# Rocket Nozzle Shape and Length

How is the efficiency of a particular nozzle shape with a given length calculated?

To clarify what I'm not asking for: I understand the idea behind the de Laval nozzle, where you constrict the flow when it's subsonic and then expand it when it's supersonic in order to convert pressure in the combustion chamber into exhaust velocity. I understand the equations that relate the diameter of the nozzle at a particular point down the principle axis with the temperature / pressure / velocity at that same point. Essentially, everything said in this page and all the equations there make sense to me.

But what I haven't been able to get any material on (even when following the citation chain back to old papers from the 40s) is how the diameter should change with axial distance. The maths for a de Laval nozzle, as in the link above, tells me what the ratio of throat diameter to exit diameter should be, but not if the nozzle should be 1m, 10m, or 100m long. I understand that a perfect nozzle should be infinitely long and infinitely wide at the end. I can compute the (theoretical) losses from it not having an infinite exit diameter easily enough, but I don't have a clue how to do the same when limiting it to a finite length. I don't even know what branch of physics would give the relevant constraints here: I'm guessing something to do with supersonic fluid dynamics and preventing instabilities or flow separation, but my search hasn't given me the actual equations that are relevant here. Just methods for geometric constructions of specific nozzle shapes alongside a stated efficiency.

Two things that do come up is that a cone with ~15° half-angle is about 95% efficient, while the 'standard' bell shape is 99% efficient. But I have no idea where these numbers come from. I'm not even sure what the efficiency is measured relative too (an infinite nozzle? A nozzle with the same exit diameter but infinite length? The ideal nozzle bounded by the same size)?

I'd be happy with a method to compute the efficiency of a conical nozzle for different lengths and half angles under even the most idealised (spherical cow in a frictionless vacuum kind of world) of situations, although a reference to a more complete mathematical treatment would be nice.

• Are you familiar with the method of characteristics as applied to nozzle design? Commented Jul 21 at 17:49
• I am not, although it is a phrase that I've seen turn up a few times - but never with a citation that helped me understand what it was. Commented Jul 21 at 17:54
• I'm not sure it's really what you are looking for but there is some info here mae-nas.eng.usu.edu/MAE_5540_Web/propulsion_systems/section8/… It also gives references. Commented Jul 21 at 18:02
• That looks like it has what I'm looking for, but it's hard to get much out of the slides without more context, I'll try to track down the textbook being referenced. Am I right to think it's Computational Fluid Dynamics by Anderson et al? Commented Jul 21 at 18:36
• Searching on "nozzle design using method of characteristics" will likely turn up a lot of stuff, maybe some is better explained. I am not sure which Anderson text is referenced. Commented Jul 21 at 18:44

Nozzle design is complicated and your question packs a lot into it, touching on many of the complications.

My comment was intended to address the part of your post that states "But what I haven't been able to get any material on (even when following the citation chain back to old papers from the 40s) is how the diameter should change with axial distance." At least in my day this was done using the method of characteristics. It is probably done with CFD now.

Nozzle efficiencies are also complicated and there are several that can be in play. If you want an overall efficiency maybe the best is energy conversion efficiency, which I will spend the rest of this answer writing about.

Quoting from Sutton 4th edition p.77

The flow in a real nozzle differs from that of an ideal nozzle because of friction effects, heat transfer, imperfect gases, nonaxial flow, and nonuniformity of working substance and flow distribution. The degree of departure is indicated by the energy conversion efficiency of a nozzle, which is defined as the ratio of the kinetic energy per unit of flow of the jet leaving the nozzle to the kinetic energy per unit of flow of a hypothetical ideal jet leaving an ideal nozzle that is supplied with the same working substance at the same initial state and velocity and expands to the same exit pressure as the real nozzle.

$$e = \frac {(v_2)_a^2} {(v_1)_a^2 + c_p(T_1 - T_2)}$$

Where:

• $$e$$ = energy conversion efficiency
• $$v$$ = velocity
• $$c_p$$ = specific heat at constant pressure
• $$T$$ = temperature
• Subscripts: $$1$$ is the nozzle inlet, $$2$$ is the nozzle exit, $$a$$ = actual (the unexpanded equation has $$i$$ in it for ideal, but that is replaced by the enthalpy change in this version, so you don't see an $$i$$)

• Sutton p. 90 and surrounds (general nozzle theory)
• Hill and Peterson Chapter 11.3 (general nozzle theory)
• Shapiro Dynamics and Thermodynamics of Compressible Fluid Flow Chapter 15 (Method of Characteristics)
• If I understand correctly, both the Method of Characteristics and CFD are computational ways to solve a boundary problem? What I don't really get is the physics of the problem that it being solved. Is it what's the maximum gradient at which the nozzle can expand before flow separation happens? A sort of continuous Prandtl-Meyer where the aim is to avoid shockwaves? Or is there another physical idea at play here that I'm missing? Commented Jul 21 at 19:05
• $\theta = v/2$ on page 30 & 31 of the slides I think is the heart of the general answer I'm looking for (in terms of broad physics rather than specific engineering methods), but I'm uncomfortable with equating angles with velocities purely for dimensional reasons, and the derivation escapes me without a better definition of the terms. But I'll have to study this closer before I ask any more stupid questions. Commented Jul 21 at 19:07
• It looks like the version of Sutton available online, which is often worse than my paper copy, didn't get this nozzle stuff dumbed down. So check out page 90 etc here.web.archive.org/web/20140722111108/http://web.mit.edu/e_peters/… Commented Jul 21 at 19:11
• There is a good chapter on nozzles (11.3) in Hill & Peterson which talks about method of characteristics some (again, this is newer than my paper copy) soaneemrana.com/onewebmedia/… Also in Shapiro Chapter 15 which gives the full treatment on MoC. There was a copy of that at the Internet Archive but it's not available now. archive.org/details/dynamicsthermody0000shap/page/n9/mode/2up Commented Jul 21 at 19:34
• Ok, thanks, I think I have enough homework for weeks lol. Page 77-82 of Sutton linked above has the qualitative ideas and answers I was looking for for Bell nozzles, and enough of a hint that I could derive the theoretical efficiency of a conic nozzle myself. Commented Jul 21 at 19:57

From a search for "Rao rocket nozzle" (on startpage com) I was able to download two of Rao's papers (Jet Propulsion journal of June 1958 as rao_1958.pdf and ARS Journal of Nov. 1961).

NASA publication 19830016278.pdf Perfect Bell Nozzle Parametric and Optimization Curves of May 1983 by Tuttle and Blount is available from ntrs.nasa.gov

The Reaction Research Society, apparently amateur rocketry oriented, has an interesting paper in 2023: https://www.rrs.org/2023/01/28/making-correct-parabolic-nozzles

Looks like the following pairs well with the previous for determining the shape of the bell: http://www.aspirespace.org.uk/downloads/Thrust%20optimised%20parabolic%20nozzle.pdf

Also try a 2019 paper (in English, from China) New Contour Design Method for Rocket Nozzle of Large Area Ratio from https://onlinelibrary.wiley.com/doi/epdf/10.1155/2019/4926413

More than enough reading in these papers.

I'm answering my own question not to one-up the very useful material already posted, but because I think a high-level overview of why bell nozzles are bell shaped and why they have the dimensions they have might be helpful to other people. The previous answers show that there's little available online that isn't highly technical or in graduate-level textbooks, so this is my attempt at a summary of what's happening. It's deliberately short on technical maths, but high on physical principles.

## Expanding from Hot to Cold

(Going over more basic stuff than the question asked for the sake of completeness)

The purpose of a rocket nozzle is to turn the hot, high pressure, quasi-stationary gases in the combustion chamber into a fast flowing gas moving directly away from the rocket. Thermodynamically, this comes down to turning the energy locked up in the temperature and pressure of the gas (both the kinetic energy of the molecules bouncing around randomly, and any internal energy from their rotations/vibrations), into macroscopic kinetic energy where all the gas particles are moving in the same direction (collimated, to borrow optics parlance). This is useful because while the former has 0 net momentum, the latter has a lot of net momentum which, through Newton's Third Law, becomes thrust for the rocket.

This collimation is done by moving the gas out of the combustion chamber through a specially shaped "pipe" - the nozzle. This pipe has cylindrical symmetry and first narrows down so that Bernoulli's principle (modified to take into account that gases are compressible) makes the gas speed up. Until the narrowest point of the pipe, called the neck, where it reaches mach 1: the speed of sound for the exhaust gases at the temperature and pressure at the neck. The compressibility of the gas means that constricting further than this chokes the flow rather than causing it to speed up. Instead, the thing to do is to expand it back: like opening the valve on a pressurised can of deodorant.

Credit: https://www.geeksforgeeks.org/bernoullis-principle/. Bernoulli's principle in the first part of a rocket nozzle. Conservation of mass flow requires $$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$$. This means that as the pipe gets narrower, the bulk fluid velocity must increase (the density increases a little, but not enough to compensate entirely for the reduction in cross sectional area when subsonic). Energy is conserved because this increase in bulk velocity comes from better aligning the random velocity of the constituent particles of the fluid; which is equivalent to saying the pressure of the fluid drops. This stops happening at Mach 1, where the compressibility dominates and narrowing the pipe makes the flow choke rather than speed up.

Skipping over some moderately involved maths, we find that: How much of the pressure is "converted" into collimated momentum depends on the ratio between the cross-sectional area of the neck and the final opening of the nozzle. The greater this expansion ratio, the more efficient the nozzle and the higher the thrust and ISP. Note that the geometry between the combustion chamber and the neck is not critical, because subsonic gas flow is well behaved - it's what happens after the neck which is most important and will be looked at in the following sections.

Before we get to that, we should note that this expansion can't be performed completely, because infinitely wide nozzles are difficult to construct. In addition, when burning in atmosphere this is not desirable anyway: the exhaust gas should have enough pressure left to prevent the ambient atmospheric air from flowing up into the nozzle and causing all sorts of instabilities. The result is that, even for vacuum engines, other design considerations (the mass of the materials, and the need to fit the nozzle in the rest of the rocket) means that a specific expansion ratio is picked. To give a feeling for the order of magnitude, this is ~20 for sea level engines and ~50+ for 2nd stage engines, but with substantial variations.

In what follows, I'll assume that we've fixed the desired expansion ratio, and will now ask: How do we connect the narrow neck and wide nozzle exit together? How far apart do they need to be?

## Conical Nozzle

The simplest way to connect a small circle at one end with a wide circle at the other is with a (truncated) cone. The longer the cone, the smaller the half-angle because they are linked by simply geometry as $$L = \frac{R_e - R_t}{\tan(\alpha)}$$ Where $$R_t$$ is the radius at the throat ($$A_t = 2\pi R_t$$) and similarly at the exit for $$R_e$$.

$$A_t$$ is the throat cross-sectional area, $$A_e$$ the exit cross-sectional area - their ratio is fixed according to the considerations of the previous section.

This leaves us with one parameter to play with, how does varying it effect the nozzle's performance? Looking at the diagram above, we see that the gas flow out of the nozzle is not completely collimated! The central part of the flow is moving in the desired direction, but the outwards parts are moving at an increasing angle which reaches $$\alpha$$ at the edge. This leads to some loss of efficiency as momentum (and hence thrust) is wasted as cosine losses. Integrating this loss leads to the expression: $$\lambda = \frac{1}{2}\left(1 + \cos{\alpha}\right)$$

which is the ratio between the momentum of the outflowing gas in this conical nozzle, compared to one that was infinitely long ($$\alpha\to0$$) but with the same expansion ratio. That is, $$\lambda$$ is the theoretical efficiency of the nozzle geometry.

This shows that a small $$\alpha$$ is desired (less cosine losses from a better collimated flow), but this requires a longer $$L$$ because the expansion ratio is fixed. However this is undesirable because a longer nozzle is heavier and harder to fit into a rocket. Hence, we need to compromise between the two. In practice, $$\alpha=15^\circ$$ is a good trade off that gives $$\lambda = 98.3\%$$ and $$L \approx 4 R_e$$.

But can we do better? Can we make a nozzle that has the same expansion ratio but is both shorter and has better collimated flow at the end?

## Towards a Bell Nozzle

We've reached the limit of what simple conical geometry can achieve, but why not use a more complicated shape?

The large initial expansion angle ($$\alpha_1$$) allows for a shorter nozzle and the smaller final angle ($$\alpha_2$$) results in well collimated flow. The ideal end result is to both increase $$\lambda$$ whilst having $$L'\lt L$$

The issue with this is that supersonic gasses do not like going around corners (circled in green). Forcing it to do just that can lead to dangerous instabilities. Even if these are avoided, supersonic flow around areas of tight curvature causes expansion (or compression) shockwaves which absorb energy and so reduce the total efficiency of the nozzle. The solution, therefore, is to replace these corners and straight lines with smooth curves.

Hence the final task is, for a fixed expansion ratio and nozzle length $$L'$$, to draw a contour such that the effective $$\alpha_1$$ is large and $$\alpha_2$$ small in order to minimise both the cosine losses of the outflow and the energy loss from shockwaves in the supersonic flow. This requires some hefty mathematical techniques that I won't go over here.

## The Bell Nozzle

A bell shape nozzle is the solution to the problem above. Variants exist based on what physical effects are taken into account and the exact design goals (including having a shape which is easy to fabricate in practice), but the Rao nozzle is a very widespread and efficient one. The inner most green circle in the diagram above is replaced by a small circular arc - the high pressure in the region means that the gas can take relatively tight corners without too much problem, while the outer part of the nozzle is a parabola with low curvature that smoothly connects to it. This also allows a partial cancelation between the expansion shockwaves where the nozzle curves outwards, and the compression shockwaves where it curves back inwards.

The end result is that a Bell Nozzle gives a maximum expansion angle on the order of $$\alpha_1 \approx 30^\circ$$, a final exit angle of about $$\alpha_2 \approx 10^\circ$$ for a total nozzle length of 80% or under of an equivalent conical nozzle. Below is a diagram and table from Sutton (referenced in @Organic Marble's answer) showing a comparison of different geometries.

Looking at the above, we can see that for an 80% bell nozzle with an expansion ratio of 50 the efficiency is $$98.8\%$$. We can calculate that $$0.6\%$$ of the potential momentum is wasted as cosine losses and hence that another $$0.6\%$$ is dissipated through shockwaves within the nozzle. The end result is both better performance and a more compact size than a conical nozzle: a rare case of win-win in this game of trade-offs.

## Reality

Everything above, including the details of the maths skipped, is done in a pretty idealised physical model. In reality there are many practical factors that need to be taken into account: the viscosity of the fluid, imperfect thermalisation within it, inhomogeneities through out, instabilities in combustion, ongoing chemical reactions, erosion (through both friction and heat transfer) of the nozzle, and much more. These are all important, but act more as tweaks to the principles laid out above than drastic revisions of them. One consequence of these additional affects is that the realised efficiencies of nozzles will be lower than the theoretical efficiencies mentioned above. The references given in the other answers lay bare both the maths I glossed over, and all of these engineering complexities. Despite (or because of) the lack of these full technical details, I hope that someone will find these notes as helpful to read as I did to write.

## tl;dr

Bell nozzles are good because they get wide quickly, have almost parallel sides at the end, and do this with low curvature where it matters. Depending on the exact design, they have about $$98-99\%$$ theoretical efficiency compared to an infinitely long nozzle (that has the same exit radius), whilst having the distance from the neck of the nozzle to the exit of about $$2.2-3$$ times the exit radius.