The Fuel Tanker Problem – Tristan J. Newman

Introduction:

In space, the effective currency governing how far or fast a spacecraft can go is called delta-v (Δv), or change in velocity. In effect the total delta-v of a rocket is the measure of how much its speed

would increase by expelling its entire mass of fuel through an engine. Of course, external forces such as gravity and atmospheric drag will cause the dissipation of kinetic energy from the rocket, so

the actual delta-v required to achieve orbit from the surface of a body is always larger than the orbital velocity at the desired altitude.

Because of the first (energy can be neither created nor destroyed) and second (there is no process

which can completely convert energy from one form to another without dissipation losses) laws of

thermodynamics, delta-v requires the expulsion of fuel through a physical engine. A physical engine

has an intrinsic property called specific impulse (ISP) that is the measure of how much thrust can

be generated per unit weight of fuel burned per unit time. ISP itself has units of time, but it is incorrect to think of it as a measure of time. However, when multiplied by an acceleration (typically

the acceleration due to gravity at the surface of the Earth, or 9.8 meters per second per second), one

gets a value commonly referred to as exhaust velocity (ve), the effective velocity of the exhaust gases that pass through the nozzle of the physical engine.

The Tsiolkovsky Rocket Equation (TRE) uses the exhaust velocity of a physical engine to calculate

the amount of delta-v attainable by a rocket during a maneuver (or series of maneuvers) given the

mass of the rocket both before (m0) and afterwards (mf):

(TRE)

(

)

TRE will become important later when discussing mass ratios. For now, some terms must be

defined before we can comprehensively discuss the titular Fuel Tanker Equations.

Definitions:

 Fuel Tanker – A fuel tanker in this context is any spacecraft which is capable of making a

round trip from a refueling point to a cargo loading point, where it will take on a certain

pre-measured quantity of fuel and bring it back to the refueling point. A single stage and

single engine (or configuration of engines) is used for the entire mission cycle.

 Refueling Point – The refueling point is the point in space at which the fuel tanker offloads

its cargo, and replenishes its own tanks of burnable fuel.

 Cargo Loading Point – The point at which non-burnable fuel (cargo fuel) is loaded onto the

tanker. No refueling takes place at the cargo loading point.

 Mission Cycle – A mission cycle is a round trip from the refueling point to the cargo loading

point and back again.

 Burnable Fuel – Burnable fuel is the fuel that the tanker is allowed to burn during its designated mission cycle.

 Non-Burnable (Cargo) Fuel – Non-burnable fuel (cargo fuel) is fuel that the spacecraft is

not allowed to burn during its designated mission cycle. A certain amount of cargo fuel is

designated to become burnable fuel at the refueling point.

Figure 1 - Typical mission cycle for a fuel tanker type spacecraft.

Mission Constraints

There are two primary mission constraints and for a fuel tanker type spacecraft.

1. It must be able to complete a mission cycle on one engine or set of engines configured in a

single stage. This is the reusability requirement, as a tanker should be useful for multiple

mission cycles without need of replacement.

Furthermore, each leg of the mission cycle must have consistent delta-v requirements – the

amount of burnable fuel allowed is determined by the delta-v requirements for a given trip,

and requiring the fuel tanker to expend any more or less delta-v may disturb the

equilibrium of delivering and reusing cargo fuel for each mission cycle.

Quantitatively, this implies that the specific impulse is constant and the same for both legs

of the mission cycle. If ISP is constant, and the delta-v requirements for each leg is known

then it follows from TRE that ( ⁄

of each

), where R is the mass ratio ⁄

leg of the mission cycle.

Therefore, if and

are defined to be the tanker’s mass at the start and end of the first

leg respectively, and and

are the tanker’s mass at the start and end of the second leg

respectively, then the following ratios must be constant.

(1)

(

)

(2)

(

)

2. The tanker should be designed such that it burns through all of the burnable fuel in its stores before reaching the refueling point. The only fuel that is not burned is the cargo fuel.

Then, after refueling it must be able to offload a predetermined amount of fuel, using the

remainder to refill its own burnable fuel storage tanks.

Taking this requirement into consideration, the amount of burnable fuel can be expressed in

terms of the total amount of cargo fuel to be carried, and the amount of cargo fuel that is to

be offloaded at the refueling point.

(3)

Where and

are the respective amounts of fuel burned during the initial and final

legs of the mission cycle.

Mass Targets for Mission Cycle Completion

As mentioned in the introduction, delta-v can only be achieved through reduction in mass. TRE

shows that the primary driver for delta-v is mass difference: a high initial mass and a low final mass

yield a larger delta-v. However, using more efficient engines (i.e. higher ISP) will yield greater delta-

v for a given mass of fuel. It was shown in equations (1) and (2) that the mass ratio of the tanker at

the end of the mission cycle leg to that at the start can be expressed entirely as a function of exhaust

velocity (which is directly proportional to ISP), and the desired amount of delta-v to be achieved for

that leg.

Since the ratios are constant known values, this means that the masses at each point of the mission

cycle must also be constant known values. When engineering the spacecraft, these are known as

mass targets for the mission.

The first mass target is the initial mass of the tanker when unladen with cargo but fully stocked in

burnable fuel. To derive this, we must consider the definition set forth for the mass ratio of the initial leg

Between the point when the spacecraft had mass to when it had mass

, the amount of fuel

burned had a mass of . Therefore

can be written as

(4)

( )

By applying the same algebra to the second leg, it follows that the second mass change is

(5)

( )

Summing equations (4) and (5) yields the left hand side of equation (3), therefore

(

) ( )

Since the second mission constraint requires that no burnable fuel be loaded back onto the

spacecraft at the cargo loading point, the only mass change that occurs between and

is the

onloading of

in cargo fuel (i.e.

). Therefore equation (3) becomes

(

) ( ) ( )

(

) ( ) ( )

All that remains is to solve algebraically for .

[(

) ( ) ] [ ( )]

[

] [ ]

[

]

[

]

Note that the quantity

can

( ⁄ ) ( ⁄ ) ( ( )⁄ )

be interpreted physically as the mass ratio over the course of the entire mission cycle if the fuel

tanker does not take on cargo.

(6)

[

]

Because

, the second mass target is

(7)

[

]

Furthermore, since

, the third mass target can be expressed as

[

]

Which can be simplified algebraically to

[

( ) ( ) ]

[

]

(8)

[

]

Finally, the mass ratio R2 can be used to find the fourth and final mass target

(9)

[

]

To summarize, the four mass targets for mission cycle completion are:

[ ]

[

]

[

]

[

{

]

The Cargo-Offload Relation

Although useful in designing a spacecraft, the mass targets by themselves do not provide concrete

values without knowing what to input for

and

. These are as much properties of the

spacecraft as the mass targets themselves are. The exact values of these depend on the mass of the

rest of the spacecraft, which includes all structure and fuel tanks.

Structure mass is represented solely by the value

which includes the aggregate mass of all

non-fuel carrying components such as command/control units, RCS fuel, engine mass, decouplers,

connections, and landing legs.

The mass of fuel carrying elements can be regarded with two separate values: those for burnable

fuel carrying elements, and those for cargo carrying elements. This mass is a linear function of the

mass of fuel carried. For burnable fuel:

( )

(10)

( )

Where

is the burnable fuel tank factor, or the total mass of fuel tanks which

⁄

must be added to accommodate one unit mass of burnable fuel. There also exists a similarly-defined

cargo fuel tank factor given by

.

⁄

(11)

The initial fully fueled, cargo unladen mass can therefore be expressed in terms of the individual

masses for the fundamental components of the spacecraft: Structure, burnable fuel tanks, and cargo

fuel tanks (plus an additional term for the total mass of fuel carried).

(12)

Substituting equations (10) and (11) into (12) yields

(13)

( )

Because equation (13) is equal to (6)

( )

[

]

Unifying each term on basis of mass yields

( ) ( )

[

]

[

]

(14)

[

]

[

]

Equation (14) is called the Cargo-Offload Relation, as the total mass distribution of the tanker must

satisfy it in order to be effective. Generally, either the cargo mass or the offload mass is known. For

example, if a fuel tanker is designed to remove all the fuel from a surface base in one mission cycle

then

is known. However, if the mission plan calls for a tanker to completely refill the stores

of an orbiting space station then

would be the fixed quantity.

As an exercise, it can be shown that similarly expressing the tanker’s mass at the end of the mission

cycle ( ) in terms of its structural components, and then equating that with equation (9) will also

yield equation (14).

Profitability Analysis and the Positive Performance Condition

Recall that the second law of thermodynamics dictates that no process can perfectly convert one

form of energy into another without some losses. This is particularly true in the case of a fuel tanker

spacecraft. When discussing profitability in this context, the reference is to the mass of cargo fuel

offloaded at the refueling point compared to the total mass of cargo fuel carried there.

(15)

With this in mind, consider again equation (14). The terms

,

and

are called design

constraints, and are intrinsic properties of a physical tanker. Ideally they are as low as possible,

but they cannot ever be zero. However, setting them equal to zero and analyzing the simplified equation provides a useful set of bounds for

and

. In such a case, equation (14)

becomes

[

]

]

[

[ ]

With the constraint set forth in equation (15), this becomes

[ ]

[ ]

(16)

[ ( ) ]

Where (1- P) is a positive quantity, since P must be less than one due to the second law of thermodynamics. Furthermore, since

is a positive known quantity, the leading coefficient

must be zero.

( )

( )

In this instance it is more convenient to express P in terms of the mass ratios for each leg of the

mission cycle.

(17)

(

)

Equation (17) is the maximum profitability that can be attained by a fuel tanker. Because this assumes the spacecraft (excluding the fuel) is entirely made of massless parts, it is impossible to

build a spacecraft with the same delta-v requirements, and cargo load capacity that has a

profitability exceeding this value.

It can be simply shown that

and

, which is in

for any combination of values for

holding with the first law of thermodynamics – energy can neither be created nor destroyed. Since

would imply that no fuel was burned during the mission cycle, and that kinetic energy

was created from nothing.

Conversely, equation (17) has no built-in minimum. From a practical standpoint however, when

the spacecraft has entered a design-based profitability deficit. Physically this means

that based on the current delta-v requirements, a fuel tanker designed to carry a certain amount of

fuel would burn more fuel than it would deliver.

It is possible to calculate the relationship between the values of and that would yield a deficit

by setting equation (17) less than or equal to zero.

(

)

( )

In order to avoid a deficit, the tanker design must satisfy the inverse:

(18)

Inequality (18) is called the positive performance condition for an ideal fuel tanker. The values of

and

that satisfy it represent the mission cycle delta-v’s for which the spacecraft will deliver a

positive profitability rating. (18) provides a useful upper bound for a design’s delta-v requirements.

However, in reality the weight of spacecraft components is not negligible. The propellant mass fraction of a spacecraft – the total mass of fuel in a spacecraft compared to its initial mass – is an important determiner of how efficient a spacecraft is. As a point of reference, the Apollo Lunar Module had a PMF of approximately 0.7, and its ascent stage had a PMF as low as 0.5

[http://www.braeunig.us/space/specs/lm.htm].

Condition (15) can be applied to equation (14), and after some algebra which is not in itself particularly insightful, will yield an expression for the profitability of a physical fuel tanker:

(19)

( )( )

( )

Where is called the fuel tank factor and is defined by

and can be physically

interpreted as the factor by which a given quantity of fuel increases the total mass of the tanker.

is called the structural inefficiency of the design given by

. For an

⁄

ideal tanker, and .

Note that equation (19) simplifies to equation (17) for an ideal tanker. The terms that represent the

mass contributions of structural components have a damping effect on the end profitability.

Intuitively, this would mean that as a heavier spacecraft carrying the same amount of fuel will in

fact burn more of it to achieve a given delta-v.

As before, equation (19) can be used to determine the positive performance condition. This time

the result is generally applicable to a physical fuel tanker:

(20)

(

)

( )

Like equation (19), (20) simplifies to the ideal model (18) when the mass of structural components

is neglected.

Figure 2 - Example positive performance regimes for an ideal and physical fuel tanker. [http://www.gcalc.net]

Parting Observations

The purpose of an orbital fuel tanker is to move a quantity of fuel from one location and supply it to

another. It has been shown here that doing so will never result in the full amount being transferred,

while still maintaining the reusability of the tanker. For larger quantities of fuel being moved by

cheaply constructed disposable transports, the profitability is greater over the short term.

However, to conduct multiple return trips reusability is the better investment – for a well-designed

fuel tanker, the profitability will eventually overcome the costs associated with its construction and

launch.

Although the equations outlined here do not concretely outline exactly how such a tanker could be

built, they do provide the means to determine whether a prospective design has the capability to

fulfill its mission requirements and objectives.

If the delta-v requirements are known then the profitability equation, positive performance

condition, and cargo-offload relation will help to determine the cargo and offload ratings for a particular design. Then, the mass target equations can be used to guide the construction of the spacecraft on a by-weight basis.

Together, these equations will help designers reduce the number of iterations designers will need

to undergo before design functionality convergence is achieved.