Here is a video of the assembled rocket motor being tested. These tests were conducted late in the fall of 2014. I have since machined new fuel grains and am beginning to consider the design of a test stand for the motor. Unfortunately, the video became corrupted in the post-processing stage, and is quite shaky. I have acquired a new camera, and will be re-running these tests shortly. I will be sure to continue posting these videos.
I recently completed my first few propellant burn tests. I conducted these tests with the following goals:
- Verify that the rocket engine can be started simply
- Test the screw valve used to control the oxygen flow rate for ease of use and reliability
- Ensure that there are no leaks in the oxygen delivery system
Here is a sample of some of the tests I ran:
Overall, I am very happy with the results of the test. I was able to start the engine with no more than a lighter and there were no discernible leaks in the oxygen lines. After running ten burn tests, I have decided that the screw valve on the regulator is not an ideal way to regulate the oxygen flow. I will be looking into putting a ball valve into the oxygen lines so that I can shut off the oxygen flow more quickly in the case that there is a need to do so.
As soon as I finish the machining, I will run some tests with the fuel grain mounted in the frame and with the nozzle, so stay tuned!
The nozzle will have a conical profile due to its proven performance and ease of manufacture. The throat was designed with a diameter of 0.1875 inches based on the size limitations placed on the nozzle by the structural components. Following Rocket Propulsion Elements’ guideline, the exit diameter of the divergent nozzle is five times the throat diameter, 0.9375 inches. The half-angle, α, of the divergent nozzle is 12°, while the convergent nozzle has a half-angle of 45°. The convergent half-angle is not as important as the divergent nozzle’s, since almost any convergent nozzle can allow flow to reach sonic pressure at the throat if there is enough pressure in the combustion chamber. An angle of 12° was chosen for the diverging side to give as much axial flow as possible, even though it comes at the cost of more weight due to extra length. The optimization of thrust vs. weight was not considered for this nozzle, since the prototype will not see flight and excess weight is not an issue for static testing.
HRE I’s nozzle has been designed to accept a perfluoroelastomer (Simriz ®) O-ring with excellent thermal and chemical stability to keep the combustion chamber pressure as high as possible. The nozzle will be held in place by sandwiching it between a structural end plate and the fuel grain. I will be machining the nozzle from AISI 1018 low-carbon, cold rolled steel. This steel is easy to machine, cost effective, and has a very high melting point. It is unlikely that the throat will start to degrade and break down from thermal stress if it is machined with a good finish, but if it does I will look at the option of inserting a graphite throat-piece.
With a small throat diameter, HRE I’s fuel grain will be able to have a fuel port large enough to allow for the chamber flow velocity to be neglected. If the ratio of fuel port cross-section to throat area is less than four, the effects of chamber velocity on the chamber pressure can no longer be neglected. A small ratio means that gases will accelerate themselves down the fuel port. The energy used to accelerate these gases comes at the price of chamber pressure, which is one of the driving variables in the thrust equation. This design avoids performance losses due to chamber velocity effects.
Using SolidWorks, I conducted a FloXpress analysis of the nozzle. The inlet and outlet conditions used were as follows:
- Inlet (Combustion Chamber): Air, 80 psi, 350°F
- Outlet (Atmosphere): 14.7 psi
The analysis produced the following flow visualization:
The estimated average flow exit velocity is 11000 inches/second, while the highest flow velocity achieved is estimated to be 23000 inches/second. The visualization shows no recirculation, which was a recurring issue that had to be overcome as I worked to design the nozzle. The flow appears to separate from the nozzle walls quite early. This analysis does not take into account any thermal effects unrelated to the gas properties, so it assumes a steady-state nozzle temperature and pressure. During testing I will observe the flow and compare it to this analysis to note any similarities and differences.
In this review of nozzle performance calculation, I will not be going through derivations of the formulae or discussing their origins. An intermediate level understanding of thermodynamics and general physics is assumed, and the formulae are presented are those that apply to simple conical nozzles. I have made the decision to use a conical profile nozzle for HRE I based on ease of manufacture, simplicity, and suitability to small rocket engines. All the following formulae come from the third chapter of Rocket Propulsion Elements by Sutton and Biblarz.
Nozzle exit velocity for ideal and non-ideal rockets is described by the equation:
From this relation we can identify the relationship between exhaust velocity, the pressure differential, and working-fluid properties. The thrust produced by a nozzle is made up of two components which add to each other; pressure thrust and momentum thrust. Pressure thrust is the additional thrust gained due to the pressure difference between the nozzle exit plane and the atmospheric pressure. The following equation for thrust illustrates the effect of changing altitude on the amount of thrust produced by the rocket motor. For a simple and easily manufactured conical nozzle, a correction factor λ is multiplied onto the momentum flux term to adjust the ideal parameter for the amount of non-axial flow. The nozzle half-angle variable is α.
This correction factor is always less than 1 for values of α greater than 0, since any deviation from perfectly axial flow will yield less thrust in the axial direction.
The specific impulse of the rocket motor is then:
Design charts of thrust coefficient versus nozzle area ratio at a specified k are available, such as those available on TU Delft’s Aerospace Engineering website.
Other Performance Correction Factors
Other correction factors that should be applied when attempting to precisely determine expected thrust include:
- velocity correction factor (ζv) – the square root of the energy conversion efficiency
- discharge correction factor (ζd) – the ratio of actual mass flow rate to ideal mass flow rate
These two correction factors combine to form the thrust correction factor:
Using this thrust correction factor, we are arrive at the predicted thrust value:
Table of Symbols (click to enlarge)
Nozzle Flow Theory
This write-up on nozzle flow theory is based off of two main sources. The primary source is my notes from a fluid dynamics class on incompressible flows I recently took under Professor Gwynn Elfring at The University of British Columbia, while the secondary source is Rocket Propulsion Elements by George P. Sutton and Oscar Biblarz. Information from these two sources will be mixed and matched throughout this write-up, but I will do my best to distinguish between the two when appropriate by using the superscripts 1 and 2 to denote Dr. Elfring and Sutton & Biblarz’s works, respectively.
Thrust is produced when the momentum-flux generated by gases leaving the rocket engine creates a reactive force on the rocket itself, driving the rocket in the opposite direction. The use of a properly designed nozzle increases the momentum-flux of the gases, leading to increased thrust. A converging-diverging nozzle converts a large amount of thermal energy into kinetic energy through an isentropic, fully reversible flow process. 2 When a converging nozzle is used to direct pressure-driven flow from a high pressure environment to a low pressure environment, the flow accelerates through the nozzle. This results in a reduced pressure in the nozzle. The pressure in the nozzle can never be less than the atmospheric pressure and can also never drop below the sonic pressure, p*, since the speed of the flow through a converging nozzle is never greater than Mach 1. 1 Two rules for the exit pressure of a converging nozzle can be derived from these facts:
- If the sonic pressure is less than the atmospheric pressure, then the exit pressure is equal to the atmospheric pressure1
- If the pressure at the nozzle throat is equal to the sonic pressure (sonic pressure is always achieved at the narrowest part of the nozzle), then the exit pressure will be the sonic pressure1
If condition two is met an overpressure condition will be present at the nozzle exit, which causes the flow to expand as it exits the nozzle throat.1
Converging-diverging nozzles are able to further accelerate the flow coming out of the converging portion of the nozzle. This is only possible, however, if the overpressure condition is present at the exit of the converging nozzle (the throat). If the flow through the throat of the nozzle is not sonic, then the exit pressure will be the atmospheric pressure, similar to the convergent nozzle. If the pressure difference between the combustion chamber and atmospheric pressure is large enough, sonic flow will be achieved in the throat of the nozzle. As the flow exits the throat into the divergent nozzle it continues to accelerate, causing the pressure in the divergent nozzle to drop below the sonic pressure. This pressure can continue to drop until it is below the atmospheric pressure. 1 As the flow exits the nozzle at sub-atmospheric pressure, it experiences a shock transition as it is forced to compress back to atmospheric pressure (over-expanded flow). Not enough of a pressure differential results in the aforementioned type of shock, while too large of a pressure differential gives a supersonic choked flow. Choked flow occurs when the exit pressure is higher than the atmospheric pressure, resulting in shocks caused by the expansion of the nozzle flow into the atmosphere (under-expanded flow). In design of nozzles, it is desirable to have the exit pressure be as close to the atmospheric pressure as possible to prevent an over or under-expanded flow.1, 2
Any surface within the nozzle which blocks the flow path will locally cause the kinetic energy in the flow to be converted back into thermal energy nearly equal to the stagnation temperature and pressure.2 The massive amount of kinetic energy present in the flow translates into enough thermal energy to destroy the nozzle at stagnation conditions. For this reason, the inner surfaces of the nozzle must be free from protrusions and have a high quality finish.
Calculating Rocket Performance
Calculation of ideal rocket performance is possible under the following list of assumptions found in Rocket Propulsion Elements:
- The working substance is homogeneous
- The working fluid is completely gaseous
- The working fluid behaves like an ideal gas
- The flow is adiabatic, ie: there is no heat transfer through the rocket walls
- Boundary layer effects are neglected
- There are no shock waves or discontinuities in the flow
- Propellant flow is steady and constant
- Exhaust gases leaving the rocket have velocity only in the axial direction
- The working fluid velocity, density, pressure, and temperature are all uniform on all planes normal to the longitudinal axis
- The combustion reaction is steady state and transient phases are neglected
- Stored propellants are at room temperature and cryogenic propellants are at their boiling points
With these assumptions, idealized theoretical values for motor performance can be calculated. Adapting these values to reflect real performance parameters is then a matter of applying appropriate correction factors. Performance parameters of real rocket motors usually only vary between 1-6% from ideal parameters, thus the application of correction factors to ideal values has become an accepted method of rocket performance analysis.2
In the next few days I will be posting a summary of nozzle flow and thermodynamic calculations outlining the steps I am taking to design the first nozzle that will be used to run fuel performance tests.
Bear with me as I am currently putting together a really large post on nozzle flow and theory. I have recently begun work on designing HRE I‘s nozzle, and have become a bit bogged down by the large amount of theory. Right now I plan on fabricating a non-optimized convergent-divergent conical nozzle which will allow me to develop a proof-of-concept prototype and conduct some propellant testing. Following this I will be refining the nozzle design and working to achieve more thrust and efficiency.
Poly Methyl Methacrylate
A common material used by hobbyists as a fuel grain in hybrid rocket engines is poly methyl methacrylate, or PMMA. Widely known by trademark names such as Plexiglas, Lucite, and Perspex, PMMA is a common thermoplastic and is easily found in sheet form. I recently purchased an 8 foot long cast PMMA rod for use as the fuel grain in the first few tests of HRE I. Notwithstanding a few chemistry and materials engineering classes, my background in polymers is rather lacking. This drove me to do some investigation into PMMA and its behavior during combustion.
As its name suggests, PMMA is the polymer form of methyl methacrylate. Each monomer is comprised of an α-methyl, methyl-ester, and methylene bridge (methanediyl group). The monomers that make up PMMA are linked through addition, meaning that a hydrogen atom must be removed from each monomer for a polymer chain to be formed. The two carbons then bond, creating a new methylene bridge. The Chem Polymer Project lists several chemicals which react with PMMA, including: ethyl and methyl alcohols, chlorinated hydrocarbons, esters, ketones, organic acids, aromatic solvents, and lacquer thinners.
Methyl Methacrylate Monomer
Poly Methyl Methacrylate
If one were to assume a one-step finite-rate forward reaction, combustion of PMMA would take place as follows:
However, when solid phase polymers are exposed to high temperatures, degradation reactions (following several reaction pathways) begin to occur which release combustible gaseous products. The mixture of these degradation products with an oxidizer makes for a highly combustible combination of gases. A paper by the National Institute of Standards and Technology states that PMMA undergoes a reversal of the polymerization reaction when it breaks down under thermal energy. This produces a large yield of methyl methacrylate monomers. The NIST also indicates that abnormal structures within PMMA (such as unsaturated end groups or head-to-head linkage within the chain) cause a decrease in thermal stability and ignition time, as well as an increase in burn rate. These abnormal structures are a common byproduct of the polymerization process of PMMA. As a thermoplastic, PMMA becomes pliable near its melting point, 140°C, and can be allowed to cool back to its original form. Thermoplastics are able to do this since no irreversible chemical reactions occur while it sets.The ignition temperature of PMMA in the presence of open flame is around 290°C.
Chem Polymer Project: http://chempolymerproject.wikispaces.com/Plexiglass-E-MgEb
Image 1 and Pichai Rusmee’s University of Utah Dept. of Mech Eng. Lab Notes: http://www.mech.utah.edu/~rusmeeha/labNotes/degradation.html
Paper by the National Institute of Standards and Technology: http://fire.nist.gov/bfrlpubs/fire95/PDF/f95104.pdf