REFERENCE: Specific Impulse

Rocket Thrust:    F = ṁ × V_e + Δp × A_e

DEFINITIONS:  

     ṁ  = mass flow rate;  
     V_e = velocity;  
     Δp = pressure difference; 
     A_e = area

Equivalent Velocity:  
     V_eq = V_e + (Δp × A_e) / ṁ

Total Impulse:  
     I = F × Δt =  ∫Fdt =  ∫(ṁ × V_eq) dt =  m × V_eq

Special Impulse: Total Impulse/Mass:  
     I_sp = I/(m×g₀) =  V_eq / g₀

SPECIFIC IMPULSE	NASA Glenn Center

Source Documents
NASA: Specific Impulse
Answers.com: Specific Impulse
Wikipedia: Specific Impulse
While impulse is a change in momentum, specific impulse (commonly abbreviated I_sp) is impulse per unit mass of propellant. I_sp is the efficiency of a propulsion system; thus, I_sp is useful for comparing propulsion systems; much like "miles per gallon" can compare different car models.  As I_sp increases, propellant requirement decreases; thus, the propulsion system is more efficient.  

An ion engine might have an I_sp  from 2,000 to 10,000 seconds.  

Thrust and specific impulse are different concepts. Thrust is the aft directed force which propels a rocket forward. Typical rocket engine generates thrust through the reaction of a gaseous propellant exiting exhaust nozzle at high speed. The gas accelerates to the the rear (aft) and the rocket accelerates forward (opposite direction). Rockets use a propulsion system to accelerate a propellant. From his second law of motion, Newton defines a force to be the change in momentum of an object with a change in time.  Momentum is the object's mass times its velocity. In spacecraft, specific impulse is the change in momentum per unit weight-on-Earth of the propellant; this equates to particle exhaust speed (V_e) divided by acceleration due to gravity (g₀) near Earth surface (see below table).

SPECIFIC IMPULSE (I_sp) EXAMPLES:

Propulsion Technology	Effective Exhaust Velocity (Ve)	Specific Impulse (Isp)
Turbofan jet engine	29,000 m/s	3,000 sec
Solid Rocket Booster	2,500 m/s	250 sec
Liquid O2-H2	4,400 m/s	450 sec
Ion thruster	29,000 m/s	3,000 sec
VASIMR	30,000–120,000 m/s	3,000–12,000 s
Dual-stage 4-grid ion	210,000 m/s	21,400 sec
Ideal photonic rocket	299,792,458 m/s	30,570,000 sec
Given	Observed	Ve  ÷ g0 g0 ≈ 9.807 m/s²

When dealing with a gas, the basic thrust equation is given as:
F = (ṁ_e × V_e) - (ṁ₀ × V₀) + [(p_e - p₀) × A_e]

DEFINITIONS

F: force of thrust

ṁe: exit mass flow rate

Ve: the exit velocity

ṁ0: free stream mass flow rate

V0: free stream velocity

Δp: pressure difference

Ae: engine area

For rocket engines, with no free stream air in the propulsion system; the thrust equation simplifies:
F = ṁ × V_e + Δp × A_e Divide both sides by ṁ, mass flow rate:
F / ṁ = V_e + Δp × A_e / ṁ

Define a new term, equivalent velocity (V_eq) as the velocity on the right side:
V_eq = V_e + (p_e - p₀) × A_e / ṁ

DEFINITIONS

pe: end pressure

p0: initial pressure

Δp: pe -p0       pressure difference.

Rocket thrust equation becomes:
F = ṁ × V_eq The total impulse (I) of a rocket is defined as the average thrust times the total firing duration (Δt):
I = F × Δt

DEFINITIONS

te: end time

t0: initial time

Δt: te - t0       time difference,       total duration of event

Since the thrust may change with time, define an integral equation for the total impulse.

I = ∫F dt Substitute as shown:

I =  ∫(ṁ × V_eq) dt

The mass flow rate, ṁ, is the rocket's exhaust mass per time. Assume equivalent velocity remains constant, integrate the equation to get:

I = m × V_eq

where m is the total mass of the propellant. 
Divide this equation by the weight of the propellants to define the specific impulse. The word "specific" just means "divided by weight". The specific impulse I_sp is given by:

I_sp =V_eq / g₀

where g₀ is the gravitational acceleration constant (9.80665 m/sec²). Now, substitute for the equivalent velocity in terms of the thrust:

I_sp = F / (ṁ × g₀)

Mathematically, I_sp is a ratio of thrust to propellant mass flow. A quick check of the units for Isp shows that:

I_sp = m/sec / m/sec² = sec

SIDEBAR Newton's 2nd Law of Motion defines force as mass times acceleration. F = m × a Acceleration is velocity per time:   a = v / t Momentum is an object's mass times its velocity. p = m × v Thus, force can also be defined as momentum per time:    F = m × v / t = p / t   In spacecraft, specific impulse (Isp) is the propellant's change in momentum per unit Earth weight; this is particle exhaust speed (Ve) divided by acceleration due to Earth gravity (g0). Isp = Ve / g0

The higher the rocket's specific impulse (fuel efficiency), the farther the spacecraft can go with a given amount of fuel. With an ion thruster's small level of thrust, it must exceed 10,000 hours for speeds needed to reach asteroids, comets, outer planets and their moons.

Why use specific impulse?  I_sp has following benefits.
1) Quickly determine thrust. If we know the weight flow rate through the exhaust nozzle, we know the thrust value. 
2) Quickly compare engine efficiencies. Let two different rocket engines have different I_sp values. The engine with the higher value is more efficient because it produces more thrust for the same amount of propellant. 
3) Quickly analyze rocket thermodynamics. I_sp units (seconds) are the same in either English units or metric units. 
4) Quickly "size" an engine during preliminary analysis. The result of our thermodynamic analysis yields a certain I_sp value. The rocket weight will define the required thrust value. Dividing thrust value by  I_sp will tell us required mass flow of propellants. This information determines the physical size of the engine.

SLIDESHOW

---

---

---

---

VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

---

---

---

---