Sod's 1D Shock Tube Solution

I have nearly completed my 2D gas dynamics code. A rotated Roe-HLL approximate Reimann solver is implemented on a structured trapezoidal grid with MUSCL upwind reconstruction and UMIST limiter to ensure a 2nd order TVD scheme. A RK-4 algorithm is used for time integration. Here is the output computed for Sod's Shock Tube Problem as a test case where P1=10,000 Pa and P2=100,000 Pa at t=.002 seconds. These results are consistent with the exact solutions.

Constants:
MM =  30 g/mol
Specific heat ratio = 1.4
Initial temp = 298 K

 Grid:
nx = 800
xmax = 4 m
nt =  10,000
tmax = .01 sec

I am still trying to work out some bugs in its application to 2D axisymmetric flows.

Second Test Firing

Paraffin-A was used for the second firing of the hybrid rocket motor. As with the first test using Paraffin-C, the burn rate was far too high for the current setup. Even when a paper core liner was used to inhibit regression before initiation of oxidizer flow, the results with the lower melting point fuel were more severe, causing the motor casing to burn through in one area. Here is the test video:

 

Prior to firing, I fixed the problems that prevented proper data collection for test 1. As it turns out, I had flipped the positive and negative output terminals on P2 so that no data was recorded for the first test. With that rectified, we see the pressure time trace both within the motor and upstream of the injector. 

Pressure-Time Trace for Test #2

 With the data collected, I was able to run my analysis code. Some modifications involving the burn duration and combusted oxidizer mass were implemented, yielding the expected results for port radius regression and C*. Hybrid rocket operation in which paraffin was burned with nitrous oxide occurred on the interval 44.64 seconds to 46.70 seconds. This is seen in the plots of burn rate pressure and oxidizer flux dependence. A heavily pressure-dependent burn is observed before the start time, indicating the combustion of the APCP pre-heater grain. On this interval, a RK-4 algorithm was used to iteratively plot the port radius as a function of time.

Port Radius vs. Time

Burn rate dependencies were plotted on this interval. The least squares algorithm was used with log-log transformed variables in order to derive the burn rate relation. We see that the regression rate exhibits a highly oxidizer flux dependent trend (r^2 = .9245) and no pressure dependence trend.

Regression Rate Flux Dependence

Log(rdot,) = -7.975265 + 0.4652322*Log(Gox,)

rdot = .343864 * Gox^.4652322

a = 0.343864

n = 0.4652322

r^2 = .9245

Regression Rate Pressure Dependence

Based on the computation, we also find:

Characteristic Exhaust Velocity C* = 2544.294 m/2

Orifice Discharge Multiplier (CD * sqrt[P1 - Pc] = mdotox)

CD = 6.609519E-5

Combusted Oxidizer Mass = 148.32 g

Simply based on the total combusted oxidizer mass (148g) and the fuel mass (169g), we see that the optimal O/F ratio of ~7.0 was not met. When we plot the O/F ratio over time, we see that it tends to oscillate rapidly about a constant value of .881161.

O/F Ratio vs. Time

Clearly, the fuel's regression rate outpaced the rate of oxidizer injection, resulting in an extremely inefficient O/F ratio for the fuel. If paraffins are to be used in amateur rocket motors, their regression rates must be slowed or more effective means of oxidizer injection must be explored. Due to the destruction of the motor casing, no more trials involving raw paraffin will be performed. Rather, methods of slowing its burn rate using a cloth matrix will be explored.

Scott Fintel Correspondences

Firiday February 10, 2012

Dear Mr. Fintel,

I thought you might be interested to know that the first test of the paraffin-N2O hybrid rocket motor was a success! You can watch the video on the blog or on youtube at Hybrid Rocket Motor Test 1 However, I do have some concerns that I wish to discuss. The first is that while I got a pressure-time trace for the upstream nitrous Injector pressure, I did not get one for the motor's internal pressure. After firing, I checked that the pressure tap on the forward closure was not plugged with wax. It wasn't. What do you think happened? Might I increase the currently small diameter of the pressure tap? It could also have been an instrumentation error with the sensor itself. Also, It appears that quite a bit of wax was consumed before injection began, causing all of the fuel and also the liner to be completely consumed even with an under-filled nitrous tank (437g). I think I might attempt to thin the walls of the thermoplastic tubing, initiating nitrous flow sooner. I also plan to substantially increase the injector orifice area. I will likely be posting the full report of the firing along with the acquired data under "collected data" later today.

After testing each of four plain paraffins with 2% carbon black as an opacifier, I plan on testing each paraffin with the addition of a polymer in order to achieve a more controlled burn rate. Thanks again for all of your help.

Sincerely,
Sam

Friday February 10, 2012

Hello Sam,

Congratulations on your successful test!

It was obvious from the video your wax burned/melted much faster than you would have wanted. I know some people have had good results by melting and mixing wax with hot melt glue. That increased the melting point of the fuel and decreased regression rate. I'm not sure what type of wax you used, but there are types of wax with higher melting points that may work better as hybrid fuel, I think one type of wax called hurricane wax has a higher melting point.

I made a hybrid fuel grain many years ago by drawing cut newspaper through melted wax and then wrapping it on a coring rod as it came out of the wax. I never did test that fuel grain but I think it would have worked well. Even a thick cardboard tube soaked in melted wax should give good results.

Keep up the good work and let me know when you update your web site with the test data.

Cheers,

Scott