For this experiment, an assumption was made regarding the ambient temperature and pressure for this lab pressure being Pa= 101.325 KPa and temperature being Ta=14°C respectively
For this experiment, the theoretical and experimental values for pressure and thrust will be examined and compared in order to analyze the behavior of air through a convergent nozzle.
Firstly, we are dealing with compressible flow so that means density is rapidly changing in the conservation of mass equation as shown below:
When a fluid starts to move at speeds comparable to the speed of sound we start to observe compressible effects, and the relationship is modeled by the following equation which is a combination of two equations:
In order to determine when the flow is choked, the following equation is used which relates the atmospheric to tank pressure ratio:
The right side gives 0.528 which is a constant, so as long as the Pressure ratio is less than 0.528 there is choked flow, when the pressure ratio is higher, then the flow is considered unchoked. Knowing when the flow is choked vs unchoked is needed in order to solve for the theoretical pressures using isentropic and isothermal assumptions.
For Isothermal Expansion, the following assumptions are made:
1) slow discharge
2) T0 is constant
3) = constant
we get the following equation:
Experimental Setup and Procedure:
For the outline of the procedure please refer to Fluids Laboratory Manual . There were no deviations in this experiment.
Results: Based on the results, the experiment was choked between 3 to 82 seconds. Since starting transient is over after the first 3 seconds, the data is taken from that point onwards. These results were done for 10 second intervals despite the fact that the calculations were done for all the points. Refer to the Appendix for all values.
Table 1: Percent Error for Calculated values for Isothermal and Isentropic Assumptions compared with Experimental Tank Pressures.
Time(s) Exp. Tank Pressure (Pa) Percent Error (%) Isothermal Theoretical Pressure(Pa) Percent Error (%) Isentropic Theoretical Pressure(Pa)
3 696894 6.1 657052 19.1 585224
10 600919 0.02 590282 39.4 431011
20 493498 2.7 507489 75.4 281433
30 413657 5.4 437073 - 185810
40 352225 6.5 376834 - 123884
50 301548 7.2 325053 - 83344
60 260318 7.2 280459 - 56538
70 225154 7.0 242004 - 38663
80 196128 6.1 208797 - 26641
82 191025 5.8 202721 - 24750
Based on the results for Isothermal, the error seems to increase but reaches a maximum at 50 seconds then begins to decrease at 80 seconds. This result shows that due to pressure change rapidly increasing, the isothermal pressure becomes inaccurate. However, over time it is the more appropriate value due to the slower discharge nearing the end of the experiment. As for Isentropic, as time increases the percent error between the Isentropic and Tank Pressure increases. The reason for this is that there are fewer compressibility effects as time goes by so the flow is more constant thus invalid.
The following Figure (2) is a graph that models the relationship between Tank Pressure, Isothermal