Thrust is the force which moves a
rocket through the air. Thrust is generated by the rocket engine
through the reaction of
accelerating a mass of gas. The gas is accelerated to the the rear
and the rocket is accelerated in the opposite
direction. To accelerate the gas, we need some kind of propulsion
system. We will discuss the details of various propulsion systems
on some other pages. For right now, let us just think of the
propulsion system as some machine which accelerates a gas.
From Newton's second law of motion, we
can define a force to be the change in momentum of an object with a
change in time. Momentum is the object's mass times the
velocity. When dealing with a gas, the basic thrust
equation is given as:
F = mdot e * Ve - mdot 0 * V0 + (pe - p0) * Ae
Thrust F is equal to the exit mass flow rate
mdot e times the exit velocity Ve minus the
free stream mass flow rate mdot 0 times
the free stream velocity V0 plus the pressure difference across the
engine pe - p0 times the engine area Ae.
For liquid or solid
rocket engines, the propellants, fuel and oxidizer, are carried on board.
There is no free stream air brought into the propulsion system,
F = mdot * Ve + (pe - p0) * Ae
where we have dropped the exit designation on the mass flow rate.
Using algebra, let us divide by mdot:
F / modt = Ve + (pe - p0) * Ae / mdot
We define a new velocity called the equivalent
velocity Veq to be the velocity on the right hand side of the above equation:
Veq = Ve + (pe - p0) * Ae / mdot
Then the rocket thrust equation becomes:
F = mdot * Veq
The total impulse (I) of a rocket is defined as the average thrust
times the total time of firing. On the slide we show the total time as
"delta t". (delta is the Greek symbol that looks like a triangle):
I = F * delta t
Since the thrust may change with time,
we can also define an integral equation for the total impulse. Using the symbol
(Sdt) for the integral, we have:
I = S F dt
Substituting the equation for thrust given above:
I = S (mdot * Veq) dt
Remember that mdot is the mass flow rate; it is the amount of exhaust mass
per time that comes out of the rocket. Assuming the equivalent velocity remains
constant with time, we can integrate the equation to get:
I = m * Veq
where m is the total mass of the propellant. We can divide this equation
the weight of the propellants to define the specific impulse.
The word "specific"
just means "divided by weight". The specific impulse Isp is given by:
Isp = Veq / g0
where g0 is the gravitational acceleration constant (32.2 ft/sec^2 in English units,
9.8 m/sec^2 in metric units).
Now, if we substitute for the equivalent velocity
in terms of the thrust:
Isp = F / (mdot * g0)
Mathematically, the Isp is a
of the thrust produced to the weight flow of the propellants.
A quick check of the units for Isp shows that:
Isp = m/sec / m/sec^2 = sec
Why are we interested in specific impulse?
First, it gives us a quick way to determine the thrust of a
rocket, if we know the weight flow rate through the nozzle.
Second, it is an
indication of engine efficiency. Two different rocket engines have
different values of specific impulse. The engine with the higher value
of specific impulse is more efficient because it produces more thrust
for the same amount of propellant.
Third, it simplifies our
mathematical analysis of rocket thermodynamics.
The units of specific impulse are
the same whether we use English units or metric units.
Fourth, it gives us an easy
way to "size" an engine during preliminary analysis. The result of
our thermodynamic analysis is a certain value of specific impulse. The
rocket weight will define the required value of thrust. Dividing the
thrust required by the specific impulse will tell us how much weight flow
of propellants our engine must produce. This information
determines the physical size of the engine.
There is a similar efficiency parameter called the
which is used to characterize turbine engine performance.
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