pressure (Pa)
Po = 1 # External pressure (Pa)
ri = 1 # Inner radius (m)
ro = 1.1 # Outer radius (m)
r = (ro-ri)/2 # Average radius (m)
t = 0.1 # Wall thickness (m)
l = 5 # Length (m)
# Aluminum Properties (1060)
rho_Al = 2700 # Material density (kg/m^3)
S_Al = 255*10**6 # Yield stress - Al. (Pa)
# Misc. Parameters
FS = 1.1 # Factor of safety
g = 9.81 # Gravity (m/s^2)
# note** in python statements are always delimited by an ':'
if (r/t) <= 10: # Thin-Wall Condition
Hoop_Stress = (Pi*r)/t
Radial_Stress = -Pi/2
Longitudinal_Stress = (Pi*r)/(2*t)
else:
Hoop_Stress = (Pi*ri**2 - Po*ro**2 - ri**2*ro**2*(po-pi)/r**2)/(ro**2 - ri**2)
Radial_Stress = (Pi*ri**2 - Po*ro**2 + ri**2*ro**2*(po-pi)/r**2)/(ro**2 - ri**2)
Longitudinal_Stress = (Pi*ri**2)/(ro**2 - ri**2)
# where is sigma_name, what does it do?
Sigma = abs([Hoop_Stress, Radial_Stress, Longitudinal_Stress])
#solveset from the sympy package is like syms in matlab, but what are you solving for? hoop, radial and longitudnal stress already have values?!
[Hoop_Stress, Radial_Stress, Longitudinal_Stress, Sigma_Name] = solveset(Hoop_Stress, Radial_Stress, Longitudinal_Stress, Sigma_Name)
Max_Stress = max(Sigma)
Sigma_Max = find(Sigma==Max_Stress) # Maximum stress
Max_Stress_Type = Sigma_Name(1, Sigma_Max)
if Sigma_Max<(S_Al*FS): # Stress Analysis
print('Stress_Analysis:\n\n Pass\n')
else:
print('Stress_Analysis:\n\n Fail\n')
Volume_Al = pi*l*(ro**2-ri**2)+2*pi*ri**2 # Total Volume (m^3)
Mass_Al = rho_Al*Volume_Al; # Total Mass (kg)
Weight_Al = Mass_Al*g # Total Weight (N)
Carbon Fiber & Nomex Pressure Vessel
The Carbon Fiber & Nomex pressure vessel conditions will be set up in a similar way.
The weight of the vessel is constructed as follows; Weight of Carbon Fiber; $$ \rho _{CF}g\pi\lt_{CF}\left ( r_{CF,i}^{2} +r_{CF,i}^{2}\right )=W_{CF} $$
Weight of Nomex; $$ \rho _{Nom}g\pi l t_{Nom} r_{Nom}^{2}=W_{Nom} $$
Weight of End Caps; $$ 2\rho _{Al}\pi l\pi r_{i}^{2}=W_{Al} $$
And the total; $$ W_{CF}+W_{Nom}+W_{Al}=W_{total} $$
The stress will be a similar inequality as above. However, the actual values are difficult to estimate. I may need to perform FEA, find some better research, or estimate by superposition. **Need to discuss.
In [ ]:
import math as m
# Carbon Fiber & Nomex Pressure Vessel
Pi = 10 # Internal Pressure (Pa)
Po = 1 # External Pressure (Pa)
t_CF = 0.01 # Carbon Fiber Thickness (m)
t_Nom = .01 # Nomex Thickness (m)
r_CF_i = 1 # Inner Radius (m)
r_CF_o = 1.1 # Outer Radius (m)
r_Nom = 2 # Average Radius (m)
l = 5 # Length (m)
rho_CF = 2700 # Material Density (kg/m^3)
S_CF = 300*10**6 # Yield Stress - Aluminum (Pa)
rho_Nom = 2700 # Material Density (kg/m^3)
S_Nom = 150*10**6 # Yield Stress - Aluminum (Pa)
FS = 1.1 # Factor of Safety
g = 9.81 # Gravity (m/s^2)
S_CF = 30 # Yield Strength (N)
S_Nom = 13 # Initial Tear Strength (N)
'''
S_Total # Total Yield Stress (To be Determined)
Sigma_Total # Total Max Stress (To be Determined)
'''
Volume_CF = pi*l*t_CF*(r_CF_i**2+r_CF_o**2)# Volume of Carbon Fiber (m^3)
Volume_Nom = pi*l*t_Nom*r_Nom**2 # Volume of Nomex (m^3)
Mass_CF = rho_CF*Volume_CF # Mass of Carbon Fiber (kg)
Mass_Nom = rho_Nom*Volume_Nom # Mass of Nomex (kg)
Weight_Total =(Mass_CF+Mass_Nom)*g # Total Weight of Vessel (N)