Solving Space Mechanics Homework: Dry Mass Fraction, F

[Biosyn]

Homework Statement

The dry mass fraction, F when multiplied by the propellant mass gives the dry mass of the vehicle (not counting its payload). This dry mass faction F, is a function of system design and lightness of the materials employed for its construction. So if F = 0.1, a ship carrying 90 tonnes of propellant would have a dry mass of 9 tonnes. If the mass ratio of the system had to be 10 to perform a certain ΔV, this would allow the craft to carry 1 tonne of payload. But if F = 0.11, the dry mass of the system would be 9.9 tonnes and the payload would fall by a factor of 10, to 0.1 tonne. If F = 0.12, the dry mass of the vehicle would be 10.8 tonnes, and the mission would be impossible even without any payload.

I need help understanding where 0.1 tonne comes from. I know how they got 1 tonne of payload.

Homework Equations

$\frac{M+P}{P}$= e$^{ΔV/M}$

M = Dry mass of vehicle without payload
P = Mass of propellant

The Attempt at a Solution

Let x = payload

(9 + 90)/9 - X = 10

X = 1 tonne

(9.9 + 90)/9.9 - X = 10

X = 0.09090909~ tonne

 

[Simon Bridge]

You mean the 0.1T cargo for F=0.01?

mass ratio has to be 10 to make a delta-vee.
then a ship with 90T of fuel can still do the delta-vee if the combined dry-mass and cargo-mass is smaller than a certain amount. What is this amount?

What is the dry-mass for a ship with F=0.01 and 90T of fuel?
How much does this leave for cargo?

 

[Biosyn]

You mean the 0.1T cargo for F=0.01?

mass ratio has to be 10 to make a delta-vee.
then a ship with 90T of fuel can still do the delta-vee if the combined dry-mass and cargo-mass is smaller than a certain amount. What is this amount?

What is the dry-mass for a ship with F=0.01 and 90T of fuel?
How much does this leave for cargo?

Sorry, where did you get F=0.01 from? I'm confused.

The amount of payload for a ship with F=0.1 and 90T of fuel will be 1T.

I'm not sure where they came up with 0.1T of cargo for a ship with F=0.11 , 90T of fuel , and 9.9T of dry mass.

The mass ratio is 10 = (M+P)/M right?

 

[Simon Bridge]

F=0.11 is the example given but you seem to have found that confusing. The math is the same for any F. Here's the bit I guessed we were talking about:

But if F = 0.11, the dry mass of the system would be 9.9 tonnes and the payload would fall ... to 0.1 tonne.

mass ratio = R
mass of fuel = P
dry mass = D
cargo mass = C
then mass-ratio is R = (P+D+C)/(D+C)

so for P=90T, D+C=10T to get a mass-ratio: 100/10 = 10
as a consequence: C = 10-D.

If you like: since F=P/D, R=[(1+F)P+C]/[FP+C] solve for C
But the above example does it in 3 steps ... thus:

1. Given R and P you get D+C.
2. Given F and P you get D
3. combine 1 and 2 to get C.

 

[Biosyn]

F=0.11 is the example given but you seem to have found that confusing. The math is the same for any F. Here's the bit I guessed we were talking about:


mass ratio = R
mass of fuel = P
dry mass = D
cargo mass = C
then mass-ratio is R = (P+D+C)/(D+C)

so for P=90T, D+C=10T to get a mass-ratio: 100/10 = 10
as a consequence: C = 10-D.

If you like: since F=P/D, R=[(1+F)P+C]/[FP+C] solve for C
But the above example does it in 3 steps ... thus:

1. Given R and P you get D+C.
2. Given F and P you get D
3. combine 1 and 2 to get C.

Does
$\frac{90T + 9.9T + C}{9.9T + C}$ = 10T ?

Then solving for C
...

C = 0.1T

 

[Simon Bridge]

Does
$\frac{90T + 9.9T + C}{9.9T + C}$ = 10T ?

Then solving for C
...

C = 0.1T

Now you know where that 0.1 comes from?

 

[Biosyn]

Now you know where that 0.1 comes from?

Yup, but uh..I'm not entirely sure where 10T comes from.

Is D+C always going to be 10T when P=90T?EDIT:
Or is it C = 0.1 because the units cancel out. Therefore that's why the Ratio is equal to [(1+F)P+C]/[FP+C].
Sorry, I think I'm just going around in circles. ._.

 

[Simon Bridge]

There's a bunch of definitions:
Dry mass, and mass ratio. These are separate.
The 10T comes from a mass-ratio of 10, when the propellant mass is 90T.
(You'd think a MR of 10 would mean "10:1" wouldn't you? But it doesn't.)

Using above notation:
if M = D+C+P is the total mass of the craft.
MR = M/(M-P)

So a 100T ship with MR=10 has P=90T and D+C=10T.

Rocket design is actually quite complicated and these wee numbers help engineers make design decisions.