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# - * - coding: utf-8 - * -
"""
Created on Sun May 12 20:08:52 2019
@author: MAC
"""
import numpy as np
from numpy import linalg as LA
# Constants
T_SIDE = 86164.10 # sidereal day (s)
MU = 3.9860e14 # earth standard gravitational parameter (=G * M_earth) (m3.s-2)
R_T = 6378e3 # earth radius (m)
G0 = 9.81 # standard gravity (m.s-2)
R = 287 # specific gas constant (J.Kg-1.K-1)
def initial_vel(latitude,retrograde=False):
"""
Initial velocity (m.s-1) depending on the launch point latitude (°)
:param latitude: the launch point latitude in °
:return: the initial velocity in m.s-1
initial_velocity = (2 * np.pi / T_SIDE) * R_T * np.cos(np.deg2rad(np.abs(latitude)))
if retrograde:
return -initial_velocity
else:
return initial_velocity
def final_target_vel(h, a):
"""
.. math:: \\sqrt {2 \\cdot MU \\cdot (\\frac {1}{R_T + h} - \\frac {1}{2 * a})}
:param h: injection altitude (m)
:type h: float
:param a: orbit semi major axis (m)
:type a: float
:return: orbit absolute injection velocity (m.s-1)
:rtype: float
v_square = 2 * MU * ((1 / (R_T + h)) - (1 / (2 * a)))
assert v_square >= 0, 'error: square root only on positiv values'
return np.sqrt(v_square)
"""Optimise mass loading in the case of two stage rocket
.. math:: structural_coefficient = \\frac {strcture_mass} {structure_mass + propellant_mass}
:param isp1: specific impulse of the first stage (s)
:type isp1: float
:param isp2: specific impulse of the second stage (s)
:type isp2: float
:param eps1: structural coefficient of the first stage
:type eps1: float
:param eps2: structural coefficient of the second stage
:type eps2: float
:param Vc: target critical velocity (m.s-1)
:type Vc: float
:param Mp: payload mass (Kg)
:type Mp: float
:return: mass of the first and second stage (Kg)
:rtype: float
"""
omega_tab=np.linspace(0.001,0.9,100) # array of omega to test
qp_tab=[]
for omega in omega_tab:
q2 = omega + eps1 * (1 - omega)
q2 = np.power(q2,isp1 / isp2) * np.exp(Vc / (G0 * isp2))
q2 = 1 / q2
q2 = 1 - q2
q2 *= omega / (1 - eps2)
qp = omega - q2
qp_tab.append(qp)
qp=max(qp_tab)
omega=omega_tab[qp_tab.index(qp)]
q2=omega - qp
q1=1 - q2 - qp
M1=(Mp / qp) * q1
M2=(Mp / qp) * q2
""" Compute atmospheric properties according to the International \
Standard Atmosphere (ISA) model
:param altitude: altitude in meter
:type altitude: float
:return: T, P, rho : temperature (K), pressure (Pa), density (Kg.m-3)
:rtype: float, float, float
assert altitude < 84852 , str(altitude) + ' is higher than max altitude of the ISA model'
h0_tab = np.array([0, 11e3, 20e3, 32e3, 47e3, 51e3, 71e3, 84852]) # base altitude (m)
a_tab = np.array([-6.5e-3, 0, 1e-3, 2.8e-3, 0, -2.8e-3, -2.0e-3]) # lapse rate (K.m-1)
# the base properties of each range are tabulated to reduce computational cost
T0_tab = np.array([288.15, 216.65, 216.65, 228.65, 270.65, 270.65, 214.65]) # base temperature (K)
P0_tab = np.array([101325, 22632, 5474.9, 868.02, 110.91, 66.939, 3.9564]) # base pressure (Pa)
index = 0
while altitude > h0_tab[index + 1]:
index += 1
a = a_tab[index]
T0 = T0_tab[index]
P0 = P0_tab[index]
T = T0 + a * (altitude - h0_tab[index]) # Toussaint equation
P = P0 * np.exp(-G0 * (altitude - h0_tab[index]) / (R * T))
P = P0 * np.power(T/T0, -G0/(a*R))
rho = P / (R * T)
def laval_nozzle(Pt,Tt,Pamb,Ath,Ae,gamma,R,tolerance=1e-5):
"""Compute the thrust assuming a started nozzle
.. math:: Thrust = massFlowRate \cdot exitVelocity + (exitPressure - ambiantPressure) \cdot exitArea
:param Pt: total pressure in the combustion chambre (Pa)
:type Pt: [type]
:param Tt: total temperature in the combustion chambre (K)
:type Tt: [type]
:param Pamb: ambient pressure (Pa)
:type Pamb: [type]
:param Ath: throat section area (m2)
:type Ath: [type]
:param Ae: exit section area (m2)
:type Ae: [type]
:param gamma: ratio of specific heats (adimentional)
:type gamma: [type]
:param R: specific gas constant (J.Kg-1.K-1)
:type R: [type]
:param tolerance: [description], defaults to 1e-5
:type tolerance: [type], optional
:return: Thrust (N), Specific impulse (s)
:rtype: [type]
A_ratio = lambda M,gamma: (1 / M) * np.power((2 / (gamma + 1)) * (1 + ((gamma - 1) / 2) * M * M), (gamma + 1) / (2 * gamma - 2))
# compute the nozzle exit section area corresponding to the adapted regime
Madapted = np.sqrt((2 / (gamma - 1)) * (np.power(Pamb / Pt,(1 - gamma) / gamma) - 1))
Aadapted = Ath * A_ratio(Madapted,gamma)
if Aadapted > Ae: # the nozzle is under-expanded
# compute the exit Mach number iteratively using dichotomy algorithm
A_ratio_target = Ae / Ath
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
while np.abs(error) > tolerance:
if error > 0:
Me_lower = Me
else:
Me_upper = Me
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
Te = Tt / (1 + ((gamma - 1) / 2) * Me * Me) # exit static temperature
Pe = Pt * np.power(Tt / Te,gamma / (1 - gamma)) # exit static pressure
elif Aadapted < Ae: # the nozzle is over-expanded
Mshock = 1
step = 1
A_ratio_target = Ae / Ath
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
while np.abs(error) > tolerance:
if error < 0:
Me_lower = Me
else:
Me_upper = Me
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
Te = Tt / (1 + ((gamma - 1) / 2) * Me * Me)
Pe = Pt * np.power(Tt / Te,gamma / (1 - gamma))
while (Pe - Pamb) * 1e-5 > tolerance:
Mshock += step
Pt_ratio = np.power(((gamma + 1) * Mshock * Mshock) / ((gamma - 1) * Mshock * Mshock + 2),gamma / (gamma - 1)) * np.power((gamma + 1) / (2 * gamma * Mshock * Mshock - gamma + 1),1 / (gamma - 1))
Pi = Pt * Pt_ratio
Ai = Ath / Pt_ratio
A_ratio_target = Ae / Ai
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
while np.abs(error) > tolerance:
if error < 0:
Me_lower = Me
else:
Me_upper = Me
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
Te = Tt / (1 + ((gamma - 1) / 2) * Me * Me)
Pe = Pi * np.power(Tt / Te,gamma / (1 - gamma))
if (Pe > Pamb) and (A_ratio(Mshock,gamma) > Ae / Ath):
# the normal shock is outside the divergent and all the flow inside is supersonic
A_ratio_target = Ae / Ath
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
while np.abs(error) > tolerance:
if error > 0:
Me_lower = Me
else:
Me_upper = Me
Me = (Me_upper + Me_lower) / 2
error = A_ratio_target - A_ratio(Me,gamma)
Te = Tt / (1 + ((gamma - 1) / 2) * Me * Me) # exit static temperature
Pe = Pt * np.power(Tt / Te,gamma / (1 - gamma)) # exit static pressure
break
if Pe < Pamb:
Pe = Pe_old
Mshock = Mshock_old
step /= 2
else: # the nozzle is adapted
Me = Madapted
Te = Tt / (1 + ((gamma - 1) / 2) * Me * Me) # exit static temperature
Ve = Me * np.sqrt(gamma * R * Te) # exit velocity
# maximum mass flow rate in the case of a started nozzle
Q = np.power(2 / (gamma + 1),(gamma + 1) / (2 * (gamma - 1))) * np.sqrt(gamma / (Tt * R)) * Pt * Ath
Thrust = Q * Ve + (Pe - Pamb) * Ae
isp = Thrust / (Q * G0)
return Thrust,isp
def adiabatic_flame(mech,T,P,X):
"""Compute adiabatic flame temperature
:param mech: chemical mechanism in cti format
:type mech: [type]
:param T: initial temperature (K)
:type T: [type]
:param P: pressure (Pa)
:type P: [type]
:param X: initial chemical composition
:type X: [type]
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"""
# work in progress
#%% Trajectory tools
def solve_traj():
"""
"""
# .......
#%% Astrodynamics tools
def state_to_orbital_elem(r,v):
"""
Convert state vector to orbital elements
inputs:
r : coordinate vector (m)
v : velocity vector (m.s-1)
output: {'a','e','i','capital_omega','small_omega','nu'}
where:
a : semi_major axis (m)
e : eccentricity (adimentional)
i : inclination (deg)
capital_omega : argument of the ascending node (deg)
small_omega : argument of the perigee (deg)
nu : true anomaly (deg)
NB: - An internal quantity (small) is considered for the case of undefined
classical orbital elements. When an element is undefined it is set to
zero and other(s) following it are measured according to this convention
- The function stops when r and v are aligned
- This function is widely inspired by the Matlab function written by
Prof. Josep Masdemart from UPC
"""
orbel={'a':0,'e':0,'i':0,'capital_omega':0,'small_omega':0,'nu':0}
small = 1e-10
h = np.cross(r,v) # angular momentum
rm = LA.norm(r) # module of the coordinate vector
vm = LA.norm(v) # module of the velocity vector
hm = LA.norm(h) # module of the angular momentum vector
if hm < small:
print('rectilinar or collision trajectory')
return
n = np.array([-h[1],h[0],0])
nm = LA.norm(n)
if nm < small:
n = np.array([1,0,0])
nm = 0
else:
e = ((vm * vm - MU / rm) * r - dotrv * v) / MU
em = LA.norm(e)
if em < small:
e = n
em = 0
else:
orbel['a'] = (hm * hm / MU) / (1 - em * em)
aux = h[2] / hm
if np.abs(aux) > 1: aux = np.sign(aux)
# all the angles are first computed in rad
orbel['i'] = np.arccos(aux)
if nm > small:
aux = n[0]
if np.abs(aux) > 1: aux = np.sign(aux)
orbel['capital_omega'] = np.arccos(aux)
if n[1] < 0: orbel['capital_omega'] = 2 * np.pi - orbel['capital_omega']
if em > small:
aux = np.dot(e,n)
if np.abs(aux) > 1: aux = np.sign(aux)
orbel['small_omega'] = np.arccos(aux)
if e[2] < 0: orbel['small_omega'] = 2 * np.pi - orbel['small_omega']
aux = np.dot(e,r) / rm
if np.abs(aux) > 1: aux = np.sign(aux)
orbel['nu'] = np.arccos(aux)
if em > small:
if dotrv < 0: orbel['nu'] = 2 * np.pi - orbel['nu']
if np.dot(h,ha) < 0: orbel['nu'] = 2 * np.pi - orbel['nu']
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# convert the angles to deg
orbel['i'] = np.rad2deg(orbel['i'])
orbel['capital_omega'] = np.rad2deg(orbel['capital_omega'])
orbel['small_omega'] = np.rad2deg(orbel['small_omega'])
orbel['nu'] = np.rad2deg(orbel['nu'])
return orbel
def propagate_orbit():
"""
"""
#%% Thermodynamic cycles
# the following functions provide the pressure, the mixuture composition
# and the initial temperature in the combustion chambre and properties of some
# components of the cycle.
def pressure_fed():
"""
"""
def electric_turbo():
"""
"""
def open_expander():
"""
"""
def closed_expander():
"""
"""
def dual_expander():
"""
"""
def tap_off():
"""
"""
def gas_generator():
"""
"""
def ox_rich_staged_combust():
"""
"""
def full_flow_staged_combust():
"""
"""
if __name__ == "__main__":