An ideal converging-diverging (CD) nozzle accelerates a subsonic flow to supersonic by converging the flow to sonic speed at the throat and diverging it until supersonic speed is reached. Such nozzles are commonly employed in rocket engines, where the primary objective is to convert pressure energy into kinetic energy to generate thrust. For optimum performance, it is necessary to analyze the inlet and outlet pressure. The paper studies how the nozzle operates under different outlet pressure (back-pressure) conditions while keeping the inlet pressure constant at 50 bar. Analytical calculations were performed to predict the critical back-pressure, and it was found that a normal shock appears in the nozzle when the back-pressure lies between 9.49 and 49.74 bar. Optimal expansion is achieved when the back-pressure is approximately 0.684 bar, corresponding to a supersonic Mach number of 3.47. These analytical predictions are compared with CFD simulations performed in ANSYS Fluent using the SST k–ω turbulence model. Shock waves were observed at 45, 35, and 15 bar back-pressure, while at 5 bar the flow was fully expanded and shock-free. The maximum Mach numbers obtained numerically for the respective back-pressures were 0.826, 1.318, 2.35, and 2.4. The percentage deviations from the ideal analytical Mach number (3.47) were 76.2%, 62.0%, 32.3% and 30.5%, respectively. The results indicate that increasing the pressure ratio between the inlet and backpressure shifts the shock wave towards the outlet of the nozzle and ultimately eliminates it at sufficiently low pressure. This work directly quantifies shock location and Mach number deviations from ideal isentropic theory by plotting a graph of Mach number along the centerline.