Research Status Update

The four paraffins to be used in initial testing have arrived, meaning that the initial experimentation phases can begin. General melting points are known, however, the research requires more exact values for a numerical comparison of regression rate data. I have begun collecting melting point data on the paraffins by two distinct methods.

The first is a direct method of slow heating in a MEL-TEMP apparatus. Here, a small sample of paraffin is carefully forced to the bottom of a capillary tube and placed into a slot in a voltage-controlled heating block. A thermometer is used to monitor the internal block temperature, which is assumed equal to the instantaneous temperature of the paraffin sample given sufficiently slow heating allowing for diffusion and reduced thermal gradients. A viewing lens is used to monitor the phase of the sample. The temperature at which the sample begins to melt and is fully melted are recorded as a range on the order of single degrees. The actual melting point lies on this range, making the average a good estimator.

The second method is the determination of melting point based on a heating curve. A thermal probe is used to plot the temperature of a sample of liquid paraffin as it cools. Provided the assumption of high rates of heat transfer between solid and liquid phases leading to low thermal gradients within the sample, temperature will remain constant during the phases change process. It is critical that the solid phase undergo minimal cooling in the presence of the liquid which heats the solid. The interval is found in which the graph of temperature is most level, indicating the change and the temperature at which it occurs. Based on a preliminary trial, shown bellow, the assumptions needed for accurate measurement may not be met due to the low conductivity of the paraffin. The rounding at the bounds of level intervals is evidence of thermal gradients present within the sample, meaning that the propagation of the solid phase was uneven within the sample. This phenomenon was observed during the trial. The downward slopes of intervals considered "level" indicates that heat was lost from the solid at a greater rate than the liquid could supply it, also indicating higher thermal gradients.

Cooling Curve for n-Paraffins A and B

Cooling Curve for n-Paraffins C and D

Note: The two "level" portions of the red temperature plot for n-paraffin "D" may indicate a bimodal distribution of carbon chain lengths in the sample. All other curves appear to be well behaved. Rounding at the boundaries of "level" intervals may also be the result of a distribution of carbon chain lengths about a mean.

I will be posting all data collected to date. I am presently designing the primary testing apparatus including the hybrid rocket motor device, nitrous oxide feed system, and test stand. The injector will be ordered first along with a male coupler for the nitrous tubing and the aluminum blank for the forward closure. Based on the dimensions and threading of the injector and coupler, the injector face and forward closure will be machined. Also, I will be purchasing and calibrating the two pressure transducers to be used in the experiment. This will require its own apparatus consisting of a variable pressure vessel in which the pressure data is known and compared to voltages across the transducers.

ACSI Update: 2D Grids for Gas Dynamics Solver

Based on my extensive literature review, I have chosen a finite volume method with a structured grid for solving the equations of gas dynamics within a hybrid rocket motor. I will be using the Monotone Upstream-centered Scheme for Conservation Laws (MUSCL) in conjunction with AUSMPW+ approximate Riemann flux solver at the cell faces. These methods have demonstrated an enhanced capacity for capturing the oblique shocks and discontinuities present within a typical rocket motor and supersonic nozzle. Recently I have been investigating these methods in order to gain a more rigorous understanding of their function. You can download the documents I have read on my "Research Documents (CFD)" page under the heading "MUSCL Reconstructed AUSM Flux Splitting Scheme." So far, I have constructed a computational grid for an arbitrary radius function since the motors to be simulated are axisymmetric. The following diagram presents the theory behind the grid's construction. The code is also provided.

      PROGRAM NavierStokesSolver2D

      IMPLICIT NONE

      DOUBLE PRECISION Q(5, 4, 101, 101, 1001), n(4, 2, 101, 101),

     .L(4, 101, 101), A(101, 101)

     .X(5, 101), Y(5, 101, 101), T(1001)

      DOUBLE PRECISION Xmax, R(3, 101), Tmax, dx, dy(3, 101), dt

     .radius, height

      INTEGER i, j, k, Imax, Jmax, Kmax

     

      !Setting up the grids

      dt = Tmax / (Kmax-1)

      dx = Xmax / Imax

      DO i=1, Imax

          X(1, i) = (i - 1) * dx

          X(2, i) = (i - 1) * dx

          X(3, i) = (i - .5) * dx

          X(4, i) = i * dx

          X(5, i) = i * dx

         

          R(1, i) = radius(X(1, i))

          R(3, i) = radius(X(5, i))

          R(2, i) = (R(1, i) + R(3, i)) / 2

         

          dy(1, i) = 2 * R(1, i) / Jmax

          dy(2, i) = 2 * R(2, i) / Jmax

          dy(3, i) = 2 * R(3, i) / Jmax

         

          DO j=1, Jmax

              Y(1, i, j) = (j - 1) * dy(1, i) + height - R(1, i)

              Y(2, i, j) = j * dy(1, i) + height - R(1, i)

              Y(3, i, j) = (j - .5) * dy(2, i) + height - R(2, i)

              Y(4, i, j) = (j - 1) * dy(3, i) + height - R(3, i)

              Y(5, i, j) = j * dy(3, i) + height - R(3, i)

          END DO

      END DO

     

      !Finding cell areas, side lengths, and normal vectors

      DO i=1, Imax

          DO j=1, Jmax

          A(i, j) = dx * (dy(1, i) + dy(3, i)) / 2

          L(1, i, j) = dy(1, i)

          L(2, i, j) = sqrt((X(2,i) - X(3,i))**2 +

     .          (Y(2,i,j) - Y(3,i,j))**2)

          L(3, i, j) = dy(3, i)

          L(4, i, j) = sqrt((X(1,i) - X(4,i))**2 +

     .          (Y(1,i,j) - Y(4,i,j))**2)

             n(1, 1, i, j) = -1

             n(1, 2, i, j) = 0

             n(2, 1, i, j) = (Y(3,i,j) - Y(2,i,j)) / L(2, i, j)

             n(2, 2, i, j) = (X(2,i) - X(3,i)) / L(2, i, j)

             n(3, 1, i, j) = 1

             n(3, 2, i, j) = 0

             n(4, 1, i, j) = (Y(1,i,j) - Y(4,i,j)) / L(2, i, j)

             n(4, 2, i, j) = (X(4,i) - X(1,i)) / L(4, i, j)

          END DO

      END DO

         

      END PROGRAM NavierStokesSolver2D

   

      FUNCTION radius(x)

          IMPLICIT NONE

          DOUBLE PRECISION radius, x

          radius = 1

          RETURN

      END

     

Reseach Plan

A: Focus/Question: Regression rate behavior of liquefying hybrid rocket fuels with varying melting points that form hydrodynamically unstable melt layers resulting from sheer stresses induced in a combustion port boundary layer. How are the ballistic coefficients in the axially average regression rate equation dependent on the melting points of n-paraffin fuels using nitrous oxide oxidizer?

B: Hypothesis/Goal: If there is a relationship between the melting point of n-paraffins and regression rate, then as melting point decreases an increase in regression rate will be observed at some or all oxidizer flux levels. Ultimately, the burn rate performance of liquefying fuels should be maximized at little or no expense to energetic performance as quantified by density specific impulse or mechanical properties of the fuel.

C: Methods: (Preliminary: These will likely change as the experiment is conducted and refined)

Propellant Grain Casting:

1. Cut enough 3” sections of propellant casting tube to completely fill the intended casing.
2. Mass each casting tube.
3. Tape each casting tube to a casting mandrel assembly.
4. Melt the mass of paraffin in grams corresponding to 4.5 times the length of the motor casing in centimeters.
5. With the aid of a spoon and gloves, pour the molten propellant evenly into the casting assemblies. Avoid trapping bubbles or allowing the paraffin to cool before the casting assembly is filled.
6. Allow the fuel to cool. During cooling, the grains should be centrifuged to ensure good bonding with the liner.
7. Remove the tape from casting assemblies and apply a twisting motion to remove the mandrel from the fuel port cast into each grain.
8. Determine the fuel port diameter, outer diameter of the fuel, fuel grain volume, and mass of the individual fuel grains. Record these data both in a log and in sharpie on the side of each grain.

Thermodynamic Prediction Using RPA Code or CAE

1. Select a range of O/F ratios and possible chamber pressures, dividing each range into several intervals of equal size.
2. The following procedure iterates thermodynamic calculations for each O/F ratio under each possible chamber pressure in order to obtain every possible combination of the interval elements for pressure and O/F.
3. At the given pressure, vary the O/F ratio and record the effect on Isp, C*, Tc, and k.
4. Graph the results to determine the O/F ratio that produces the maximum C* and Isp at the given pressure.
5. Repeat with varying pressures and graph the optimal C* and Isp with respect to chamber pressure. The C* at any given mixture ratio should not be highly dependent on pressure!

Injector Flow Calibration

1. Before each firing, mass the oxidizer tank before and after the filling operation is carried out.
2. During each firing, record the chamber pressure and the nitrous oxide pressure in feed system over time.
3. Numerically integrate the square root of the pressure drop across the injector over the entire duration of the firing. This value is proportional to the total oxidizer mass flow.
4. Determine the constant of proportionality for each firing.
5. Average the values for each firing to arrive at a coefficient relating the time-dependent oxidizer mass flow rate to the instantaneous pressure drop across the injector.

Firing and Data Acquisition

1. Using a razor saw, carefully slice one of the fuel grains to adjust the overall length of the fuel to fit the motor casing.
2. Mass the fuel grains. Measure the total fuel port length.
3. Measure the nozzle throat diameter.
4. Slide the fuel grains into the liner and assemble the fuel section of the motor. This includes the nozzle, retention rings, pre-combustion chamber, post-combustion chamber, and pressure transducer tap.
5. Mass the combustion chamber assembly.
6. Mass the oxidizer feed system tank.
7. Fill the tank completely with liquid nitrous oxide, allowing air to vent out of the top until the liquid reaches the venting port. Close the venting port at this time. Close the fill valve
8. Mass the filled oxidizer feed system tank
9. Secure the electrically actuated solenoid valve and pressure transducer tap upstream to the injector assembly.
10. Connect the oxidizer tank to the solenoid valve.
11. Connect the compressed air feed system to the oxidizer tank.
12. Secure the constructed motor to the test stand.
13. Insert an igniter and pre-heater grain through the nozzle and to the back of the combustion port.
14. Open the compressed air valve and pressurize the oxidizer.
15. Move to a safe distance (100ft) trailing the connections to the two pressure transducers, solenoid valve, and igniter.
16. Connect the transducer leads to the amplifier and computer interface. Start the data acquisition software.
17. Remotely fire the igniter. Once lit, quickly activate the solenoid valve, allowing nitrous oxide to flow.
18. Record the chamber pressure versus time trace for the firing along with the pre-injector oxidizer pressure trace.
19. When oxidizer flow stops, immediately close the solenoid valve.
20. Allow the motor to cool.
21. Mass the combustion chamber assembly.
22. Remove the fuel grains and mass them.

Data Analysis: The averaged characteristic velocity approach will be taken in order to numerically approximate the instantaneous regression rate and internal ballistics during any given trial. Pressure drops across the injector orifice are used to calculate instantaneous nitrous oxide mass flow rates once a proportionality constant has been calibrated by the above procedure. Based on time-averaged characteristic velocity information, nozzle throat cross-sectional area, instantaneous oxidizer mass flow rate data, and a chamber pressure versus time trace, a Runge Kutta-4 time integration scheme is adopted to iteratively determine the internal ballistics over the duration of the trial through application of an augmented regression rate relation. An iteratively converging least squares algorithm is derived in order to approximate the ballistic coefficients in the O/F shifted regression rate equation. The ballistic coefficients are compared across the range of melting points, resulting in a best-fit curve which approximates the relationship between an n-paraffin's melting point and a given ballistic coefficient. The mathematical construct relevant to this approach is detailed in my Tuesday, October 18, 2011 post titled Mathematical Construct of Data Acquisition

Bibliography: (Documents so far pertaining to liquefying hybrid rocket fuels)

 
Karabeyoglu, M. A., Greg Zilliac, Brian J. Cantwell, Shane De Zilwa, and Paul Castelluci. "Scale-Up Tests of High Regression Rate Liquefying Hybrid Rocket Fuels." The American Institute of Aeronautics and Astronautics (2003). Print.

 
Brown, Timothy R., and Michael C. Lydon. "Propulsion." Testing of Paraffin-Based HybridRocket Fuel Using Hydrogen Peroxide Oxidizer. Proc. of 2005 Undergraduate Space Research Symposium and Career Fair, University of Colorado, Boulder Colorado. Colorado Springs: United States Air Force Academy Department of Astronautical Engineering, 2005. 2005 Undergraduate Space Research Symposium and Career Fair. Colorado Space Grant Consortium, 5 Apr. 2005. Web. 16 Sept. 2011. <http://spacegrant.colorado.edu/COSGC_Projects/symposium_archive/2005/docs/115.pdf>.

Arena, Zach, Alexander Athougies, and Alden Rodulfo. Hybrid Rocket Motor. Diss. California Polytechnic State University, 2010. Print.

 
Karabeyoglu, M. A., B. J. Cantwell, and D. Altman. "Development and Testing of Paraffin-Based Hybrid Rocket Fuels." The American Institute of Aeronautics and Astronautics (2001). University of Minnesota. Stanford University, 8-11 July 2001. Web. 4 Oct. 2011. <www.d.umn.edu/~rrosandi/Hybrids/Combustion/AIAA2001-4503.pdf>.

 
Chiaverini, Martin J., Nadir Serin, David K. Johnson, Yeu-Cherng Lu, Kenneth K. Kuo, and Grant A. Risha. "Regression Rate Behavior of Hybrid Rocket Solid Fuels." Journal of Propulsion and Power 16.1 (2000): 125-32. Planete Sciences. Pennsylvania State University. Web. 15 Sept. 2011. <www.planete-sciences.org/espace/basedoc/.../RegressionRateGOx-HTPB.pdf>.

 
Lohner, Kevin, Jonny Dyer, Eric Doran, Zachary Dunn, and Greg Zilliac. "Fuel Regression Rate Characterization Using a Laboratory Scale Nitrous Oxide Hybrid Propulsion System." Kevin Lohner: Selected Publications. Proc. of 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Sacramento, California. Stanford University, 12 July 2006. Web. 16 Sept. 2011. <http://www.stanford.edu/~klohner/Publications/JPC_hybrid_propulsion.pdf>.

Sutton, GP., & Biblarz, O. (2010). Rocket Propulsion Elements. Hoboken, NJ: John Wiley & Sons Inc.

Karabeyoglu, M. A., David Altman, and Brian J. Cantwell. High Regression Rate Hybrid Rocket Propellants. The Board of Trustees of the Leland Stanford Junior University, assignee. Patent US 6684624 B2. 3 Feb. 2004. Print. 

 

Lazarev, Alexander, and Alon Gany. Experimental Investigation of Paraffin-Fueled Hybrid Combustion. Tech. Print. 

 

Karabeyoglu, M. A., Greg Zilliac, Paul Castellucci, Paul Urbanczyk, Jose Stevens, Gohkan Inalhan, and Brian J. Cantwell. "Development of High-Burning-Rate Hybrid-Rocket-Fuel Flight Demonstrators." AIAA/ASME/ASEE Joint Propulsion Conference, Huntsville, AL, July 2003. Print. 

  

Karabeyoglu, M. A., Brian J. Cantwell, and Greg Zilliac. "Development of Scalable Space-Time Averaged Regression Rate Expressions for Hybrid Rockets." AIAA/ASME/ASEE Joint Propulsion Conference, Tucson, AZ, July 2005. Print. 

 

Grosse, Matthias. "Development Work on a Small Experimental Hybrid Rocket." AIAA/ASME/ASEE Joint Propulsion Conference, Seattle, WA, July 1997. Print.

Change In Project's Direction

The acquisition of 5-AT and high-test peroxide proved to be outside the allotted project budget. I will proceed with my alternate project. The new project still centers around the regression rate of liquefying hybrid rocket fuels, so the experimental design, mathematical construct, and much of the procedure will remain the same. However, the independent variable is changed from the concentration of an additive to the melting point of the hydrocarbon fuel. Due to the formation of a hydrodynamically unstable melt layer at the fuel surface, the melting point of the fuel should play a critical role in the rate of regression. The relationship is theorized in the "Development and Testing and Testing of Paraffin-Based Hybrid Rocket Fuels." The melting point of a linear hydrocarbon is decreased as the number of double bonds in the chain increases. Also, longer hydrocarbon chains are responsible for higher melting points. Along with paraffin, several other saturated and unsaturated fats will be tested to achieve a variety of melting points. Sugars will also be investigated. The addition of a -nitro group onto the sugar may increase its energy content and burn rate. If there is time, this will also be investigated. I hypothesize that:

If there is a relationship between melting point and regression rate, then as melting point decreases an increase in regression rate will be observed at some or all oxidizer flux levels. 

The oxidizer will instead be nitrous oxide, a standard in the field of amateur hybrid rocketry. Nitrous oxide is safe and easy to handle. Also, it is self-pressurizing due to its vapor pressure of 747psi at 20 deg C. This will simplify oxidizer injection techniques, but limit the achievable chamber pressure. A cavitating venturi will still be used to measure and control oxidizer flow rates over time.