The Graben:. Nozzle

Convergent Divergent Nozzle:

The convergent divergent nozzle (CDN) is located at the aft end of the combustion chamber. The purpose the CDN is to control the expansion of the exhaust products in such a way that it efficiently converts the energy produced in the combustion chamber into thrust. Through empirical testing, it has been found that 65% to 75% of the total thrust is developed by accelerating the exhaust products to sonic velocities at the nozzle throat. The rest of the thrust is from the nozzle expansion cone. The CDN must be designed to smoothly accelerate the exhaust products to produce the desired thrust. Since this will be the only source of thrust, a considerable amount of time will be spent on the research and design of the CDN for this rocket configuration.

Nozzle Configuration:

The two primary CDN configurations used in modern rockets are known as submerged and external. For the submerged configuration the nozzle entry, throat, and part or all of the exit is found within the combustion chamber. The external configuration is a classic de Laval CDN that is mounted at the aft end and is entirely external to the combustion chamber. Due to the complexity of the submerged configuration, the CDN of choice will obviously be the external design.

Important sections of a typical CDN.

Aerodynamics:

The CDN will consist of three primary sections the convergent, the throat, and the divergent sections. The convergent section will accelerate the subsonic flow from the combustion chamber while decreasing the gas pressure, thus increasing the kinetic energy of the gas and transferring it to the launch vehicle. The throat section will support the transonic flow; ideally the throat velocity will be the speed of sound (mach 1, 1,087 ft/sec). The divergent section will then be used to further accelerate the supersonic flow and bring the gas pressure to match the outside pressure. If the exit pressure does not match the outside pressure, then the CDN will either be under expanded and create expansion waves for a lower outside pressure or oblique shock waves for a higher outside pressure. A good design will smoothly accelerate the exhaust gas throughout the nozzle, thus no shocks will exist in the CDN. The graphic below shows the pressure velocity relation of a ideal CDN, ultimately this is what we want to achieve.

Generic pressure velocity profiles for de Laval nozzles.

CFD velocity distribution of Stock CDN system.

CFD pressure distribution of Stock CDN system.

The Nozzle Entrance Design Considerations:

For the external configuration, the contraction ratio at the entrance is typically determined by the chamber design. The inlet half angle is usually between 30 and 60 degrees with most designs near 45 degrees. The steeper the angle the greater the chance of erosion occurring particularly at pressures greater than or equal to 1500 PSI. The entrance geometry should be designed such that the length and erosion rate are minimized. If erosion control is the primary concern, then half angles of less then 30 degrees should be employed. If length is a concern, then a compromise will have to be found.

Throat Design Considerations:

The throat is the region of the nozzle with the smallest cross sectional area. The throat geometry should provide a smooth transition from subsonic to supersonic flow. The design should minimize the rate of potential erosion while maintaining the desired CDN performance, ease of fabrication and nozzle alignment. The throat can be a relatively long and cylindrical throat or it can be very short and nearly circular. Typically the cross sections of the up and downstream sections should be no less than 0.5 the throat radius, larger radii can be used to provide smoother transitions when feasible. Although a circular throat section will work just fine in short nozzles, there are some advantages to using a cylindrical throat such as:

Divergent Exit Design Considerations:

The exit configuration should be designed to maximize performance while staying within the constraints of the CDN. There are two different exit configurations used in the CDNs, these are contoured and conical configurations. The contoured exit configuration is used because it will tend to turn the flow such that the exhaust products exit nearly axially, thus reducing divergence losses. The contoured nozzle exit will tend to be shorter than conical exits that deliver the same thrust and will tend to improve the specific impulse, but will not greatly affect the overall performance of the CDN… increases of only 0.5 – 1.0 percent have been estimated. Taking this into account and adding in the complexity of machining the contoured CDN, the final design will most likely be a conical configuration. For this low altitude design, empirical test have shown that expansion ratios of 7 to 10 are ideal and the difference between the conical and contoured exit configuration is negligible.

Conical exits should have a half angle that is optimized for the design altitude and throat-to-exit length. The half angle typically will fall within a range of 6 to 28 degrees, most designs are 15 to 17.5 degrees. Divergence loss in a conical section is estimated as:

To minimize the divergence loss a small half angle should be used, but the smaller the angle the longer the divergent section will have to be. For a given throat-to-exit length, ambient pressure, and chamber pressure, the half angle that maximizes the thrust coefficient (CFdel) is estimated as:

Empirical tests have produced a performance maps similar to the one below which can be used in the initial design stage.

Delivered thrust coefficient of a conical nozzle
as a function of expansion ratio, exit half
angle, and exit length normalized on throat radius.
Form NASA SP-8115.

The contoured exit is used only when performance maximization is critical to the design. In this case the 0.5 to 1.0 percent improvement in specific impulse will be paramount. Another advantage will be realized in the length of the diverging section, a contoured section will be shorter than the conical section that delivers the same specific impulse. The material cost and weight will be minimized, but the potential for erosion will be greater in contoured sections. Typically the initial divergence angle should be limited to a maximum of 32 degrees with the difference between the initial and exit angles being less then 12 degrees. The most common internal divergence angle is between 20 and 26 degrees, if the difference is greater than 12 degrees the losses in the delivered specific impulse will increase proportionally.

The Physics of the CDN:

Analysis:

The analysis of the CDN is possible through experimentation and theoretical modeling. The experimental evaluation will be discussed later when we have experimental data. So, to evaluate the initial design theoretically we must make specific assumptions to simplify the computational fluid dynamics. The primary assumptions are:

Since the flow is assumed to be one-dimensional, we can utilize the continuity equation, conservation of momentum, the first and second laws of thermodynamics , and the equation of state to analyze the flow and environment throughout the CDN as a function of the cross sectional area. To reduce the amount of mathematical knowledge required the derivations will be skipped and only the final formulae will be presented with explanations. To fully appreciate the mathematics, it would be good to spend the time deriving and studying the equations.

The Basics:

The three basic equations of motion are the continuity, momentum, and energy equations. To define these for one-dimensional steady state flow, a small control volume must be defined as shown below and the equations are simplified accordingly.

Taking advantage of the assumptions, the only forces acting on the control surface are pressure forces. The momentum equation can then be defined as:

Again using the assumptions, if there is no external heat transfer and no work, the energy equation becomes:

If we combine the continuity and momentum equations we can derive a formula that describes the flow through the CDN as a function of the change in area. This shows the influence of sonic velocities on the flow properties.

The formula reveals that at subsonic velocities, a decrease in area will result in an increase in pressure and velocity while an increase in area results in a decrease in pressure and velocity. For velocities greater than mach 1, a decrease in area will result in a decrease in pressure and velocity while an increase in area results in an increase in velocity and pressure. This relation indicates that the subsonic flow entering thin CDN from the combustion chamber can only be accelerated to sonic velocities in the convergent section, thus allowing for further acceleration in the divergent section.

Stagnation Properties:

The stagnation properties are used to define a reference state for compressible flow. Stagnation properties are attained by bringing the flow at a given point adiabatically to rest. Now we can relate the conservation of energy to the stagnation pressure and temperature at any point within the CDN. To find this relation we need to look at the change in enthalpy. Enthalpy is the sum of the internal energy and the product of pressure and volume, enthalpy can be thought of as the thermodynamic potential of the system. The change in enthalpy is defined as:

After some substitution and simplification, the stagnation temperature as a function of mach number is:

If we assume that the flow in the combustion chamber is zero or negligible, then the temperature and pressure within the combustion chamber are the stagnation temperature and stagnation pressure. As the gas is accelerated through the CDN, the static temperature and pressure will tend to decrease. If the flow is adiabatic and reversible, then the stagnation temperature, pressure, density, and enthalpy at any cross section will equal the temperature, pressure, density, and enthalpy in the combustion chamber. Taking advantage of this situation, we can use the above equations to determine the pressure and temperature at any cross section assuming the mach number and stagnation properties are known.

Mass Flow Rate:

The mass flow rate, in general, is defined as:

Substituting in the pressure and temperature equations, the mass flow rate related to mach number is:

The mass flow rate through the CDN must be constant, so we can now solve for the critical cross sectional area where the velocity is mach 1, and determine the ratio of areas.

This ratio will have two possible isotropic solutions, one for subsonic and one for supersonic velocities. When we graph the ratio of areas versus mach number, we get a cross sectional trace of the ideal CDN required to accelerate the gas to sonic velocities at the throat.

A graph of the area ratio versus mach number.

Internal mach number and pressure relation as a
function of position in the SS-design CDN.

Another useful equation that we can directly measure and relate to the theoretical equations above is the exhaust velocity. The equation for exhaust velocity is obtained through the combination of the conservation of mass equation and the energy equation. After some substitutions and simplifications the exhaust velocity becomes the function of the pressure ratio, specific heat, and chamber temperature and is defined as:

If we were to derive this equation then there would be a term in the square root for the gas approach velocity. In most cases the combustion chamber cross sectional area is much greater than the throat cross sectional area, so the approach velocity is relatively small and can be considered negligible. The chamber temperature can be measured directly, but if that is not possible then the stagnation temperature can be substituted instead. According to this equation it should be easy to see that an increase in the temperature or a decrease in the molecular mass of the propellants will increase the exhaust velocity and in turn increase the specific impulse. We could also adjust the pressure ratio and specific heats, but the contribution will be significantly less than a change in temperature and molecular mass.

Thrust:

The thrust is a performance parameter that corresponds to the force imparted to the rocket through the CDN. The thrust is produced by the CDN smoothly converting the thermal energy in the combustion chamber into kinetic energy by accelerating and expanding the exhaust gas. The thrust can be estimated as:

From this equation we can see that for a given CDN and mass flow rate the thrust will vary with the altitude of the rocket. As the altitude increases, the thrust will increase assuming no shocks develop in the CDN. The maximum amount of thrust will only occur when the ambient pressure is zero. For an ideal rocket, the thrust equation can be expanded to:

From this equation we can see that if we design the CDN such that the exit pressure is equal to the ambient pressure, the thrust of the ideal rocket will be directly proportional to the throat area and combustion chamber pressure. Another performance parameter commonly used related to the thrust is the thrust coefficient. The thrust coefficient represents how much the thrust is amplified by the CDN as compared to the thrust produced if the combustion chamber pressure acted over the throat area only. It is defined as the thrust divided by the combustion chamber pressure and throat area and can be calculated as:

Just as with the thrust equation for the ideal rocket, the thrust coefficient peaks when the exit pressure equals the ambient pressure, this is the optimum thrust coefficient. Using the thrust coefficient allows us to simplify the ideal thrust equation to:

This simplification becomes extremely usefully when evaluating the CDN experimentally because the throat area, combustion chamber pressure, and thrust can all be easily measured. For a given CDN the throat area will be fixed, so the only parameter that can be altered will be the chamber pressure. By modifying the chamber pressure and measuring the thrust, we can calculate and plot the thrust coefficient curve to find where it peaks. It should be noted that even though the thrust coefficient is a function of combustion chamber pressure, it is not directly proportional to the chamber pressure. It is directly proportional to the throat area, if the throat area can be easily modified. Typical values of the thrust coefficient range from about 0.8 to 1.9 and not only can it be used to optimize the system, but it can also be used as a correction factor for the rocket at a given altitude.

CDN Expansion:

When designing a CDN, the goal is to design it in such a way as to accelerate and expand the working fluid isentropically at some target ambient pressure. In a properly designed CDN the fluid will be accelerated to the speed of sound at the throat. In the divergent section of the nozzle the fluid will expand and continue to accelerate, this results in a drop in pressure. So, a well designed nozzle will drop the pressure to equal the target ambient pressure at the exit of the CDN. Since the ambient pressure will change with altitude, there will be a time when the nozzle becomes under or over expanded. When the CDN becomes under or over expanded the thrust will suffer some penalties, so these conditions should be explored. Below is a graph showing the distribution of pressure through a CDN for various flow conditions.

The distribution of pressure through a CDN for various flow conditions.

Under-expanded nozzles discharge the working fluid at an exit pressure greater that the external pressure. Typically this is the result of the exit area being too small. Over-expanded nozzles have a lower exit pressure than the external pressure. Typically this is the result of to exit area being too large.

When a nozzle is under-expanded, curve 10, it will flow full and have external expansion waves at the nozzle exit. Since the expansion of the gas is incomplete, the thrust coefficient and specific impulse will be less than ideal.

When a nozzle is slightly over-expanded, curve 8, it will flow full until the exit pressure reaches between 25% and 40% of the external pressure, curve 7. Again, the expansion will be inefficient and external expansion waves will be observed. The thrust coefficient and specific impulse will again be lower than ideal.

When the nozzle is over-expanded flow, curves 5 – 7, separation will occur in the divergent portion of the nozzle. This discontinuity will be axially symmetric and will cause the diameter of the supersonic jet to be less than the nozzle exit diameter. This means that at the nozzle exit, the flow will be supersonic surrounded by an annular section of subsonic flow. As the external pressure decreases with altitude, the shock wave will travel downstream in the divergent section of the nozzle. Internal shock should be avoided when possible because they will reduce the thrust and introduce stresses in the nozzle. Although the internal shocks can’t be directly observed, external shock waves will be observed in the plume.

When the exit pressure is nearly the same as the chamber pressure, subsonic flow will exist throughout the entire nozzle, curves 1 - 4. When the pressures are equal, there will be no flow at all. These conditions are typically found for short periods of time during starting and stopping. Even though the flow is subsonic, non-axially symmetric local separations may occur within the nozzle. There can cause large momentary side forces on the nozzle during starting and stopping transients.

Influences of Chamber Geometry:

There are an infinite number of possible chamber geometries. Typical geometries are cylindrical with various lengths. One thing to consider when designing the combustion chamber is that the gas in the chamber heats up and accelerates toward the nozzle entrance the expanding gas will cause a pressure drop resulting in some energy loss. This pressure drop will result in lower pressures at the nozzle entrance, thus lowering the thrust and specific impulse. The pressure drop and energy loss will tend to increase with the length of the chamber, so whenever possible the shorter the chamber the better in this respect. The gas acceleration typically is adiabatic but not isentropic in the chamber, and the losses are maximized when the chamber diameter and nozzle diameter are equal. So, it is best not to use long narrow combustion chambers whenever possible. In relation to nozzle design, when the combustion chamber cross sectional area is 4 times larger than the throat area, the pressure drop and energy loss will be greatly reduced.

Real Nozzles:

Typically, when a nozzle is designed everything is assumed to be essentially ideal. Unfortunately, hardly anything in the real world is ideal. So, below is a list of the differences that has been observed when testing and comparing real nozzles and ideal nozzles. Some of these can be incorporated into the design and others should just be noted for future reference.

In most cases an empirically determined correction factor can be used to modify the design algorithms.

Divergence losses occur in the divergent section of the nozzle and vary as a function of the cosine of the divergent angle. The losses can be reduced by using bell-shaped nozzle contours.

A low nozzle contraction ratio result in pressure losses in the combustion chamber and thus reduces the exit velocity and thrust.

There are indeed boundary layer effects. Wall friction can reduce the exhaust velocity by 0.5% – 1.5%.

Incomplete combustion will introduce solid particles or liquid droplets into the exhaust gas, thus introducing losses up to 5%.

Particles introduced into the exhaust gas will cause the gradual erosion of the throat. This will reduce the combustion chamber pressure and cause small losses of about 0.7% in the specific impulse.

Unsteady combustion will cause an oscillating flow, thus resulting in losses.

Chemical reactions may occur within the nozzle. This will change the gas properties and can introduce losses, typically around 0.5%.

During pressure transients during starting and stopping, the nozzle will not be operating under the conditions it was designed for, thus this will introduce small losses.

Non-uniform gas composition due to incomplete mixing, incomplete combustion, and turbulence will introduce losses.

Gas properties are typically not ideal, thus the differences may be up to 0.2% - 0.7%.

Typically the nozzle will be designed to operate at a specific altitude or ambient pressure. If no altitude compensation is used then losses of up to 15% may be measured at other altitudes.

Other Design Considerations:

Nozzle to Chamber Attachments:

The four common methods are the bolted joint, the threaded attachment, the snap joint, and the key/ lock wire joint. Below is a table of some of the pros and cons for each joint.

Thermal Design:

Using the equations above, the theoretical temperature at any cross section can be calculated. The nozzle walls will obviously have to withstand relatively high temperatures, high gas velocities, chemical erosion, and high stresses. The material chosen must have a high heat transfer rate and must be strong enough to withstand the pressures at operating temperatures.

The maximum temperature in the CDN will indicate the need for a cooling system if the CDN material can not support the thermodynamic loads. The typical cooling system is a known as a cooling jacket may be used. The cooling jacket fits around the outside on the combustion chamber and nozzle. The jacket will have a system of channels for some type of coolant to flow through and the maximum temperature during operation will dictate the number of channels and flow rate required. If a cooling jacket is required some guidelines to aid in the design are listed below:

To limit the temperature to acceptable levels, another option is to use a thermal liner and or throat insert. A thermal liner is a material that forms the aerodynamic contour of the nozzle and is exposed directly to the exhaust products. A thermal insulator is placed behind the liner to protect the structural case from high thermodynamic loads. The throat insert is a erosion resistant liner places in the throat of the CDN. These are typically used in large commercial rocket nozzles, but can easily be utilized in amateur rockets. The six groups of materials used for throat inserts and thermal liners are:

Although most of these materials and methods of application tend to be very expensive, many amateur rocket engines use graphite CDNs because of its thermal properties.

Final Nozzle:

Below are images of the final CDN and the mounting flange. The CDN and flange were machined from ANSI 1018 Steel. This material was used primarily because of its machinability and availability. The mounting flange is attached to the combustion chamber using a bolt joint to allow for an accurate alignment. In between the mounting flange and combustion chamber is a high temperature Garlock gasket (see the combustion chamber page) to ensure a snug fit and to prevent leakage through the bolt holes. One side of the CDN is threaded to allow for an easy way to change out nozzles, if another geometry is required. The flange is threaded such that the nozzle sits flush against the combustion chamber.