Formulation:
I specify a trip. A certain distance to
be covered in a certain amount of time.
The engine provides constant
acceleration for the duration of the trip.
I assume the initial and final speed
are zero and the ship travels in infinite empty space.
The first half of the trip is going to
be acceleration.
The second half of the trip is going to
be deceleration.
I specify a certain amount of fuel
during the trip.
Fuel is a set percentage of ship mass.
This makes equations invariant to ship mass.
This is a brute force example, to remove gravity assists and everything from the equations. It's just to compute engine requirements.
Consideration:
The engine is going to operate at
constant exhaust speed.
Ship mass is going to decrease as fuel
is expelled as exhaust.
To maintain constant acceleration,
exhaust flow and thrust is going to decrease over time.
Energy is going to be expended to
accelerate the exhaust to a given exhaust speed.
The faster the exhaust, the less fuel
is needed but the more energy is needed.
The equations shows the results for the acceleration trip. Results for the deceleration trip are equivalent to a trip that begins with reduced mass, the results are very similar.
Definitions:
Definition of all quantities with units of measure
Block schematic of the system
Solution:
First step is to compute ship acceleration.
Ship acceleration is constant and depends only on trip parameters.
Now I can compute the laws of motions for
the ship. It's going to be a differential equation because thrust
depends on mass and rate of mass depends on thrust needed to maintain
constant acceleration.
From the law of motion I can estimate the exhaust speed. Using less fuel requires throwing it away at an higher exhaust speed.
Now I can compute the full set of
motion equations. They do not depend on speed and acceleration or
other trajectory parameters because I decided that I use a certain
amount of fuel in a certain amount of time as original assumption.
Now I can compute the power and energy requirements of the engine.
Expelling fuel at an higher speed require expending power.
Conclusions:
This set of equations computes the parameters of the Direct Propulsion System required to complete a trip of a given distance in a given time with a given fuel consumption.
Most important are the top acceleration of the ship and the exhaust speed of the engine.
The total energy requirement scales with the square of the average travel speed. Taking more time to do the trip has the most effect on energy use.
Recap of the resulting equations
Example:
I want to make a ship capable of making a round trip to Jupiter in 5 years allotting just 1% of the ship mass as usable fuel.
No comments:
Post a Comment