# [Propulsion] Minimum Isp

```Hey all,

Since Andrew was talking about re-starting the propulsion list
recently, here's post ;)```
```
---

Tom and i were discussing scaling laws for Isp at the
Wed. meeting. I'll give a summary, but this is just from my memory, so

As a simple 1st approach to designing a motor for orbit, we estimate
the mission requirement as a vehicle capable of achieving a certain
minimum delta-V. Meaning the relative speed the vehicle would achieve
absent losses such as gravity and air friction. For our 1st guess we
selected the minimum delta-V as ~9 km/s, around 20% more than the
orbital velocity, which is about 7.6 km/s.

(I went back and looked up our old requirement estimate,
it was 9.46 km/s.)

Delta-V can be computed in terms of the mass ratio and propellant
exhaust velocity as

1) delta-V = c * Log[m0 / mf]

Where (c) is the propellant exhaust velocity, (m0) is the initial
vehicle mass, and (mf) is the final vehicle mass at burn out.

For the record,

2) c = gee * Isp

Where (Isp) is the specific impulse, defined as Thrust/(mass flow
rate), and (gee) is the Earth's gravity, which i believe by convention
is assumed to be 9.8066 m/s^2. For calculations never use Isp
directly, always use exhaust velocity.

---

The important thing about the 1st equation is that there is an
exponential relationship between (c) and the fraction of the initial
vehicle mass that must be propellant. Further, for a given structural
technology there is an upper limit on how large the propellant
fraction can be. Therefore given the structural technology there is a
minimum Isp needed to reach orbit.

When talking with Tom at the meeting i forgot that all this applies to
single stage vehicles. Staging improves the situation somewhat, so we
should take that into account in our calculations.

---

In the rocket biz there is a standard way of expressing the propellant
fraction etc.

(mL) is the mass of the payload
(mp) the propellant mass
(md) is the dead 'weight' (Useless stuff like motors and tanks, etc. ;)

I'm not sure how well it works, but it's common to attempt to separate
the effects of payload mass from propellant and vehicle structure,
which motivates the definition of the "propellant mass fraction"

Lp === mp / (mp + md)

Eq(1) can be rewritten in terms of (Lp)

[   Lp mL + mp    ]
3) delta-V = c * Log[-----------------]
[Lp mL + (1-Lp) mp]

------

It would be nice if as part of exploring propellants we could get this
info up on the Wiki in a semi-polished form.

------

The rest of this info is taken verbatim from my old notes, much of
which were based on the talk we got from Ranier Anacker.

Typical propellant exhaust velocities: [m/s]
Hydrogen + Oxygen       3700, 4500
Kerosene + Oxygen       2700, 2800
Gasoline + Oxygen       2600, 2700
Alcohol  + Oxygen       2500, 2700
Shuttle SRBs            2500, 2600
Most Solids             2200, 2500

"Realistic Lp values are between 0.7 and 0.9."

Mass fraction of real rockets:
Vehicle                mp[kg]     md[kg]     Lp        mL[kg]
V2 (1945)              8482       3082       0.7335    1000

Shuttle SRB            502700     81900      0.8599    -
Shuttle External Tank  703100     35000      0.9526    -
Shuttle E.T.+Orbiter   703100     178300     0.7977    -
Shuttle (1995)         1708500    342100     0.8332    13000

DeltaII (1995)         207706     19359      0.9147    2000-5000
DeltaIII (2000)        262749     27468      0.9054    8000

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