Thrust, Rocket Equation, Specific Impulse, Mass Ratio
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Thrust, Rocket Equation, Specific Impulse, Mass Ratio. Thrust. Thrust comes from: a) Increase in momentum of the propellant fluid (momentum thrust) b) Pressure at the exit plane being higher than the outside pressure (pressure thrust). Where does the thrust act?

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Thrust
Thrust

Thrust comes from:

a) Increase in momentum of the propellant fluid (momentum thrust)

b) Pressure at the exit plane being higher than the outside pressure (pressure thrust).

Where does the thrust act?

In the rocket engine, the force is felt on the nozzle and the combustor walls,

and is transmitted through the engine mountings to the rest of the vehicle.

Effective Exhaust Velocity


Delta V and Mass Ratio

Consider a rocket with effective exhaust velocity ce. As propellant is blasted out the exhaust nozzle, the mass of the vehicle decreases. This is substantial in the case of the rocket as compared to air-breathing engines, because all the propellant comes from inside the vehicle. From Newton's Second Law, 

.

where M1 is the initial mass, which includes the propellant, and M2 is the mass after the propellant has been used up to achieve the velocity increment DV.


Specific Impulse and Mass Ratio

Specific Impulse of the system is 

where g is the standard value of acceleration due to gravity at sea-level (9.8m/s2). Note that the unit of Specific Impulse is seconds. Using this definition, 

Note: Some organizations

express Specific Impulse without dividing by g

Mass Ratio of a rocket is


Example:

For missions from Earth's surface to escape from earth's gravitational field, Mass Ratio is large.

For specific impulse of 390 s, g = 9.8 m/s2, and DV = 11,186 m/s (36700 fps), the mass ratio is 18.67.

This means that the rocket at launch time must be at least 18.67 times as big as the spacecraft which is left after all the fuel is burned. To get a high specific impulse like 390 s, we have to use a costly system like liquid hydrogen - liquid oxygen.

For earth orbit, the velocity increment DV needed is 25,000 fps, while 36,700fps will enable escape from Earth's gravitational field.


Newton's Law of Gravitation

To find the velocity increment required for various missions, we must calculate trajectories and orbits. This is done using Newton's Law of Gravitation:

 Here the lhs is the "radial force" of attraction due to gravitation, between two bodies; the big one of mass M, and the little one of mass m.

The universal gravitational constant G is 6.670 * 10-11 Nm2/kg2.


Rocket equation including drag and gravity
Rocket Equation Including Drag and Gravity:

Ref: Hill & Peterson, Chapter 10.

The rate of acceleration of the vehicle is

.

Neglecting the air drag and gravity terms, we get the

Ideal Rocket Equation


Drag term in the rocket equation

Inclination, deg. to flt. direction,

CD low-speed

CD peak @1.2

CD @ Mach 2

0

0.06

0.15

0.13

4

0.185

0.16

8

0.23

0.2

Drag Term in the Rocket Equation

where atmospheric density above Earth varies roughly as

With density in kg/m3 and h in meters, a = 1.2 and b = 2.9 x 10-5

 Roughly, density at 30,000 meters is about 1% of its sea-level value.

Drag Coefficients (typical)

Note: drag coefficient peak is reached at around Mach 1.2.


Gravity term
Gravity Term

At 100 miles above the surface the change from the surface is still only about 5%.


Example
Example

Specific impulse of 390 s,

g0 = 9.8 m/s2,

and DV = 11186 m/s (36700 fps),

Mass ratio is 18.67.

This means that the rocket at launch time must be at least 18.67 times as big as the spacecraft which is left after all the fuel is burned. To get a high specific impulse like 390 s, we have to use a costly system like liquid hydrogen - liquid oxygen.

Velocity increment DV for Low Earth Orbit: ~ 25,000 fps,

Escape from Earth's gravitational field ~ 36,700fps


Single stage sounding rocket
Single Stage Sounding Rocket

Altitude at burnout, assuming it goes straight up:

Neglecting drag

Ifis constant,

(Sounding rocket: not quite single-stage)

NASA Goddard Space Flight center

http://www.gsfc.nasa.gov



Expression for maximum altitude reached
Expression for Maximum Altitude Reached

Note that at burnout, the sounding rocket is still moving fast upward.

Equating kinetic energy at burnout with change in potential energy of the final mass


Example

Values given:

Ce = 3000 m/s

R = 10

tb = 30s

Hmax = ??


Chemical rockets
Chemical Rockets

Burn time of existing rockets is ~ 30 to 200 seconds.

Payload Mass

Structure (incl. engine) Mass

Propellant Mass

Definitions

Payload ratio

Structure coefficient

Thus,



Multistaging 2
Multistaging - 2

Payload of stage is mass of all subsequent stages.

Structural coefficient of stage:

If stage contains no propellant at burnout:


Multistaging 3
Multistaging - 3

Mass ratio of stage

i.e.,

.

Similar stages: same and


Multistaging 4
Multistaging - 4

If are equal,


Multistaging 5
Multistaging - 5

Structural coefficient




The idea of “Thrust Coefficient”

where F is the thrust, At is the nozzle throat area and P0 is the combustion chamber stagnation

pressure. Basically a higher thrust coefficient means a better usage of the available stagnation

pressure in converting to thrust.

The thrust coefficient has values ranging from 0.8 to 1.9.

Note also that a plot of thrust coefficient vs. altitude for a given nozzle will give the variation of

thrust with altitude for a given chamber pressure and nozzle throat area.

The thrust coefficient is also used to compare different nozzle designs for given constraints.

In the following we will use gas dynamics to derive expressions for the thrust coefficient in terms

of gas properties.


Thrust coefficient 1
Thrust Coefficient - 1

where At is nozzle throat area and p0 is chamber pressure (N/m2)

Thus,

For sonic conditions at the throat,

and


Thrust coefficient 2
Thrust Coefficient - 2

Using isentropic flow relations,

and Thrust Coefficient

Depends entirely on nozzle characteristics. The thrust coefficient is used to evaluate nozzle performance.


Characteristic exhaust velocity c
Characteristic Exhaust Velocity c*

Used to characterize the performance of propellants and combustion chambers independent of the nozzle characteristics.

whereis the quantity in brackets. Note:

So

Characteristic exhaust velocity

Assuming steady, quasi-1-dimensional, perfect gas. The condition for maximum thrust is ideal expansion: nozzle exit static pressure being equal to the outside pressure. In other words,


End of section 2
End of Section 2

  • We’ll end Lecture 2 here, and go on to discuss orbits before getting back to compressible

  • Flow and chemistry considerations. The purposes are:

  • To enable mission calculations.

  • To give everyone a chance to look at the content so far and see how much they need review of compressible flow material. Please browse the web links in the first lecture.


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