**4. Results, analysis and discussion**

#### **4.1 Calculation on dry, steady flow**

In order to test for the accuracy of the numerical code, the model was applied to a few test cases at dry condition in a convergent-divergent nozzle. Three cases were calculated namely the subsonic-supersonic flow, purely subsonic flow and the flow involving shock. In this test, the inlet temperature and total pressure for all cases are set to 421.3K and 72,700 Pa respectively while the back pressure is varied according to the desired cases. For purely subsonic case, the back pressure is set to 68,000 Pa (giving a pressure ratio of 1.069) while for the supersonic flow, linear interpolation is employed for the determination of the back pressure. On the other hand, the back pressure for the shock case is set to 60,400 Pa.

Fig. 4 illustrates the general Mach number distribution along x direction for all cases. The numerical results are compared with the exact solutions which are given by solid lines close to each case respectively. It can be clearly seen that all numerical results show a slight deviation from the exact solutions except for the supersonic case where both numerical and exact solutions agrees very well with each other. For the pure subsonic flow case, the Mach number increases with distance until it reaches a peak value of 0.57 at the throat. Downstream of the throat, the Mach number decreases in the divergent section due to the decrease in the flow velocity. Comparison with the exact solution shows a slightly higher value and this could be attributed to the numerical error presented in the model. For the subsonic-supersonic case, the Mach number increases along the nozzle. The numerical solution for this case shows good agreement with the exact solution. The Mach number increases starting from the nozzle inlet to the nozzle exit. In order to emulate the shock case, the exit pressure is reduced slightly below the pressure imposed on the subsonic case. A

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 451

Fig. 5. Pressure variations using numerical solution and exact solution (dry case).

Based on the prediction made on dry condition on a few test cases, it has been demonstrated that the numerical scheme is able to calculate the compressible flow properties with good accuracy. The next step involves the calculation of two-phase flow properties using similar scheme but with the addition of the wetness treatment. The numerical scheme was applied to three cases involving nucleating steam flows in one-dimensional nozzles. These nozzles are those of (Binnie and Wood, 1938), (Krol, 1971) and (Skillings, 1987). The exact boundary conditions imposed for these cases are summarized in Tab. 1. For all tests, the flow with

**Nozzle P0 (Pa) T0 (K) Tsat(P) (K)** 

Case2 294,000 453.00 405.96

Case 2 35,440 349.00 346.13

Binnie and Wood (1938) 143,980 391.87 383.28 Krol (1971) Case 1 221,000 423.00 396.60

Skillings (1987) Case 1 32,510 357.20 344.60

In the one dimensional calculation, the number of node used in all cases is 125. In addition to the comparison made with the experimental data (where available), the 2D results of the same method are also plotted for comparisons. The 2D calculation was carried out in a few nozzle configurations namely (Binnie & Wood 1938), (Kroll, 1971) and (Skilling, 1987). For each of these nozzles, structured mesh with consistent size of 125 × 15 is adopted and is

**4.2 Steady flow wet steam simulation** 

Table 1. Boundary conditions for different test cases.

condensation takes place.

illustrated in Fig. 6.

normal shock-wave was found slightly downstream of the throat in the divergent section of the nozzle. The flow is supersonic upstream the normal shock-wave, whereas it becomes subsonic downstream the normal shock-wave.

Fig. 4. Comparisons of Mach number using numerical solution and exact solution (for dry case).

In addition to the Mach number, the pressure variation along the distance is also investigated and this is illustrated in Fig. 5. For the subsonic case, the pressure decreases along the nozzle length due to the reduction in flow area (where the velocity increases). Minimum pressure is calculated at the throat and the pressure increases again downstream of the throat due to the increase in flow area. For the supersonic case however, the pressure decreases starting from the nozzle inlet towards the nozzle exit. Downstream of the throat where choking occurs, the velocity of the fluid further increases thus reducing the pressure further. On the other hand, for the shock case, the pressure distribution is similar to the supersonic case except for a location downstream of the throat where sudden jump in pressure is observed indicating shock. It is also worth mentioned that prior to the shock, obvious oscillations can be seen due to the adoption of the central discretization scheme in the calculation which is highly unstable. Although artificial viscosity has been added to the equations, it is believed that the value is either too small or too large to improve the stability of the solutions. Thus further investigation is needed to eliminate or at least minimize the oscillation. Downstream the shock position, the pressure further increases towards the nozzle exit.

normal shock-wave was found slightly downstream of the throat in the divergent section of the nozzle. The flow is supersonic upstream the normal shock-wave, whereas it becomes

Fig. 4. Comparisons of Mach number using numerical solution and exact solution (for dry case).

In addition to the Mach number, the pressure variation along the distance is also investigated and this is illustrated in Fig. 5. For the subsonic case, the pressure decreases along the nozzle length due to the reduction in flow area (where the velocity increases). Minimum pressure is calculated at the throat and the pressure increases again downstream of the throat due to the increase in flow area. For the supersonic case however, the pressure decreases starting from the nozzle inlet towards the nozzle exit. Downstream of the throat where choking occurs, the velocity of the fluid further increases thus reducing the pressure further. On the other hand, for the shock case, the pressure distribution is similar to the supersonic case except for a location downstream of the throat where sudden jump in pressure is observed indicating shock. It is also worth mentioned that prior to the shock, obvious oscillations can be seen due to the adoption of the central discretization scheme in the calculation which is highly unstable. Although artificial viscosity has been added to the equations, it is believed that the value is either too small or too large to improve the stability of the solutions. Thus further investigation is needed to eliminate or at least minimize the oscillation. Downstream the shock position, the pressure further increases towards the

subsonic downstream the normal shock-wave.

nozzle exit.

Fig. 5. Pressure variations using numerical solution and exact solution (dry case).
