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.