ME 239: Rocket Propulsion Rocket Flight Performance ...jmmeyers/ME239/Slides/06 - Gravity free-drag...

ME 239: Rocket Propulsion

Rocket Flight Performance: Gravity-free

Drag-free Space Flight(Follows Section 4.1)

J. M. Meyers, PhD

1

University of Vermont

Mechanical EngineeringME 239: Gravity-free Drag-free Space Flight

J. M. Meyers, Ph.D.2

Here we will address the flight performance of rocket propelled vehicles:

• Spacecraft

• Space Launch Vehicles

• Missiles

• Projectiles

Rocket motors provide thrust (a force) needed to accelerate (o decelerate!), overcome drag forces

and/or change direction/orientation

Flight missions can be classified into several flight regimes:

• Flight within Earth’s atmosphere (launch systems, atmospheric missiles)

• Near space environments (very high altitude and Earth orbit)

• Lunar and planetary flight (Earth escape)

• Deep space exploration (Sun escape)




4.1) Gravity-free Drag-free Space Flight

• Far away from any massive body ⇒ no gravity

• Far away from any sensible atmosphere ⇒ no aerodynamic forces

• Only force applied is from rocket motor

4.2) Forces acting on a vehicle in the atmosphere

• Near a massive body ⇒ gravity

• Inside a sensible atmosphere ⇒ aerodynamic forces

• Thrust from rocket motor

4.3) Basic relations of motion:

• For vehicles flying within the proximity of a massive body

• Neglect other heavenly body influences

• Drag and Lift relevant

• Recall particle motion dynamics

4.4) Space Flight

• Newton’s Law of gravitation defines the attraction between two bodies in space

• Supplemental outside notes and material will be given here

4.5) through 4.9) is mostly conceptual apart from 4.7) which deals with vehicle configurations and

handling of vehicle staging (quite interesting!)

Outline of Chapter



J. M. Meyers, Ph.D.

4.1) Gravity-Free Drag-Free Space Flight

4

Assumptions for this analysis:

• Outer space environment (no drag and lift aerodynamic forces)

• Far away from any star or other massive body (no significant gravitational acceleration)

• Flight direction is the same as the thrust direction (along axis of nozzle which passes through

the vehicle’s mass center)

• Propellant mass flow (�� ) is constant of burn duration (��)

• Propulsive force, �, is also constant over burn time



J. M. Meyers, Ph.D.


5

Recall from our definition of thrust:

� = �� + �( − �)and effective exhaust velocity:

� = � +( − �) ��

� = ��

Let � be the instantaneous vehicle mass and � be the vehicle velocity. From Newton’s 2nd Law we have:

� = ��

Assuming start-up and shut-down transients are very short in duration compared to overall burn time the

instantaneous mass of the vehicle can be expressed as a function:

�0 ≡full vehicle initial mass

�� ≡propellant mass

�� ≡propellant burn time

� ≡instantaneous time

� = �0 −�� = �0 1 − ��

�0��



J. M. Meyers, Ph.D.


6

Earlier we defined the propellant mass fraction and vehicle mass fraction as:

� = �0 1 − � �� = �0 1 − 1 −��

��

� = ��0

= propellantmasstotalinitialvehiclemass MR = �+

�0= finalvehiclemass

totalinitialvehiclemass

� = 1 − MR = 1 − �+�0

And as: �0 −�� = �+

An illustrative representation of the various masses can be seen in Figure 4-1 (reproduced on next slide)

�0 represents the sum of useful propellant plus empty final vehicle mass including payload and

stores

�+ represents the sum of inert masses of engine system, residual propellant, GNC

electronics/mechanics, related equipment, payload (everything but fuel!)



J. M. Meyers, Ph.D.


7

4.1 Definitions of various vehicle masses





For constant propellant mass flow �� and finite burn time �� the total mass of the propellant is:

�� = �� The instantaneous vehicle mass is then:

� = �- −�� = �- −��

From Newton’s 2nd Law:

where � represents some time

between motor start and motor stop

� = �� = �

�� = ��

�� = �� - −�� =

�(��/��)�- 1 − ��-��

��

�� = ��/��1 − ��/�� = ��/��

1 − ��/��





Integrating:

/�� = / ��/��1 − ��/�� ∆� = −�ln(1 − �)+�-

Not described in book. How is this done?

/�� = / ��/��1 − ��/�� − �- = − ��

�� ln(1 − � �

�� ) 1213− −��

�� ln(1 − � �

�� ) 12-

�� − �- = −�ln(1 − � �� ) 1213

− −�ln(1 − � �� ) 1214

�� − �- = −�ln(1 − �) + �ln(1 − � �-�� )For initial time

�0 = 0

�� − �- = −�ln(1 − �)

∆� − �- = −�ln(1 − �)Here �� is also referred to as

the velocity increment ∆� ∆� = −�ln(1 − �)+�-





∆� = −�ln(1 − �)+�- = � ln (�-/�5)+�-

• ∆� in your text is referred to as the “velocity increment”

• However, it is also quite common to refer to this term as “delta vee” from an orbital mechanics

standard as this term is denoted as ∆6• The term V in our discussions for this course has be reserved for specific volume

• We will still occasionally utilize the “delta vee” phrasing but bear in mind this represents ∆�• If the initial velocity is zero (�- = 0) then the integration yields a velocity at thrust termination of:

�� = ∆� = −�ln(1 − �)= −� ln (MR)= � ln (1/MR)= � ln (�-/�5)

Eq. 4-6

Eq. 4-5

• Eq. 4-6 represents the maximum velocity increment (“delta vee”) that can be obtained in a gravity-

free vacuum with constant propellant mass flow after starting from rest.




Fig 4-2: Maximum vehicle velocity in a gravitationless, drag-free space for different mass ratios and specific impulses (plot

of Eq. 4-6). Single-state vehicles can have values of 1/MR up to about 20 and multistage vehicles can exceed 200.


• The effects of variations in �, 78, and � are quite apparent in Figure 4-2





• The concept of maximum attainable delta vee in a gravity-free environment may seem inapplicable to

atmospheric flight near a massive body

• Actually, the variations still playa role in flight environments where gravity and aerodynamic forces play

an appreciable role

• The derived �� relations are quite useful in understanding the influence of basic design parameters and

in comparing :

• One propulsion system to another

• One flight mission to another

• One proposed upgrade to another

• Possible design modification/improvement





EXAMPLE PROBLEM: Problem 4-1 in text

For a vehicle in gravitationless space, determine the mass ratio to boost the vehicle velocity by

a) 1600 m/sec and b) 3400 m/sec if the effective motor exhaust velocity is 2000 m/sec. If the

initial total vehicle mass is 4000 kg, what are the corresponding propellant masses?

a) 1600 m/sec

�� = ∆� = 1600m/sec “boost vehicle” ⇒ �- = 0�/8:�� = ∆� = � ln (1/MR) :∆;/< = ∆� = 1/MR MR = :=∆;/<⇒ ⇒

MR = :=>?--/--- MR = 0.4493⇒

MR = �+�0

= 1 −��0 ⇒ ��=(1 − MR) �0 ��=(1 − 0.4493) 4000⇒ ��=2203kg⇒

a) 3400 m/sec

MR = :=�G--/--- MR = 0.1827⇒

��=(1 − 0.1827) 4000 ��=3269kg⇒





EXAMPLE PROBLEM: Problem 4-1 in text (cont.)

So, for a vehicle with an initial total mass of �0=4000 kg in gravitationless, aerodynamic force-free motion:

It takes ��=2203kg of propellant to get the vehicle up to 1600m/sec which constitutes a mass

fraction of MR = 0.4493 which is a launch vehicle of 55% fuel

However, it takes ��=3269kg of propellant to get the vehicle up to 3400m/sec (a slightly more than

double velocity increase) which constitutes a mass fraction of MR = 0.1827 which is a launch vehicle

of 92% fuel!!!!!

The motor you choose will dictate what type of payload mass you can launch. However, the end

velocity will be heavily dependent on the amount of fuel you can launch with the vehicle





• It’s easy to see how vehicle propellant mass fraction has a logarithmic effect on vehicle velocity

• By increasing propellant mass fraction, �, from 0.80 to 0.90 the interplanetary maximum velocity is

increased by 43%:

��,K2-.L��,K2-.M

= −� ln(1 − 0.90)−� ln(1 − 0.80) = 1.4303

• A propellant mass fraction of 0.80 means only

20% of vehicle initial mass is available for

structure, skin, payload, propulsion system,

GNC, aerodynamic surfaces, TPS, etc.

• The remaining 80% is reserved for fuel





• Rule of thumb limit for single stage � is 0.95 which implies MR = 20 (MR = 1/(1 − 0.95))

• For multi-staged vehicles, MR can exceed 200!





• This influence of mass fraction on vehicle velocity also affects atmospheric flight and gravity influenced

motion of rocket propelled craft

• In each of these environments it is important to manage your mass fraction wisely

• The Plasma Test and Diagnostics Laboratory deals with aspects of this through thermal protection

system (TPS) material and geometry management

• Owing to a lack of understanding in the chemical kinetics involved, numerical simulations used to design

heat shields result in a generally conservatively-sized geometry as a factor of safety.

• But this geometry oversizing adds appreciable mass to the vehicle

• Our lab adds more understanding to the chemistry interacting with the heat shield material thereby

giving the numerical community anchor points upon which higher fidelity numerical models can be

employed to design more appropriately sized TPS




• TPS mass fraction reduction is a popular topic

• Chemistry of how aerothermodynamic environment and

high temperature chemistry affects the material

performance is not wholly understood


Carbon ablator in an air

plasma





• We can rearrange 4-6 and solve for propellant mass required to acquire a desired delta vee (Δ�) for a

given take-off mass (recall �� = �- −�5):

�� = �- 1 − :=P;/< Eq. 4-8

• For final vehicle burnout mass (�5):

�� = �5 :P;/< − 1 Eq. 4-8

• From Eq. 4-6 we see that �� ∝ � and therefore, �� ∝ 78

• Thus any improvement to 78 can be made from:

• Better propellants (increase RS)

• More favorable nozzle area ratio, T• Higher chamber pressure, (increase �1)

• The above “improvements” are only improvements bearing that the increased mass due to motor sizing

alterations does not overly reduce performance through effective propellant fraction

ME 239: Rocket Propulsion Rocket Flight Performance ...jmmeyers/ME239/Slides/06 - Gravity free-drag...

Documents

Transcript of ME 239: Rocket Propulsion Rocket Flight Performance ...jmmeyers/ME239/Slides/06 - Gravity free-drag...