Rocket Performance and Efficiency

0.1 But first, What is a Rocket Engine?

In this section we are going to talk a lot about how wecharacterize rocket engine performance. But it is helpful tostart by defining what a rocket is.

A rocket engine is a reaction engine, an engine that expelsmass to generate thrust. Rockets carry all of their own work-ing fluid, called propellant, in contrast with air-breathingengines. Furthermore reaction engines are only consideredrockets when they use gasdynamic expansion of the workingpropellant to accelerate it. This contrasts with, for instance,many types of electric propulsion thrusters which use bodyforces such as the Coulomb electrostatic force to acceler-ate gasses. And while the focus for this section will be ontrue rocket engines, some of the definitions and conceptswill apply when we talk about other reaction engines suchas electric thrusters.

Below is a conceptual drawing of the most-typical rocketengine families along with an electric thruster for good mea-sure.

Combustion

Chamber

Nozzle

Catalyst

Fuel

Bipropellant Monopropellant Hybrid

Nuclear

Thermal

Solid

Power

Electric

Note the common feature of all of these engines (with theexception of the electric thruster) is a converging / diverg-ing nozzle De Laval nozzle. Such nozzles are the essentialfeature of rocket engines as this is what enables the en-gine to gasdynamically accelerate propellant gasses from astagnant state to high velocity imparting momentum on theengine. The nozzle expansion process for most gasses inwell approximated by an isentropic expansion such that the

compressible isentropic flow equations become very usefulfor analyzing such a beast.

In this lecture we will define the key parameters that definethe performance for a rocket engine.

0.2 Effective Exhaust Velocity andSpecific Impulse

Now that we have covered the basic mechanics of thrustand the isentropic nozzle relations we will begin to showhow these apply to rockets by defining a set of convenientparameters.

Local Pressure

* e1Note: Station 1 is not the same as stagnation due to non-zero velocity in chamber. For large A1/A*, stagnation is still a good approximation.

Let’s start by remembering that, for a rocket

T = (Pe − P0)Ae + mpUe (1)

Our intuition tells us that a good parameter for the effec-tiveness of a rocket is the quotient of thrust (what we want)with propellant mass flowrate (what we have to pay):

Tm

= (Pe − P0)Ae

m+ Ue (2)

This parameter is called effective exhaust velocity,

C =Tm

(3)

and is one of the most important relations in rocketry. Itfigures prominently in the Rocket Equation

1

2

∆V = C lnMiM f

(4)

which we will be derived in a couple lectures.If we assume T and m are completely independent (which

we will see is not always true), we can also interpret C as

C =

∫Tdt∫mdt

=I

Mp(5)

which is how much total impulse we get from a unit massof propellant. This interpretation points to a very closelyrelated parameter called Specific Impulse, Isp:

Isp =Cg0

=T

g0m(6)

Notice that Isp is just C normalized by Earth’s gravita-tional acceleration with overall units of seconds. While theunit is at first a bit non-sensical, Isp makes sense when youconsider your propellant consumption as a weight flowraterather than a mass-flowrate and so you are dividing thrust(a force) by weight flowrate (force per time) to arrive at atime. The etymology of Isp is rooted in the fact that forimperial units we typically work in lbm. rather than slugs.

And while weird at first, the units of seconds do have someintuitive utility. For a rocket with Isp = 100s a unit mass,m of propellant can generate enough thrust to support itsweight in Earth’s gravity for 100 seconds or 100 times itsweight for one second.

Specific impulse is popularly spoken of as the "gasmileage" for a rocket cycle and this is fairly reasonable -it fundamentally indicates how much bang for the buck youget. I’ll jump the gun just a bit for the sake of intuition andgive some typical Isp values for different types of propulsionin Table 0.1.

This is all well and good, but all we have really done atthis point is some algebra. What we really are interested inas engineers is how do I get the most gas mileage out of myrocket. And for that discussion, we’ll first define anothercouple useful parameters.

0.3 c∗

We will go back, for a moment, to choked compressible flow.With the isentropic flow equations and the M = 1 chokedcondition, we can derive (assuming constant Cp, Cv and R):

m = ρ∗a∗A∗ = ρt

[ρ∗

ρt

]at

[a∗

at

]A∗ =

Pt A∗√RTtγ

[γ + 1

2

] 2(γ−1)γ+1

(7)remembering that the t subscript denotes the total or stag-nation condition and the c subscript denotes choked (M =1) condition. This is a very useful relationship as it allows us

to compute the mass flowrate through a choked nozzle as afunction of only nozzle throat area, A∗, ideal gas propertiesand the stangation condition temperature and pressure.

Let’s define a new parameter, called characteristic velocityand denoted c∗ using this result:

c∗ =Pt A∗

m=

√RTt

γ

[γ + 1

2

] γ+12(γ−1)

(8)

Figure 0.1: c∗ parameters of interest. m, Pt and A∗ are alleasily measureable. Tt is difficult due to the extremely hightemperatures of chemical rockets.

c∗ is useful far beyond simple choked flow considerationsbecause it can be both measured and computed. Andindeed the equation above shows this directly - the LHS isa function measure-able variables stagnation pressure Pt

1,area A∗ and mass flowrate m. The RHS is a function ofintrinsic gas properties γ, R and stagnation temperature.c∗ gives us the tools to calculate a parameter (RHS) thatwe can go and easily measure in the lab (LHS).

Given that c∗ depends on γ, R and Tt it is essentially afunction only of the working fluids thermodynamic proper-ties and stagnation (chamber) state. This is not 100% true- when we get to combustion we will see how a (weak) de-pendency on pressure comes back into c∗, but for conceptualpurposes we should think of c∗ this way - as only a functionof intrinsic thermodynamic properties of our propellants.

And so finally we come to a very interesting result regard-ing the functional dependence of c∗:

c∗ = f (Tt, R, γ) (9)

R is the specific gas constant R = RuMw

which is inverselydependent on gas molecular weight, Mw. Thus

c∗ ∝

√Tt

Mw(10)

Moreover Tt is related to the gas internal stagnation en-thalpy by ∆ht =

∫ TtT0

CpdT and so we should see Tt as arepresentation of the energy content of the rocket gasses.

1In a finite sized rocket chamber we are not actually measuringstagnation pressure because there will be non-zero gas velocity. Forreasonable contraction ratios, A1/A∗, this is a small effect and it canbe mostly corrected analytically using isentropic flow relations

0.4. C f 3

Technology Isp (s) Exhaust Velocity (m/s)

Nuclear Fusion 10,000 - 50,000+ 98,000 - 490,000+Electric Propulsion 1,000 - 10,000 9,800 - 98,000Nuclear Thermal (fission) 600 - 1,000 5,900 - 9,800Beamed Thermal (microwave / laser) 600 - 1,000 5,900 - 9,800Bipropellant Chemical Propulsion 200 - 500 2,000 - 4,900Monopropellant Chemical Propulsion 100 - 250 980 - 2,450Cold Gas Propulsion 10 - 120 100 - 1,150

Table 0.1: Table 1 - Representative effective exhaust velocities for different propulsion technologies

The effect of γ on c∗ is a bit more subtle. γ is fundamen-tally related to the number of vibratory degrees of freedoma molecule has. For monatomic systems (such as helium oratomic hydrogen) γ assymptotes to an upper limit of 1.66.For most simple diatoms (nitrogen, oxygen, hydrogen) it isaround 1.4 and for larger molecules or those with more com-plex bonding it is lower. For the conditions we are interestedin within a rocket, we wouldn’t expect to find γ much lowerthan 1.1.

1.1 1.2 1.3 1.4 1.5 1.6γ

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

c∗

Figure 0.2: c∗ shows only a weak dependence on γ

The plot below shows the effect of γ on c∗ which is fairlylimited compared with Tt and Mw. And so if you take onething away from this discussion, remember that

c∗ ∝

√∆ht

Cp Mw(11)

0.4 C f

Ok, that’s interesting, but for the purposes of rocket perfor-mance we want C not c∗. Take a look at the definition ofc∗ and it looks a lot like our definition of effective velocityand specific impulse above:

c∗ =Pt A∗

m∼ T

m= C (12)

and indeed c∗ and C are closely related.In order to see how, let’s define another new parameter,

C f that related thrust to the nozzle throat area and rockettotal pressure:

C f =T

Pt A∗(13)

In prose this says:

C f represents the amount of thrust a rocket canproduce given the stagnation pressure of its pro-pellants and a useful characteristic fluid area - thechoked area of its nozzle.

Stated differently it is a measure of how effectively wetake the stagnation pressure we generate in the chamberand turn it into thrust.

And since c∗ is also defined by chamber pressure and noz-zle throat area, C f becomes the connection between C andc∗:

C =Tm

=C f Pt A∗

m= C f c∗ (14)

But why split C into C f and c∗ this way?Remember that c∗ represents the potential of the rocket

propellants themselves to create thrust and essentially de-pends exclusively on the thermodynamic characteristics ofthose propellants. C f represents how well our nozzle canconvert the propellant’s latent utility into real thrust andthus depends almost exclusively on the physical nature ofour nozzle. And so as we go about maximizing C, we candivide that into two separate problems - one of picking pro-pellants (c∗) and the other of chosing system pressures andnozzle geometry (C f ).

Since the primary role of the nozzle is to convert gassesinto thrust C f can also be seen as a measure of the "good-ness" of the nozzle. It is called thrust coefficient.

C f can be expanded from its definition above:

C f =Cc∗

=(Pe − P0)

Aem + Ue

Pt A∗m

=Pe − P0

Pt

Ae

A∗+

Ue

c∗

⇒(

1− P0

Pe

)Pe Ae

Pt A∗+

Ue

c∗

(15)

4

It is worth noting that, like the thrust equation, there aretwo pieces to C f :

(1− P0

Pe

)Pe AePt A∗ is representative of thrust

created through pressure force and Uec∗ is representative of

the contribution of gas momentum to thrust. We will referto these two components when we discuss optimal nozzleexpansion in a minute.

Beyond this things get a little messy and different peopleattack the derivation different ways. The next steps rely onthe fact that Ue

c∗ ,PePt

and AeA∗ are all related to the isentropic

expansion of gasses through the nozzle to its exit. The exitmach number, Me thus becomes the common parameter andusing classic isentropic relations we can derive the functionalrelationship of each with Me as the independent variable:

Ue

c∗=

γMe√1 + γ−1

2 M2e

[γ + 1

2

] γ+1−2(γ−1)

(16)

PPt

=

(1 +

γ− 12

M2) −γ

γ−1(17)

Ae

A∗=

[γ + 1

2

] γ+1−2(γ−1)

(1 + γ−1

2 M2e

) γ+12(γ−1)

Me(18)

With some fancy algebra, we can combine these together toarrive at

C f =

(γ+1

2

) γ+1−2(γ−1)

Me

√1 + γ−1

2 M2e

[γM2

e + 1− P0

Pe

](19)

These results shows very clearly that Uec∗ ,

PePt

and AeA∗ are

not all independent (they all depend on Me). In fact thedimensionality of this set is one - picking a number for anyone of these directlys sets the others. Furthermore notethat these equations, unlike c∗, have no dependency onstagnation temperature or molecular weight and thusno direct dependency on propellant properties. They dodepend on γ which is a property of the working fluid but aswith c∗, the dependence is not terribly strong and is reallyset for us by the propellant choice we made in optimizingc∗.I’d like to look at how the parameter we can control di-

rectly, Ae/A∗, affects the others and C f . Using the equa-tions above, we will sweep through Me and compute theother parameters directly.

1 20.740.871.011.151.281.42

Ue

c∗

P0

Pt= 0.1

-0.2-0.10.00.20.30.4

( 1−

P0

Pe

) P eAe

PtA

∗

1 2

1.20

1.25

Cf

1 2P0/Pe

1

2

3

Ae

Ac

Figure 0.3: Relationship of the components of C f to eachother and to C f itself.

Note that the momentum component grows as exit pres-sure decreases because the gasses are accelerated further be-fore release. Conversely, the pressure component decreasesas exit pressure decreases for obvious reasons. In fact thepressure component of C f takes on negative values for exitpressures below ambient - this is called over-expansion.

Ae/A∗ increases monotonically with decreasing exit pres-sure - this simply represents the fact that to achieve lowerexit pressures we must use a larger nozzle expansion ratio.

Finally note that C f reaches a maximum where Pe/P0 =1. This is called optimal expansion. It is a fundamentalresult in rocket theory and can be stated:

Rocket effective exhaust velocity (and thereforespecific impulse) is maximized when the nozzle ex-pands the exhaust gasses such that they match thelocal pressure at the nozzle exit.

In general, we want to design a rocket nozzle to matchthe exit pressure to ambient pressure. This is difficult to dofor a rocket ascending through the atmosphere where thepressure is continuosly changing. For traditional nozzles, a

0.4. C f 5

compromise that looks at the average performance over theascent pressure profile is often chosen.

Sea level

Vacuum

Figure 0.4: Note the size difference between a sea-leveland vacuum- optimized nozzle. The rocket itself is identicalexcept for the nozzle expansion.

And since the way we lower exit pressure is with a biggerexpansion ratio, ε = Ae

A∗ , this result explains why upper stage"vacuum nozzles" are so much bigger than first stage "sea-level" nozzles as can be seen in graphic above and the side-by-side of the SpaceX Merlin 1D vacuum (left) and Merlin1D sea-level (right) engines:

Figure 0.5: SpaceX’ Merlin engine has very different sizenozzles for its first and second stage even though the engineitself is nearly identical.

So what actually happens when a nozzle is under- or over-expanded? In a subsconic (incompressivle) jet, the fluid dy-namics allow for communication of information upstream

and the pressure distribution in the nozzle would relax toa state where Pe = P0. In supersonic (compressible flow),pressure information travels more slowly than the fluid ve-locity so there is no information transfer upstream. In orderto equilibrate pressure with the ambient, a set of expansionor compression waves form from the exit of the nozzle.

Throat Area

Expansion Waves

Throat Area

Shock (compression) waves

Final Area

Under-expanded Case

Over-expanded Case

Figure 0.6: In a super-sonic exhaust stream, expansion orcompression waves form to equilibrate the jet pressure tothe surrounding pressures.

Due to flow radial inhomogeneity in the flow in real noz-zles, there will always be some equilibration wave structureat the exit. The pattern of compression and expansion wavesand associated changes in gas temperature and luminos-ity are what are generate Mach Diamonds in the exhauststream.

Figure 0.7: An example of mach diamonds in the plume ofone of Stanford’s research hybrid rockets

Finally, we note that if nozzles are sufficiently over-expanded, the flow can no longer stay attached to the nozzle

6

contour and separates. This typically happens for PeP0

< 0.4and is an undesireable condition. Because the flow is not at-tached, it may be asymmetric, generating side loads. It alsomay be unstable generating unsteady forces on the rocket.

The diagram below shows how separate flow develops ina nozzle as a function of pressures.

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3.3. ISENTROPIC FLOW THROUGH NOZZLES 67

Note that performance at optimum nozzle expansion (p2 = p3 = 0.396 atm.) this nozzle deliv-ers a thrust that is somewhat below the maximum. From Fig. 3–7 we find that for ! = 6 theoptimum CF = 1.52, which corresponds to 7.2 km elevation, whereas the maximum value isCF = 1.63 in a vacuum.

Under- and Overexpanded Nozzles

An underexpanded nozzle discharges the gases at an exit pressure greater than theexternal pressure because its exit area is too small for an optimum expansion. Gasexpansion is therefore incomplete within the nozzle, and further expansion will takeplace outside of the nozzle exit because the nozzle exit pressure is higher than thelocal atmospheric pressure.

In an overexpanded nozzle the gas exits at lower pressure than the atmosphere asit has a discharge area too large for optimum. The phenomenon of overexpansioninside a supersonic nozzle is indicated schematically in Fig. 3–8, from typical earlypressure measurements of superheated steam along the nozzle axis and differentback pressures or pressure ratios. Curve AB shows the variation of pressure with

M

FIGURE 3–8. Distribution of pressures in a converging–diverging nozzle for differentflow conditions. Inlet pressure is the same, but exit pressure changes. Based on earlyexperimental data.

Figure 0.8: Diagram illustrating the flow separation phe-nomenon. From [1]

Finally we present a very nice plot of how C f varies withall of the dependencies from [1].

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3.3. ISENTROPIC FLOW THROUGH NOZZLES 65

2

1.8

1.6

1.4

1.2

1

0.8

0.61 3 10 30 100 300 1000

CF

= A2/AtArea ratio

k = 1.20

Line of optimumthrust coefficientp2 = p3

5

10

20

50

100

200

500

1000

2000

5000

p1 /p

3 =

Region below lin

e: Incipient flo

w separation

Region

below

line: U

navoidable flo

w separation

p 1/p 3 = ∞, Maximum value = 2.246

FIGURE 3–6. Thrust coefficient CF versus nozzle area ratio for k = 1.20.

1.8

1.6

1.4

1.2

1

0.8

0.61 3 10 30 100 300 1000

CF

k = 1.30

Line of optimumthrust coefficientp2 = p3

5

10

20

50

100

200

500

1000

2000

5000

p1 /p

3 =

Region

below

line: In

cipient flo

w separation

Regi

on b

elow

line:

Una

voida

ble flo

w separation

= A2/AtArea ratio

p 1/p 3 = ∞, Maximum value = 1.964

FIGURE 3–7. Thrust coefficient CF versus nozzle area ratio for k = 1.30.

Figure 0.9: Diagram illustrating the flow separation phe-nomenon. From [1]

0.5 c∗ Efficiency

Due to imperfect mixing, combustion, chamber heat-lossand other effects the realized rocket performance is typicallyless than that computed theoretically. This will be reflected

in the rocket thrust, effective velocity and c∗ and potentiallyC f .Becuase many of the larger losses are associated with the

process of combustion in the chamber, the non-ideal perfor-mance is most easily and directly measured in c∗. It is withthis in mind that c∗ efficiency is defined:

ηc∗ =c∗measured

c∗ideal=

Pt A∗

mpc∗ideal(20)

where c∗ideal is computed using Equation (8) for simple sub-stances or, more commonly, using a complex chemical equi-llibrium code as will be discussed in the next lecture.

Optimizing ηc∗ becomes a primary activity in the develop-ment of a rocket engine, often consuming a singificant frac-tion of the development budget and time. In liquid rocketsthis might manifest as injector tuning, while in a solid rocketit is attacked by adjusting propellant composition, ratios andparticle sizes.

0.6 Putting it all together

The separation of C into c∗ and C f separates concerns be-tween propellant thermodyanmics in c∗ and nozzle physicalparameters in C f .

Let’s quickly compute C, c∗ and C f to get a sense of howwe would begin to use them practically. Assume we startwith a plenum full of room-temperature, high pressure gasand then vent it into vacuum through a nozzle.

Now let’s compute c∗, C f and then C. For now we willassume optimal expansion (P0/Pe = 1) and calculate theassociated area ratio. A useful relation we will use to deter-mine Me is:

M2e = 2

PePt

1−γγ − 1

γ− 1

= 2

(

PeP0

P0Pt

) 1−γγ − 1

γ− 1

(21)

Name γ Mw c∗ (m/s) C f C (m/s)

N2 1.40 28 434.4 1.26 546.4H2 1.41 2 1621.5 1.26 2039.2He 1.66 4 1085.2 1.25 1356.2

Table 0.2: c∗, C f , C for several pure gasses at room tem-perature, P0/Pe = 1 and P0/Pt = 0.1. Note how C f isessentially constant despite the differences in gas propertieswhile c∗ and C vary substantially.

This simple exercise confirms what we said before - c∗

depends on the nature of working fluid while C f is largelyfixed by the nozzle parameters. Another interesting conclu-sion from this is that even before we get to chemistry, we

0.7. ENERGY AND EFFICIENCY 7

can see how wonderful of a working fluid Hydrogen is due toit’s low molecular weight. Remember this because hydrogenwill emerge again and again as we talk about rockets.

If we expand Equation (2) and simplify we arrive at auseful form for C:

C =Pe − P0

Pt

Ae

A∗

√RTt

γ

[γ + 1

2

]−2(γ−1)γ+1

+√√√√ 2γ

γ− 1RuTt

Mw

[1−

(Pe

Pt

)(γ−1)/γ] (22)

In Equation (22) we can see all the impacts of propellantchoice and nozzles on effective exhaust velocity. The terminvolving Tt and Mw clearly shows the effect of propellantchoice on C, as we noted with c∗ earlier. The term involvingPePt

demonstrates the impact of the system pressures andnozzle geometry we choose. Figure 0.10 summarizes all ofthis.

(K-kg-mol/kg)

(s)

Figure 0.10: How specific impulse varies

0.7 Energy and efficiency

So far we have concentrated on propulsion primarily froma conservation of momentum perspective. But there are alot interesting observations to be made when we look at thethermodynamics of propulsion as well.

With rocket propulsion and things in space we are talkingabout a lot of energy and to put that in to context, let’ssee how much energy our Low Earth Orbit satellite ends upwith.

An object moving in a conservative potential field (likeEarth’s gravity) has total energy:

E = KE+ PE =mv2

2+

GMmr

or in mass-specific energy

ε =v2

2+

GMr

In the case of our satellite in a circular Low Earth Orbitthis comes out to:

ε =v2

2+

µ

r0− µ

r= 29.1MJ/kg+ 4.5MJ/kg = 33.6MJ/kg

Again note how much of this is due to the orbital velocityof the rocket. And to put in context just how much energythis is, it is equivalent to:

• 1000x the specific energy of a Jet airliner at 650MPH

• 5x the specific chemical energy content of nitroglycerin

• 3x the required specific energy to melt aluminum

It is no wonder that not much is left when object withthis much energy slams into the atmosphere!

0.8 Rocket efficiency

Clearly there is a lot of energy being liberated in rocketengines in order to put things into space. But how efficientare they?

There are different ways to define energy efficiency, butthe the first we’ll look at is the thermal efficiency or howeffectively a rocket takes input energy and converts it to gaskinetic energy:

ηthermal =Wexhaust

Pin=

mC2

2Pin

where Pin is the amount of energy being liberated in timeto power the rocket whether it be latent, chemical, nuclearor electrical energy. In the case of chemical rockets we candefine it as the energy released by the propellants when theycombust or:

Pin = m∆h

where ∆h is the amount of thermal energy going into therocket chamber.

Substituting Equation (22) gives

ηthermal =

2γγ−1 RTt

[1−

(PePt

)(γ−1)/γ]

2∆h(23)

Because

Tt ∼∆hCp

we can subsitute such that

8

ηthermal =γ

γ− 1RCp

[1−

(Pe

Pt

)(γ−1)/γ]

(24)

And finally recognizing that for an ideal gas

RCp

=γ− 1

γ

and for a perfectly isentropic nozzle expansion process

Te

Tt=

Pe

Pt

(γ−1)/γ

we see that Equation (23) reduces to the Carnot efficiency.Figure 0.11 shows ηthermal as a function of Pt/Pe:

0 25 50 75 100

Pt/Pe

0.3

0.4

0.5

η thermal

Figure 0.11: How specific impulse varies

This is pretty good for a heat engine and a lot of earlyrocket work focused on optimizing thermal efficiency. Andthermal efficiency will matter when we get to electric propul-sion, but it is less useful a construct for chemical rocketswhere it is largely defined by the propellant properties ratherthan the machine they burn in. Furthermore the efficiencyof accelerating gasses is not really what we care about - wecare about the efficiency of accelerating the vehicle.

To understand that, we will define another, more usefuleffiency metric, called propulsive efficiency as such

ηprop =Wr

Wr + Wexhaust=

TvTv + 1

2 m(C− v)2

or the fraction of total work that actually goes to accelerat-ing the vehicle.

TvTv + 1

2 m(C− v)2=

mCvm [Cv + (C− v)2]

→ ηprop = 2vC

1 + ( vC )

2

The overall efficiency is

η = ηthermalηprop

A small example demonstrating how η varies over ratio ofvehicle speed to exhaust speed, v

C is shown below.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0V/C

0.0

0.2

0.4

0.6

0.8

1.0

η

ηprop

η

And here we see another interesting result much like opti-mal nozzle expansion. The rocket is most efficient when theexhaust velocity equals the vehicle velocity. This is actuallyquite intuitive - in this condition the exhaust gasses are leftwith zero velocity in the static frame. There is no residualkinetic energy in the exhaust so all of it’s kinetic energy havebeen transferred to the vehicle. Anywhere else the exhaustis left with residual kinetic energy which is a loss.

Propulsive efficiency will come back in a big way whendiscuss electrical propulsion as it is very important to systemoptimization.

0.9 Energy losses

Finally let’s also look at where the inefficiencies are for atypical launch vehicle powered by a chemical engine withIsp = 340 s and moving at 5,000 m/s for intuition.

Reactionenergy

1

Incomplete combustion0.02

Heat loss0.01

Friction, etc0.01

Second Law loss (unavailable thermal energy)

0.4

Exhaust kineticenergy

0.1

Useful0.46

Energy loss diagram

Notice that almost half of the chemical energy is unavail-able for conversion to work due to second law of thermo-dynamics. This is expected given the Carnot efficiency wedeveloped earlier. The biggest contributor after second law

0.9. ENERGY LOSSES 9

loss is residual exhaust kinetic energy which we saw is some-thing is minimized when V/C = 1. The other losses arequite small and we can say

When a rocket operates with exhaust velocity nearvehicle velocity, it is a surprisingly efficient heatengine

Bibliography

[1] George P. Sutton and Oscal Biblarz. Rocket PropulsionElements. John Wiley & Sons, Inc., 7 edition, 2001.

11