Wednesday, 10 May 2017 17:38

Space mission energy and fuel calculation

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The grand and lonely journey

On 5th of September, 1977, NASA launched Voyager 1 space probe. So far it has traveled 125 AU as of August 2013, and it is still traveling at the relative speed against the Sun at 17 km per second, which is the fastest speed of an object made by mankind. The rocket engine that lift Voyager 1 was Titan IIIE. NASA did great missions which took advantage of The Planetary Grand Tour and gravitational slingshot. Using gravity energy during the mission, Voyager 1 and 2 are now into the eternity of interstellar travel of lonely and silent.

Rocket engine has made history

Rocket launch has been the only solution in space exploration programs around the world. The theory was written into text book by the Russian scientist Konstantin Tsiolkovsky. In second world war, under the leadership of Wernher von Braun, rocket technology was successfully developed into combat weaponry by Nazi Germany. After WWII, the need to develop nuclear warhead delivery system, as well as the cold war between western allies, mainly lead by the US against the communist countries lead by USSR opened the golden era of space competition. In the end, the free world, with superior financial, economy, industrial and scientific backing, won the race. Today, Voyager I is still the fastest man made object heading to the destination of eternity. Even though it was launched by rocket engine, gravity assistance made huge contribution to its mission success.

The problems of rocket technology

Space programs are expensive and dangerous, especially missions that have astronauts on board. Such missions have to consider life support, food, water etc., which add more cost. The reality is, after Apollo programs, all the manned missions are within low Earth orbits. To go to Mars, the shortest journey will take 6 months. If astronauts take landing, and stay for a period of time on the Mars, their mission can be as long as 500 days.

Speedy journey will reduce the cost because less food and water are required. But how can you make the journey faster? The obvious answer is to make longer acceleration. Best of all, to have 1 g acceleration half way, and -1 g second half journey. But can you do it?

The calculation for a speedy journey 

All successful space missions require careful planing and calculation. Somehow, with respect, the rocket equations derived by Konstantin Tsiolkovsky don't consider fuel efficiency and fuel density as factors that affect the outcome. One thing that has been completely ignored is the propellant temperature. For all the rockets engines, the gaseous jet flow has temperature around 3500k*, which is '10 to 15 times higher than the normal air temperature. The high temperature indicates energy that is wasted. One more important thing is the mass of the propellant versus the mass of a rocket or a spaceship ratio. Such ratio will affect the energy efficiency. When a given amount of fuel is used, most the energy is actually spent on propellant acceleration instead of spacecraft acceleration.

In order to find a better way to analysis the mechanical work in relation to thermal expansion energy, we will redesign a "virtual" 100% efficient rocket engine that doesn't have the thermal temperature problem as a way to gain spaceship momentum change. Below is the condition to make such "perfectly efficient" rocket engine.

  1. In space, without any other external force during the acceleration of both ejected propellant and a vehicle, with any given acceleration length, the mass of both propellant ejected from the vehicle and the remaining vehicle receive the same reaction force, whereas such action force create constant acceleration for the vehicle.
  2. As soon as the propellant exit, the energy it carried, which equals to the energy density times the mass, is completely dissipated into kinetic energy of propellant and vehicle. (E =Ed*Mass)
  3. In order to maintain vehicle constant acceleration, follow on jettison of mass by the vehicle must take into account prior lost mass. This sequential mass ejection must happen as soon as prior sequence finished.
  4. Regardless how many sequences of propellant ejection, the remaining mass of the vehicle plus fuel on board will maintain its constant acceleration.

Following are the math calculation.

Let energy density of fuel as Ed, Vehicle in space as M, jettison mass of each sequence as dm, vehicle acceleration rate asa, and ejection mass acceleration rate as a0, acceleration length as L, acceleration for each sequence as T.

For each sequence of acceleration, the energy used for both mass of vehicle and propellant will be:

Rocket-energy-and-propellant-equation-01 (1)

Where U is the work done by fuel energy, F is the force acting on vehicle and propellant.

Rocket-energy-and-propellant-equation-02 (2)
Rocket-energy-and-propellant-equation-03 (3)
Rocket-energy-and-propellant-equation-04 (4)

Replace L in equation (1) with (4), a and a0 in equation (4) with (2), (3), we get:

Rocket-energy-and-propellant-equation-05 (5)

Equation (5) represents fuel energy being converted into system kinetic energy. From equations (1), (2) and (5), we get:

Rocket-energy-and-propellant-equation-06 (6)
Rocket-energy-and-propellant-equation-07 (7)

From equation (5), we have:

Rocket-energy-and-propellant-equation-08 (8)

Replace F and dm in equation (8) with equations (2) and (12):

Rocket-energy-and-propellant-equation-09 (9)


Rocket-energy-and-propellant-equation-10 (10)

For consecutive sequential propellant ejection, the total amount of mass used for n sequences will be:

Rocket-energy-and-propellant-equation-11 (11)

For each term in summation, the first one, it can be expressed as:

Rocket-energy-and-propellant-equation-12 (12)
Rocket-energy-and-propellant-equation-13 (13)
Rocket-energy-and-propellant-equation-14 (14)
Rocket-energy-and-propellant-equation-15 (15)
Rocket-energy-and-propellant-equation-16 (16)
Rocket-energy-and-propellant-equation-17 (17)

If we use X for aLM, Y for (aL + Ed), and Z for Ed=Y we can simplify equation (11) as:

Rocket-energy-and-propellant-equation-18 (18)
Rocket-energy-and-propellant-equation-19 (19)

which can be written in closed form. For this, we consider the identity

Rocket-energy-and-propellant-equation-20 (20)

and we rewrite (18) with (20) as

Rocket-energy-and-propellant-equation-21 (21)

The vehicle velocity after n sequences of propellant ejections will be:

Rocket-energy-and-propellant-equation-22 (22)

To decelerate vehicle to ZERO velocity for the same frame of reference, the vehicle can eject mass to the opposite direction with same rate as acceleration and same sequences of propellant ejection:

Rocket-energy-and-propellant-equation-23 (23)

The mass used for deceleration will be:

Rocket-energy-and-propellant-equation-24 (24)


Rocket-energy-and-propellant-equation-25 (25)
Rocket-energy-and-propellant-equation-26 (26)
Rocket-energy-and-propellant-equation-27 (27)
Rocket-energy-and-propellant-equation-28 (28)
Rocket-energy-and-propellant-equation-28 (29)

Similar to equation (18), we can simplify equation (24) as:

Rocket-energy-and-propellant-equation-30 (30)

And proceeding in analogy with (18)

Rocket-energy-and-propellant-equation-31 (31)


Rocket-energy-and-propellant-equation-32 (32)

For a mission carries certain amount of fuel, let the fuel mass to be f:

Rocket-energy-and-propellant-equation-33 (33)
Rocket-energy-and-propellant-equation-34 (34)

When equation (33) result is negative, the upper bound of summation must be reduced in order to decelerate vehicle to ZERO for the same frame of reference.











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