So I’m reading an article on rockets in my old encyclodedia(Encyclopaedia Britannica, 1975) and have a question about a statement in the article.
It states, “The gas velocity in a properly designed converging-diverging nozzle is sonic at the throat (most narrow) portion of the nozzle and becomes supersonic as it travels through the diverging (exhaust) end of the nozzle.”
So I know that when liquids travel trough a smaller space they speed up and then slow down when traveling through a larger space. Assuming same volumn and time. I also know that gases are compressable and that liquids aren’t and the two don’t act the same. Even so it seems counterintuative to me that the exhaust gases speed up on expansion out the nozzle. I could understand if more heat was being added but I don’t think that is the case. So why does the exhaust speed up after the nozzle constriction?
In a de Laval (converging-diverging) nozzle, the diameter of the throat is sized such that the subsonic flow inside the chamber, which is at a high pressure (compared to the ambient pressure), is accelerated to sonic speed at the throat without choking the flow (causing a constriction) after which it passes into the low pressure region in the diverging sectoin of the nozzle (often called the ‘cone’ even though it isn’t strictly conical in shape) where the gas products expand. The reason they expand is because the fluid has a lot of thermal energy (also known as enthalpy) that comes from compressing the fluid through the nozzle which is then released by allowing it to expand. The expanding fluid accelerates as the enthalpy is converted to momentum via expansion, and some of this pushes out radially against the nozzle cone which is optimally shaped to convert the flow into a net axial flow, which accelerates the exhaust to supersonic speeds.
In an ideal rocket, the exhaust would convert all thermal energy that is in excess of the fluid at the exit plane matching ambient pressure and corresponding temperature of ideal expansion in the flow into axial momentum and exjecting it axially at maximum speed. In reality, it is never really practical to fully expand the flow to ambient preas the length of the cone and necessary ratio of the exit plane area to throat area results in an enormously heavy nozzle (which is problematic from both the standpoint of overall inert mass and controllability of a gymboling nozzle). In general, nozzles designed for sea level operation are limited to expansion ratios of no more than 25:1 and are spefically designed to produce underexpanded flow to prevent flow separation and buckling of the nozzle due to net inward pressure differentials, while nozzles for high altitude and vacuum operation have area ratios between 40:1 and 300:1. (While there is a measurable performance benefit to ratios up to 1000:1, packaging, mass, and controllability trades are usually negligible or negative above 300:1, even for high performance cryogenic stages.)
There are some additional assumptions which are generally applied in first-order performance analysis, including axisymmetric flow, isentropic conditions (no thermal loss to or through the nozzle, and frozen flow (no combustion beyond the throat). Since those effects are small and generally cancel out to within a couple percent of total impulse) they are generally neglected in design and are later measured or inferred from data during testing to develop flight performance models.
All rocket motors and engines, save for small pressed powder hobby motors use some kind of diverging nozzles to enhance performance regardless of whether there are single or multiple nozzles. While there can be detrimental effects to the vehicle due to plume interactions, these are typically either acoustic shock effects during liftoff or plume recirculation in vacuum conditions, both of which are dealt with by the use of acoustic protection and preventing grossly underexpanded flow respectively.
Another way of looking at it is that the individual molecules of the gas are travelling supersonically, on average, because they’re very hot. They contain a great deal of kinetic energy, but little net momentum because the particles are bouncing every which way, and therefore the gas as a whole is relatively slow. The nozzle is shaped to encourage the particles to align their trajectories in one particular direction. A necessary condition to the process is that the gas expands and decreases in temperature.
Like all thermodynamic engines, there is a lower limit to the degree to which this is possible related to the ambient temperature–that is, you can only align the particle trajectories so well, because after a certain point the exhaust gases are at a lower temperature than the air. This isn’t a big problem, though, because the starting temperature is so high.
I just wanted to point out that, for subsonic flows, nozzles actually do behave the way you intuitively expect them to.
I know you asked about rockets and the excellent answers above address that. But I just wanted to share the only thing I know about nozzles on the off chance that you or maybe somebody else reading this didn’t know it already since it doesn’t seem to be explicitly described in the thread so far.
[QUOTE=wiki]
Divergent nozzles slow fluids if the flow is subsonic, but they accelerate sonic or supersonic fluids.
[/QUOTE]
On re-reading Stranger’s reply, I see that what I’ve contributed can be found in the first sentence of his post where he describes what happens in the first part of the converging-diverging nozzle (i.e., subsonic flow is accelerated in the converging portion).
Apologies for that, but hopefully my simplistic phrasing might still be of some use to those operating at my level of understanding. : )
A small correction to my post. I said: the individual molecules of the gas are travelling supersonically, on average, because they’re very hot
That isn’t quite right as I stated, since in fact the ratio between average molecular speed and the speed of sound is fixed (roughly 3:2 for typical exhaust products). The reason is that both the speed of sound and average speed increase with the square root of temperature.
Nevertheless, it’s still true that (by one way of looking at it) the purpose of the nozzle is to take these fast-moving particles and get them moving in mostly one direction.
The video shows a CFD simulation of rocket engine startup flows. The focus appears to be the surface flow effects, with the caption even mentioning the possibly destructive transient loads they can induce in the engine bell. Indeed, this can be seen in this slow motion video of a shuttle main engine start; watch the engine bell at the left of the frame wobble and ring at around 0:52. :eek: Once those vortices get blown out and smooth, uniform flow is established, all is well.
As NASA knows, the devil is in the details. But I’m still struggling to get my head around what the basic velocity profile downstream of the throat is as the throat transitions from subsonic to sonic. There are two basic claims that are understood to be true about flow in a converging-diverging nozzle:
-when the flow through the throat is subsonic, then the downstream flow decelerates from this velocity.
-when the flow through the throat is sonic, then the downstream flow accelerates from this velocity.
This implies a step-change in downstream flow behavior when the flow in the throat goes from Mach 0.99999…to Mach 1. Wikipedia’s diagram of choked nozzle flow provides a velocity profile, showing that the velocity continues to increase downstream of the throat. What does that velocity profile look like when the pressure ratio is juuuuuust barely high enough to obtain sonic flow at the throat?
The fluid reaching the speed of sound at the throat only occurs at the critical pressure, which is just a function of the ratio of specific heats of the products for a given chamber pressure. This maximizes the amount of mass flow through the nozzle. Increasing the pressure doesn’t make it go faster (since there is nowhere to expand inside the throat), but it does keep the temperature at the throat closer to the combustion temperature, and therefore the flow has more energy to expand.
Think of it in these terms: the energy of combustion results in two “forms” of potential energy; the thermal energy of the combustion products (measured by temperature and the molecular mass of the products) and the pressure contained by the chamber. At sonic flow, the amount of fluid (mass flow rate) and pressure you can get out through the throat is maximized; increasing the pressure doesn’t affect the speed of sound. However, it does compress the fluid in the chamber, which has the effect of making it hotter, such that when the resulting mass flow exits the chamber it has more energy to expand. (Note that this discussion is for illustrative purposes only; these aren’t really two forms of energy, as anyone familiar with the ideal gas law will observe; they are two measures of the overall state of the fluid, but it may help to think about converting from thermal energy to the mechanical energy in pressurization.)
The velocity profile aft of the throat is highly dependent upon the shape of the diverging section which is optimized to allow the fluids to expand and convert that energy into momentum before the plume exits the nozzle. Too narrow or short of a nozzle, and the gases escape before you get maximum expansion for a given ambient pressure (underexpanded flow); too large, and the flow is overexpanded and you end up with a nozzle that is unsupported on the inside. In practice, all nozzles are designed for slightly underexpanded flow in order to maintain structural margins, controllability, and minimize mass. If you just cut off the nozzle at the throat (rendering it just a sonic nozzle at critical pressure) then you would see the plume expanding, but with no way to convert that to vehicle momentum. For subsonic flows (if you cut the nozzle above the throat), there is essentially no “conversion of chamber pressure to thermal energy”, so there is no additional energy to accelerate the flow.
All of this pertains to isentropic, steady-state flows. The transient conditions which occur during startup or abrupt change of flow rate, or caused by geometrical discontinuities in the nozzle and unsteady flow due to injectant or shock interactions are additional considerations which are beyond the means of simple closed form analysis to address, and require either empirical methods based upon test data or simulation using CFD to assess.
You’ve delved into a lot of specifics, but somehow I don’t see the answer to my question.
Let’s narrow this down a bit:
-assume the working fluid is air, specific heat ratio = 1.4, critical outlet/inlet pressure ratio = 0.528.
-assume a generalized converging-diverging nozzle, with steady-state flow.
For a pressure ratio <0.528, the flow is subsonic at the throat, and as we move downstream, the flow velocity decreases.
Suppose we increase the upstream pressure (constant upstream temperature) until the pressure ratio is substantially greater than 0.528. Now the flow at the throat is sonic, and as we move downstream, the flow velocity increases.
So what does the velocity profile in the downstream portion of the nozzle look like when the pressure ratio is exactly 0.528?
Actually, I need to amend that. The average pressure at the nozzle exit will be the same as at the throat (assuming fully expanded flow). The flow profile will be whatever it needs to be to achieve that equilibrium, and will depend on the actual profile of the nozzle, but it will not be faster than the speed of sound at the density of the fluid. For a supersonic nozzle, the average speed will increase and the pressure at the nozzle exit will be greater than that at the throat (again, assuming fully expanded flow). Again the specifics of the flow field will depend on the nozzle contour.