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LS2: Basic engine design - compressible flow in rocket nozzles
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- Great links
Anderson - Fundamentals of Aerodynamics (Part 3 on Inviscid, compressible flow):
A great youtube video explaining nozzles and their flow: https://www.youtube.com/watch?v=p8e8A3sdVOg
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Theory
Fuel and oxidizer flow at specific rates through the feed system (see Topic 4) and are combined in the combustion chamber of the rocket. The combustion process results in products (i.e. the products of combustion after combining fuel and oxidizer) flowing out of the combustion chamber into the rocket nozzle as a gas. The combustion chamber, therefore, determines the flow conditions at the entrance of the rocket nozzle. In this LSET, we consider the flow of the combustion products through the rocket nozzle.
Mass
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Flow
Let’s simplify the problem first by looking at a practical example. Consider the flow through a water hose. In this case, the mass flow rate (LSET 1) through the hose is determined by the spicket (hose entrance) and is constant. If we change the cross-sectional area at the exit of the hose, the velocity of the water at the exit will change correspondingly. This is illustrated in Figure 1 below.
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In the previous discussion, we discussed “low” and “high” speed flow behavior. We need a metric to define what these “low” and “high” speeds mean and consequently, where incompressible and compressible flow holds. The speed of sound is used as such a metric to distinguish between these two behaviors.
The
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Speed of
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Sound
The speed of sound is the velocity at which the sound travels through a medium. This means that its value must depend on the specific medium considered. For ideal gases, the speed of sound can directly be related to the static temperature of the gas. We model the combustion products as an ideal gas (see the section on ideal gas). We can calculate the speed of sound as follows:
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Figure 2: Sound waves propagating from a source in subsonic (left), sonic (middle), and supersonic flow (right).
Mach
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Number
The velocity of the flow, relative to the speed of sound, is called the Mach number: The equation for Mach number is: 
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In compressible flows, we usually don’t use velocity but Mach number. By using velocity, we don’t know in which regime (i.e. subsonic or supersonic) we are operating, because this all depends on what the speed of sound is. By using Mach number, we know exactly if we are subsonic or supersonic and therefore, we have a good idea of what the flow behavior is like.
Choked
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Flow
As already said, the flow behavior changes drastically going from subsonic to supersonic flow. The main example that you should remember is what happens with Mach number when the flow area changes. This is illustrated in the figure below.
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Suppose we have M<1 at the inlet of a channel and we continue to shrink the area until the flow speeds up to M=1. To increase the Mach number even further, we now have to increase the area, since the flow behavior switched from subsonic to supersonic. That means that, at the point at which the flow area is smallest in the channel, we will have M = 1. This is a very important result. In the throat of the rocket nozzle, we therefore always have M = 1, given that the exit flow is supersonic. If the exit flow is subsonic, the Mach number at the throat will be M < 1. When Mach number at the throat is sonic (i.e. M = 1), the flow through the nozzle is called “choked”. This paragraph basically summarizes why the rocket nozzle looks the way it looks. To have the exhaust velocity as high as possible, we first decrease the area of the nozzle up to M=1, and subsequently increase the area until M = 
Ideal
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Gas Law
Let’s now look at how thermodynamic quantities like pressure, temperature and density are related in a general flow. The ideal gas law gives the relation for ideal gases (such as air):
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Note that in equation (1), the R is the universal gas constant, whereas in equation (2) R is the gas constant for air. This equation says that, for example, if we heat up (i.e. increase the temperature T) a gas at constant pressure p, the density will decrease to keep the pressure constant. The quantities like pressure, temperature and density in the ideal gas law are called static quantities. These are the quantities you would go measure when you would go STAND outside with a thermometer or barometer.
Total
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Conditions
In the previous discussion, the word “stand” is highlighted. The question that we can ask is: would our pressure reading change if we stood outside compared to if we would drive our car on the highway and stick the barometer out of the window?
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In conclusion, we have to distinguish between two types of thermodynamic quantities, namely static and total quantities. Where the static quantity (e.g. pressure) does not include the Mach effect, the total quantity measures the pressure as if we would slow the flow down to M=0 (imagine decelerating the flow on our hand out the window).
Conservation
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Equations
There are 3 conservation equations that hold in any physical process:
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Total temperature is constant in any process where you don’t add heat. This relates to the conservation of energy. This means that throughout the rocket nozzle (after the combustion chamber) the total temperature will remain constant.
Total pressure is constant in any lossless process (i.e. when we don’t take friction into account). This is usually a good assumption for high-speed flows. This means that throughout the rocket nozzle (after the combustion chamber) the total pressure will remain constant.
Isentropic
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Relations
Note there are some intimidating equations ahead, but don’t fret! Knowing the derivations of these is not important for us right now. What’s really important is to understand how these quantities change as you increase/decrease different variables in the equation.
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In these equations, 
The
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Mass Flow Equation in Compressible Flows
The first equation in this LSET (i.e. the mass flow equation) is not very useful in compressible flows, since it involves velocity. A better way to express the mass flow equation is to use total quantities and Mach numbers. Rewriting the first equation using the ideal gas law, and the isentropic relations for temperature and pressure, results in:
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On the left-hand side, we see the mass flow rate of the gas and total quantities. All these quantities are set by the combustion chamber and are assumed to be constant. On the right-hand side, we see the area and the Mach number. This equation will thus allow us to calculate the Mach number at each location in the nozzle, given an area. From a design perspective, it allows us to shape the nozzle area if we want a specific Mach number distribution throughout the nozzle.
Thermodynamic
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Quantities Throughout a Rocket Nozzle
We already discussed that the thermodynamic quantities that are obtained when the flow is brought to a standstill are called ‘total’ or ‘stagnation’ quantities/conditions/states. From the conservation equations, we saw that these conditions, namely total temperature and total pressure are constant in the rocket nozzle. However, we did not talk about what happens with the static quantities throughout the nozzle.
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There is one special point in our nozzle, and that is the throat. From our previous discussion, we know that this is the point where, if the conditions are right, the flow will switch from being subsonic to supersonic. At this point, where the cross-sectional area is the smallest, M=1, or the flow velocity is exactly the speed of sound.
Coefficient of
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Thrust
After having designed the rocket nozzle shape, we must have a measure of defining its performance. In other words, how good is our rocket nozzle at expanding the combustion products, starting at the combustion chamber until the ambient? The thrust coefficient Cf is used as such measure:
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More on this in the next topic.
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