Fuel Consumption for Tactical Space Combat

The Rules

The distance scale calculated from using realistic engines assumes that a ship carries sufficient fuel for one ten-turn battle at maximum acceleration for the entire battle. In a strategic campaign, however, it may be important to know how much fuel a ship has used during the battle, so that it is known how much it brings to the next battle (if it does not have time to refuel). To facilitate this, one can simply add fuel boxes to the ship control sheet. A ship gets ten fuel boxes for each point of maximum drive rating (e.g., a thrust 4 ship gets 40 fuel boxes). When one point of thrust from the drives is used (either to accelerate or to rotate), check off one fuel box. For example, if a ship has orders to rotate port 2 facings (RP2) and accelerate 2 thrust points (MD2), it will check off three fuel boxes. When the ship is out of fuel, it can no longer maneuver.

For your information, each fuel box represents a certain number (0.4 x MASS of ship x maximum thrust rating of ship for Full Thrust) of tons of fuel, this number being based upon the mass of the ship. Thus, while a big ship needs to burn more fuel to reach a certain acceleration, this is accounted for by the fact that each fuel box represents more fuel. The mass of this extra fuel was already included in the mass of the drive system as specified in the design rules. Keep in mind that this system is not an abstraction! Using this system the mass of the ship cancels in the equations. It is therefore a simple, fast, and convenient system for keeping track of the fuel used.

The second optional rule arises from the fact that in actuality, as a ship burns fuel it masses less and therefore gets more acceleration for each fuel box burned. This can also be simply included. Divide the ship's fuels boxes on the ship control sheet as follows:

Fuel Boxes Burned Thrust points produced

1-22 1

23-24 1 1/10

25-27 1 1/9

28-30 1 1/8

31-35 1 1/7

36-41 1 1/6

42-49 1 1/5

50-62 1 1/4

63-82 1 1/3

83-124 1 1/2

125+ 2

Fuel Boxes Burned	Thrust points produced
1-22	1
23-24	1 1/10
25-27	1 1/9
28-30	1 1/8
31-35	1 1/7
36-41	1 1/6
42-49	1 1/5
50-62	1 1/4
63-82	1 1/3
83-124	1 1/2
125+	2

For example, if a ship has burned 32 fuel boxes prior to the start of the turn, it could burn 7 fuel boxes (assuming it's a thrust 7 ship!) and get 8 thrust points. Always round fractions down and the "fractional" thrust points do not carry over from turn to turn (though if one doesn't mind the extra bookkeeping this can be done as well - the effect is so small it's really not worth it). Note that these extra thrust points only come into play for ships of thrust 5 and higher (remember, a ship only has 10 x thrust rating fuel boxes). See the example card for one way to keep track of this conveniently.

The Equations

For the curious, here are the equations used to derive the above results, so that you can find the mass per fuel box for engines of your own design or preference. These equations are not needed to use the rules above if you're willing to take those results on faith. I will assume that you have first read the article on realistic engines, as I will use those results as a starting point. For our purposes initially, the important results are:

Mp (tons) =N (drive units) * F (kN) * Dt (s)/(ve (km/s) * 1000 (kg/ton))

N (drive units) = P (percentage) / 100 * Mship (tons) / (Mengine (tons) + T (turns/game) * F (kN) * Dt (sec/turn)/(ve
(km/s) * 1000 (kg/ton)))

Mp is the mass (in tons) of fuel burned per turn per thrust point. Now define:

N_ (drive units/mass of ship) = P (percentage) / 100 / (Mengine (tons) + T (turns/game) * Dt (sec/turn) * F (kN) / (ve (km/s) * 1000 (kg/ton)))

Mp_ (tons/turn/mass of ship) = N_ (drive units/mass of ship) * F (kN) * Dt (sec/turn) / (ve (km/s) * 1000 (kg/ton))

Mp_ is the burn rate for this particular engine. Now define a "fuel box" (one box on the fuel cards as described above) as representing Mp_ times the mass of the ship of fuel. In other words:

1 fuel box = Mp_ * Mship

Not coincidentally, this is exactly the same number as Mp, the total fuel burned per turn per thrust point. But now note that every ship, regardless of its mass, will have one fuel block per maximum thrust rating of the ship for each turn of the "typical" game (defined as T in the engines article). For the case of Full Thrust, if a typical game is 10 turns, a thrust 6 ship will have 60 fuel boxes, a thrust 4 ship will have 40 fuel boxes, etc., independent of the ship's mass. Each fuel box, however, represents a different mass of fuel depending upon the size of the ship. Large ships need more fuel to achieve a certain acceleration, but this is exactly accounted for by the larger amount of fuel represented by each fuel box. Remember that the mass of the fuel for T turns (10 in this case) is included in the initial purchase of the drive itself. The only time you need to know the mass of fuel represented by each fuel box is if you wish to purchase extra fuel for your ships.

Now for the second rule, which allows ships to accelerate more for each fuel box burned because the ship is actually massing less as the fuel is burned. [A side note: equations involving exponentials do not work well in HTML. I recommend writing the equations out to really see what they are.] Basically, we want to know how much mass we must burn before each unit of fuel produces 1 1/10 thrust points (in other words, burning 10 fuel boxes produces 11 thrust points), how many before each unit of produces 1 1/9 thrust points, etc. The breakpoints (1 1/0, 1 1/9, etc.) are chosen arbitrarily, but there's no reason not to keep it simple. The best way to show this is with an example. Let's see how many fuel boxes we have to burn to achieve 1 1/10 or 110% of our "baseline" fully-fueled acceleration.

Using the rocket equation (see the engines article), the acceleration is:

a (km/s^2) = ve (km/s) / Dt (sec) * Ln[Mship (tons)/(Mship (tons) - Mp (tons))]

where:

a = acceleration calculated for the engine
ve = exhaust velocity of the engine
Dt = turn length
Mship = starting mass of ship
Mp = mass of propellant burned

We want to solve this equation for Mp:

Mp (tons) = Mship (tons) *(1 - Exp[-1 * a (km/s^2) * Dt (sec) / ve (km/s)]

This is how much fuel we will burn per fuel box, regardless of the ship's current mass. The acceleration we actually get is therefore:

a (km/s^2) = ve (km/s) / Dt (sec) * Ln[Mcurrent (tons)/(Mcurrent (tons) - Mship (tons) *(1 - 1 / Exp[a (km/s^2) * Dt (sec) / ve (km/s)])]

where Mcurrent is the current mass of the ship (after having burned some fuel) and Mship is the starting mass, as defined above.

We want to know what is the current mass of the ship when the acceleration produced is equal to our target "breakpoint" acceleration (1 1/10 * normal, 1 1/9 * normal, etc.). We must therefore solve this equation for Mcurrent:

BP * a_normal = ve / Dt * Ln[Mcurrent/(Mcurrent - Mship *(1 - Exp[-1 * a_normal * Dt / ve ])]

=> Mcurrent = Mship * Exp[a_normal * Dt * (Bp - 1) / ve] * (Exp[a_normal * Dt / ve] -1) / (Exp[Bp * a_normal * Dt / ve] -1)

where a_normal is the normal acceleration of the fully-fueled ship and Bp is the breakpoint desired (1 1/0, 1 1/9, etc.).

The amount of fuel burned to reach this breakpoint is:

M = Mship - Mcurrent = Mship * ( Exp[a_normal * Dt * (Bp - 1) / ve] - 1) / (Exp[Bp * a_normal * Dt / ve] -1)

and the number of fuel boxes this represents is:

fuel boxes = M / Mp = (Exp[a_normal * Dt * (Bp - 1) / ve] - 1) / ((Exp[Bp * a_normal * Dt / ve] -1) * (1 - Exp[-1 * a_normal * Dt / ve]))

For example, for the D-T fusion drive, a_normal = 0.0000999 km/s^2, ve = 22 km/s and we chose Dt = 900 sec. In order to get 1 1/10 thrust points per fuel box, then, we need to burn:

fuel boxes = ( Exp[0.0000999 * 900 * (1.1 - 1) / 22] - 1) / ((Exp[1.1 * 0.0000999 * 900 / 22] -1) * (1 - Exp[-1 * 0.0000999 * 900 / 22]))

fuel boxes = 22.2, which matches the table found above.

In summary then, all you need do is use the final equation:

fuel boxes = (Exp[a_normal * Dt * (Bp - 1) / ve] - 1) / ((Exp[Bp * a_normal * Dt / ve] -1) * (1 - Exp[-1 * a_normal * Dt / ve]))

for each breakpoint you wish to have on your fuel cards. See the example card for one way to do this conveniently.

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