**4.2 Steady flow wet steam simulation**

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 condensation takes place.


Table 1. Boundary conditions for different test cases.

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 illustrated in Fig. 6.

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 453

Fig. 8(a) and 8(b) shows the pressure distribution along Krol nozzle for Case 1 and Case 2 respectively. The calculated pressure distribution for 1D and 2D cases show an excellent agreement for Case 1. However, experimental validation could not be made for this case due to the absent of experimental data. In order to increase confidence in the current scheme however, the model was validated with Case 2 where detail measurement was present. The comparison between the 1D and 2D calculation against experimental data is illustrated in Fig. 8(b). Despite the absent of data upstream of the nozzle throat, reasonable agreement

Fig. 8. Comparison of pressure distribution along Krol nozzle for (a) Case 1 (b) Case 2.

In addition to the Binnie & Wood and Krol nozzles, the mathematical model is also tested on Skilling nozzle (Skilling, 1987) at two different boundary conditions (Case 1 and 2). This is shown in Figs. 9(a) and 9(b) respectively. The general pressure distribution shows a

Fig. 7. Comparison of pressure ratio along Binnie and Wood nozzle.

was achieved downstream of the throat.

Fig. 6. 2D mesh for different nozzles (a) Binnie & Wood; (b) Kroll (c) Skilling.

Fig. 7 shows the pressure ratio distribution along nozzle length based on (Binnie &Wood, 1938). The general distribution on the pressure shows a decreasing trend due to the increase in velocity. The onset of condensation can also be observed by the sudden jump in pressure when condensation occurs and is typical for wet supersonic expansions. This is due to the release of latent heat at supersonic conditions which tends to retard the supersonic flow. The trend continues to decrease downstream of the throat towards the nozzle exit. The distributions of pressure along the nozzle predicted are in good agreement with experimental data.

Fig. 6. 2D mesh for different nozzles (a) Binnie & Wood; (b) Kroll (c) Skilling.

experimental data.

Fig. 7 shows the pressure ratio distribution along nozzle length based on (Binnie &Wood, 1938). The general distribution on the pressure shows a decreasing trend due to the increase in velocity. The onset of condensation can also be observed by the sudden jump in pressure when condensation occurs and is typical for wet supersonic expansions. This is due to the release of latent heat at supersonic conditions which tends to retard the supersonic flow. The trend continues to decrease downstream of the throat towards the nozzle exit. The distributions of pressure along the nozzle predicted are in good agreement with

Fig. 7. Comparison of pressure ratio along Binnie and Wood nozzle.

Fig. 8(a) and 8(b) shows the pressure distribution along Krol nozzle for Case 1 and Case 2 respectively. The calculated pressure distribution for 1D and 2D cases show an excellent agreement for Case 1. However, experimental validation could not be made for this case due to the absent of experimental data. In order to increase confidence in the current scheme however, the model was validated with Case 2 where detail measurement was present. The comparison between the 1D and 2D calculation against experimental data is illustrated in Fig. 8(b). Despite the absent of data upstream of the nozzle throat, reasonable agreement was achieved downstream of the throat.

Fig. 8. Comparison of pressure distribution along Krol nozzle for (a) Case 1 (b) Case 2.

In addition to the Binnie & Wood and Krol nozzles, the mathematical model is also tested on Skilling nozzle (Skilling, 1987) at two different boundary conditions (Case 1 and 2). This is shown in Figs. 9(a) and 9(b) respectively. The general pressure distribution shows a

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 455

Fig. 10. Comparison of droplet size along Binnie & Wood nozzle (Binnie and Wood, 1938).

Fig. 11. Comparison of droplet size distribution in Krol nozzle for (a) Case 1 (b) Case 2.

than Case 2, thus results in the faster droplet formation and growth.

to the one dimensional model. Based on these figures, it is also interesting to note that the temperature imposed at the inlet boundary has significant effect in delaying the onset of condensation. This has been demonstrated by Case 1 which inlet total temperature is lower

In addition to Binnie & Wood and Krol nozzles, further test is also carried out to investigate the ability of the model to calculate the droplet size distribution inside the channel. The third test case was done by adopting the Skilling nozzle where droplet size measurement was available. Fig. 12(a) and 12(b) illustrate the comparison on the droplet size along the nozzle for Case 1 and Case 2 respectively. In Case 1, the inlet total temperature was set

Fig. 9. Comparison of pressure distribution along Skilling nozzle for (a) Case 1 (b) Case 2.

decreasing trend when the flow was expected to increase in reduced cross sectional area. Comparison with the experimental result reveals good agreement. In addition, the onset of condensation was predicted correctly for both cases. From this calculation, it is evident that the location where condensation begins is highly influenced by the inlet temperature imposed on the inlet boundary.

To quantitatively validate the calculation model, comparison is also made based on the droplet size and its distribution along the duct. Fig. 10 shows the distributions of droplet radius for 1D and 2D calculations in Binnie & Wood nozzle. It can be seen that the droplet starts to form slightly downstream of the throat, grows rapidly if the surrounding parent vapour permits, and continue to grow towards the exit. It is these droplets that contribute to the wetness inside turbine channel. The droplet size information for the 2D case was taken at stream centre with the assumption of constant fluid properties along the *y*-direction. It is believed that the actual properties vary at least slightly in this direction, thus contributing to the slight deviation as compared to the 1D result. The 2D calculation shows slightly higher droplet radius after rapid growth downstream of the throat and this can be attributed to the cumulative contribution of different sets of droplet groups from the region in *y*-direction which was not accounted for the calculation.

Fig. 11(a) and 11(b) show the comparison made on the calculated droplet radius with experimental data on Krol nozzle (Krol, 1971) for Case 1 and Case 2 respectively. The general trend on the size distribution is almost identical for all nozzles, i.e. droplet grows rapidly once it is formed and continues to grow at a slower rate towards the nozzle exit. For Case 1, reasonable agreement was achieved for the 2D calculation. However, the 1D model was unable to predict the droplet size accurately despite showing similar growth profile. This discrepancy could be attributed to the two-dimensional effect in nozzle flow where the property variation in *y*-direction is important and needs to be taken into account in the calculation. In this case, the 1D calculation has over-predict the droplet radius due to the faster expansion rate for Kroll nozzle as compared to the Binnie & Wood nozzle. For Case 2, similar trend was shown at which better prediction was made by the 2D model as compared

Fig. 9. Comparison of pressure distribution along Skilling nozzle for (a) Case 1 (b) Case 2.

imposed on the inlet boundary.

which was not accounted for the calculation.

decreasing trend when the flow was expected to increase in reduced cross sectional area. Comparison with the experimental result reveals good agreement. In addition, the onset of condensation was predicted correctly for both cases. From this calculation, it is evident that the location where condensation begins is highly influenced by the inlet temperature

To quantitatively validate the calculation model, comparison is also made based on the droplet size and its distribution along the duct. Fig. 10 shows the distributions of droplet radius for 1D and 2D calculations in Binnie & Wood nozzle. It can be seen that the droplet starts to form slightly downstream of the throat, grows rapidly if the surrounding parent vapour permits, and continue to grow towards the exit. It is these droplets that contribute to the wetness inside turbine channel. The droplet size information for the 2D case was taken at stream centre with the assumption of constant fluid properties along the *y*-direction. It is believed that the actual properties vary at least slightly in this direction, thus contributing to the slight deviation as compared to the 1D result. The 2D calculation shows slightly higher droplet radius after rapid growth downstream of the throat and this can be attributed to the cumulative contribution of different sets of droplet groups from the region in *y*-direction

Fig. 11(a) and 11(b) show the comparison made on the calculated droplet radius with experimental data on Krol nozzle (Krol, 1971) for Case 1 and Case 2 respectively. The general trend on the size distribution is almost identical for all nozzles, i.e. droplet grows rapidly once it is formed and continues to grow at a slower rate towards the nozzle exit. For Case 1, reasonable agreement was achieved for the 2D calculation. However, the 1D model was unable to predict the droplet size accurately despite showing similar growth profile. This discrepancy could be attributed to the two-dimensional effect in nozzle flow where the property variation in *y*-direction is important and needs to be taken into account in the calculation. In this case, the 1D calculation has over-predict the droplet radius due to the faster expansion rate for Kroll nozzle as compared to the Binnie & Wood nozzle. For Case 2, similar trend was shown at which better prediction was made by the 2D model as compared

Fig. 10. Comparison of droplet size along Binnie & Wood nozzle (Binnie and Wood, 1938).

Fig. 11. Comparison of droplet size distribution in Krol nozzle for (a) Case 1 (b) Case 2.

to the one dimensional model. Based on these figures, it is also interesting to note that the temperature imposed at the inlet boundary has significant effect in delaying the onset of condensation. This has been demonstrated by Case 1 which inlet total temperature is lower than Case 2, thus results in the faster droplet formation and growth.

In addition to Binnie & Wood and Krol nozzles, further test is also carried out to investigate the ability of the model to calculate the droplet size distribution inside the channel. The third test case was done by adopting the Skilling nozzle where droplet size measurement was available. Fig. 12(a) and 12(b) illustrate the comparison on the droplet size along the nozzle for Case 1 and Case 2 respectively. In Case 1, the inlet total temperature was set

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 457

In this work, the modified 1D model has been applied to calculate the effect of unsteady supercritical heat addition in Skilling nozzle (Skillings, 1987). The calculation is carried out in an unsteady mode until the solution converged into a steady oscillation. Fig. 13 illustrates the effect of supercritical heat addition to the static pressure in the nozzle. To investigate the unsteady effect accurately, the variations of static pressure at a point located just downstream of the nozzle's throat is plotted. In this case, higher oscillation was observed at the beginning of the iterations and the pressure variation settles at approximately 400 iterations. From this point, regular oscillation with a frequency of approximately 302 Hz is calculated. The inside figure illustrates the static pressure after stable and consistent oscillation is achieved. The pressure distribution is plotted against time for the calculation of the frequency. Tab. 2 shows the summary of frequencies calculated based on different authors using different numerical treatment. The prediction made by the current work shows reasonably good agreement as

**Author Case Frequency, Hz Pressure** 

Skillings (1987) Experiment 380 20 Skillings (1987) 1D program 640 - Guha (1990) 1D program 540 - White and Young (1993) 2D program 420 - Yusoff et al. (2003) 2D program 393 18 Hasini et al. (2011) 1D program 302 17

Table 2. Frequency and pressure difference comparisons by different authors.

Fig. 13. Unstable supercritical heat addition effect to static pressure.

**Difference, mbar** 

compared to the experimental data based on the Skillings nozzle.

Fig. 12. Comparison of droplet size distribution along Skilling nozzle for (a) Case 1 (b) Case 2.

higher than Case 2 with the value of 357.2 K, thus there is a slight delay in the formation of droplet embryo. In this case, the droplet starts to form slightly downstream of the throat region, undergoes rapid growth and terminates at nozzle exit. Based on Fig. 12(a), the 1D model has under-predict the droplet size, while the 2D model has over-predict the size. For Case 2, the inlet total temperature is set lower (349 K), thus the formation of droplet embryo occurs earlier somewhere in the throat region and grows rapidly once the saturation condition is met. Reasonable accuracy was achieved both for 1D and 2D cases. In this case, both 1D and 2D calculations have under-predict the droplet size. Similar to Case 1, only one experimental reading was available thus qualitative comparison on droplet size distribution along the nozzle was not possible. It is difficult to quantitatively compare the distribution of droplet size along the nozzle since only one reading was made available (Skilling, 1987) for each case.

### **4.3 Unsteady supercritical heat addition**

In wet steam flows, condensation in a nozzle usually occurs in the divergent section just downstream of the throat with the consequence of the considerable release of latent heat, *Q*, to the flow. In addition, this phenomenon has the effect of moving the flow Mach number to unity. This is due to the addition of heat to the supersonic flow, which causes it to decelerate. The effect produced when Mach number equals to unity was described as "thermal choking" (Pouring, 1965) and when this happens, the flow can no longer sustain the additional heat and becomes unstable. The excess quantity of this additional heat is termed supercritical heat addition and has caused the flow to be unstable. This phenomenon commonly occurs in high speed condensing flow has been studied in detail by (Barschdoff, 1970) and (Skillings, 1987). Attempt was also made by (Guha, 1990) to model the supercritical heat addition effect by using Denton's method (Denton, 1983). The method adopts the unsteady time-marching treatment for condensing flow of steam. However, the frequency was over-predicted and ranges from 540 Hz to 650 Hz. In recent decades, the study of supercritical heat addition in high speed condensing flow was also made by (White & Young, 1993) and (Yusoff et al., 2003). The predicted frequency was calculated to be 420 Hz and 393 Hz respectively.

Fig. 12. Comparison of droplet size distribution along Skilling nozzle for (a) Case 1 (b) Case 2.

higher than Case 2 with the value of 357.2 K, thus there is a slight delay in the formation of droplet embryo. In this case, the droplet starts to form slightly downstream of the throat region, undergoes rapid growth and terminates at nozzle exit. Based on Fig. 12(a), the 1D model has under-predict the droplet size, while the 2D model has over-predict the size. For Case 2, the inlet total temperature is set lower (349 K), thus the formation of droplet embryo occurs earlier somewhere in the throat region and grows rapidly once the saturation condition is met. Reasonable accuracy was achieved both for 1D and 2D cases. In this case, both 1D and 2D calculations have under-predict the droplet size. Similar to Case 1, only one experimental reading was available thus qualitative comparison on droplet size distribution along the nozzle was not possible. It is difficult to quantitatively compare the distribution of droplet size along the nozzle since only one reading was made available (Skilling, 1987) for each case.

In wet steam flows, condensation in a nozzle usually occurs in the divergent section just downstream of the throat with the consequence of the considerable release of latent heat, *Q*, to the flow. In addition, this phenomenon has the effect of moving the flow Mach number to unity. This is due to the addition of heat to the supersonic flow, which causes it to decelerate. The effect produced when Mach number equals to unity was described as "thermal choking" (Pouring, 1965) and when this happens, the flow can no longer sustain the additional heat and becomes unstable. The excess quantity of this additional heat is termed supercritical heat addition and has caused the flow to be unstable. This phenomenon commonly occurs in high speed condensing flow has been studied in detail by (Barschdoff, 1970) and (Skillings, 1987). Attempt was also made by (Guha, 1990) to model the supercritical heat addition effect by using Denton's method (Denton, 1983). The method adopts the unsteady time-marching treatment for condensing flow of steam. However, the frequency was over-predicted and ranges from 540 Hz to 650 Hz. In recent decades, the study of supercritical heat addition in high speed condensing flow was also made by (White & Young, 1993) and (Yusoff et al., 2003).

The predicted frequency was calculated to be 420 Hz and 393 Hz respectively.

**4.3 Unsteady supercritical heat addition** 

In this work, the modified 1D model has been applied to calculate the effect of unsteady supercritical heat addition in Skilling nozzle (Skillings, 1987). The calculation is carried out in an unsteady mode until the solution converged into a steady oscillation. Fig. 13 illustrates the effect of supercritical heat addition to the static pressure in the nozzle. To investigate the unsteady effect accurately, the variations of static pressure at a point located just downstream of the nozzle's throat is plotted. In this case, higher oscillation was observed at the beginning of the iterations and the pressure variation settles at approximately 400 iterations. From this point, regular oscillation with a frequency of approximately 302 Hz is calculated. The inside figure illustrates the static pressure after stable and consistent oscillation is achieved. The pressure distribution is plotted against time for the calculation of the frequency. Tab. 2 shows the summary of frequencies calculated based on different authors using different numerical treatment. The prediction made by the current work shows reasonably good agreement as compared to the experimental data based on the Skillings nozzle.


Table 2. Frequency and pressure difference comparisons by different authors.

Fig. 13. Unstable supercritical heat addition effect to static pressure.

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 459

an example, it will be seen that the rate of growth of this droplet group is initially small but rises rapidly as soon as they grow beyond critical radius. The pattern is similar for the next 9 droplet groups but a different behavior is observed in the case of groups (11) and (12). With the depletion of the supercooling, the critical radius increases. The newly formed droplet embryos in group (11) has a radius which is less than the critical radius. Consequently, this droplet group is unable to grow and survive (due to its initial radius which is less than the critical radius) and therefore, it will evaporate. This is represented by the sudden drop in radius as in Fig. 15. Thus, with the step size adopted in the calculation, the nucleation process can be assumed to be effectively over after the formation of droplet group (10). By selecting a shorter step size, the number of droplet groups will increase but the number of droplets in each group will fall, increasing both the accuracy and the volume of algebra to be calculated. In this work, the step length was adjusted to keep the number of droplet

After the nucleation phase, the droplet groups are combined into a single population and their properties are averaged. For the example given in this work, the changes in relevant fluid properties after the nucleation phase are given in Fig. 15. It can be seen that, as expected, after the nucleation stage, the droplets grow rapidly at the beginning but with the depletion of supercooling, the rate of growth reduced and becomes extremely slow as the fluid approaches thermodynamic equilibrium. Similar profile can be seen for the wetness fraction where rapid increase in wetness is predicted at the beginning but as supercooling

groups to approximately 10.

reduced, the rate at which water forms also reduced.

Fig. 15. Change in fluid properties during the return to equilibrium.

In order to check for the accuracy between the calculation procedures in estimating fluid properties after the completion of nucleation phase, comparison is made between the

#### **4.4 Polydispersed droplet calculations**

Thus far, the calculation of droplet properties was averaged when a new droplet group formed with respect to time and space. This is done to simplify the calculation steps thus, reducing the computation time. In reality however, these droplet groups should be treated individually as their contribution to the total wetness inside the channel can be different. However, the calculation of the individual droplet spectrum is very complex and requires large computation power epecially when three-dimensional condensing flow is involved. Despite the simplications made in the calculation algorithm for the droplet properties, very little understanding was made on the effect of the calculation method in predicting the droplet properties. Thus this section aims to investigate the properties of wet steam given by individual droplet group tracking (called Non-Averaged method) and the average droplet property (called the Averaged method). The polydispersed droplet calculation is carried out where different droplet group formation and growth is tracked individually. In this study, the effect of droplet property averaging is carefully compared and investigated.

Fig. 14 shows the progressive growth of individual droplet group as superheated steam enters a constant area duct at 1 bar and initially at 44.8K supercooled. In order to observe their individual behavior, the droplets formed during each incremental step of the calculation have been regarded as a distinct group and treated separately. It can be seen that the variations in the critical droplet size, *r\** is unaffected at the beginning but increases as the fluid become less supercooled. To investigate the behavior of the droplets, the variation of the radii of each individual droplet group is plotted separately. Taking droplet group (1) as

Fig. 14. Development of the early stage of nucleation (The numbers on curves refer to specific droplet groups).

Thus far, the calculation of droplet properties was averaged when a new droplet group formed with respect to time and space. This is done to simplify the calculation steps thus, reducing the computation time. In reality however, these droplet groups should be treated individually as their contribution to the total wetness inside the channel can be different. However, the calculation of the individual droplet spectrum is very complex and requires large computation power epecially when three-dimensional condensing flow is involved. Despite the simplications made in the calculation algorithm for the droplet properties, very little understanding was made on the effect of the calculation method in predicting the droplet properties. Thus this section aims to investigate the properties of wet steam given by individual droplet group tracking (called Non-Averaged method) and the average droplet property (called the Averaged method). The polydispersed droplet calculation is carried out where different droplet group formation and growth is tracked individually. In this study,

the effect of droplet property averaging is carefully compared and investigated.

Fig. 14. Development of the early stage of nucleation (The numbers on curves refer to

specific droplet groups).

Fig. 14 shows the progressive growth of individual droplet group as superheated steam enters a constant area duct at 1 bar and initially at 44.8K supercooled. In order to observe their individual behavior, the droplets formed during each incremental step of the calculation have been regarded as a distinct group and treated separately. It can be seen that the variations in the critical droplet size, *r\** is unaffected at the beginning but increases as the fluid become less supercooled. To investigate the behavior of the droplets, the variation of the radii of each individual droplet group is plotted separately. Taking droplet group (1) as

**4.4 Polydispersed droplet calculations** 

an example, it will be seen that the rate of growth of this droplet group is initially small but rises rapidly as soon as they grow beyond critical radius. The pattern is similar for the next 9 droplet groups but a different behavior is observed in the case of groups (11) and (12). With the depletion of the supercooling, the critical radius increases. The newly formed droplet embryos in group (11) has a radius which is less than the critical radius. Consequently, this droplet group is unable to grow and survive (due to its initial radius which is less than the critical radius) and therefore, it will evaporate. This is represented by the sudden drop in radius as in Fig. 15. Thus, with the step size adopted in the calculation, the nucleation process can be assumed to be effectively over after the formation of droplet group (10). By selecting a shorter step size, the number of droplet groups will increase but the number of droplets in each group will fall, increasing both the accuracy and the volume of algebra to be calculated. In this work, the step length was adjusted to keep the number of droplet groups to approximately 10.

After the nucleation phase, the droplet groups are combined into a single population and their properties are averaged. For the example given in this work, the changes in relevant fluid properties after the nucleation phase are given in Fig. 15. It can be seen that, as expected, after the nucleation stage, the droplets grow rapidly at the beginning but with the depletion of supercooling, the rate of growth reduced and becomes extremely slow as the fluid approaches thermodynamic equilibrium. Similar profile can be seen for the wetness fraction where rapid increase in wetness is predicted at the beginning but as supercooling reduced, the rate at which water forms also reduced.

Fig. 15. Change in fluid properties during the return to equilibrium.

In order to check for the accuracy between the calculation procedures in estimating fluid properties after the completion of nucleation phase, comparison is made between the

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 461

droplet size in a few test cases only shows reasonable agreement with the measure size. The case for unsteady supercritical heat addition also reveals promising results indicating the capability of the model to calculate this phenomenon which might cause instability in turbine channel. The polydispersed droplet calculation has also been carried out using the model. Two methods of calculation were tested and compared. The first method tracks the individual droplet groups at the beginning of nucleation process. Upon the completion of nucleation, the droplet groups are combined into a single population and its properties are averaged. The "averaged" properties are repeatedly used for the subsequent steps until equilibrium. The second method however, retains the calculation of individual droplet groups and their properties and is called the "Non-Average" method. Comparison between the two methods reveals similar trend and even though the average methods slightly underpredicts the droplet radius after the completion of nucleation, the method is thought to be more suitable and economical due to considerable saving in computation time and processing power as compared to the other method. In addition, it is also anticipated that the discrepancies between the two methods are relatively small and therefore can be

Aitken, J. (1881). On Dusts, Fogs and Clouds. *Trans. Royal Society (Edinburgh)*, Vol. 30,

Barschdorff, D. (1970). Droplet Formation, Influence of Shock Waves and Instationary Flow

Becker, R. & Doring, W. (1935). Kinetische Behandlung der Keimbildung in Ubersettigten

Binnie, A.M. & Wood, M.W. (1938). The Pressure Distribution in a Convergent-Divergent

Binnie, A.M. & Greem, J.R. (1943). An Electrical Detertor of Condensation in High-Velocity

Callender, H.L. (1915). On the Steady Flow of Steam Through a Nozzle or Throttle, *Proc.* 

Campbell, B.A. & Bakhtar, F. (1970). Condensation Phenomena in High Speed Flow of

Deich, M.E.; Tsiklauri, G.V.; Shanin, V.K. & Danilin, V.S. (1972). Investigation of Flows of Wet Steam inNozzles. *Teplofizika Vysokikh Temperatur*, Vol. 1, Part. 1, pp. 122-129 Denton, J.D. (1983). An Improved Time-Marching Method for Turbomachinery Flow

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Polydispersed Droplet Spectrum in Transonic Steam Flows with Primary and Secondary Nucleation. *Applied Mathematical Modelling*, Vol. 31, pp. 1518-1533 Guha, A. (1990). The Fluid Mechanics of Two-Phase Vapour Droplet Flow with Application

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regarded as negligible.

pp.337-368

**6. References** 

"Averaged" and "Non-Average" methods. The comparison of the changes in the relevant fluid properties between the "Average" and "Non-Average" methods is shown in Fig. 16. In general, the supercooling and wetness fraction show similar trend for both methods. However, the average method seems to under-predict the droplet radius after the nucleation phase. This is probably due to the simplifications made to the governing equations and calculation procedure when numerical sensor created in the scheme automatically ignores the subsequent droplet embryos which grows for a very short period of time and later failed to survive. The cumulative error resulting from this simplification seems to increase at a very slow pace and later becomes constant. The maximum magnitude of this error is estimated to be approximately 5%. Taking into account the wetness fraction and supercooling properties predicted by both methods, and considering the amount of resources required by the Non-average method, it is anticipated that the Average method is capable of estimating the fluid properties in condensing flow. With some minor modification towards the numerical scheme, the scheme can be adopted to the real steam turbine passage for the prediction of various fluid properties in wet steam flow.

Fig. 16. Comparison on the changes of fluid properties between the "Average" and "Non-Average" methods.

#### **5. Conclusion**

The mathematical model for the calculation of high speed condensing flow has been developed. This model was applied to different test cases involving dry and wet flow expansion in converging-diverging nozzle. The mathematical prediction shows excellent agreement with experimental data for pressure along the nozzle. However, the calculated droplet size in a few test cases only shows reasonable agreement with the measure size. The case for unsteady supercritical heat addition also reveals promising results indicating the capability of the model to calculate this phenomenon which might cause instability in turbine channel. The polydispersed droplet calculation has also been carried out using the model. Two methods of calculation were tested and compared. The first method tracks the individual droplet groups at the beginning of nucleation process. Upon the completion of nucleation, the droplet groups are combined into a single population and its properties are averaged. The "averaged" properties are repeatedly used for the subsequent steps until equilibrium. The second method however, retains the calculation of individual droplet groups and their properties and is called the "Non-Average" method. Comparison between the two methods reveals similar trend and even though the average methods slightly underpredicts the droplet radius after the completion of nucleation, the method is thought to be more suitable and economical due to considerable saving in computation time and processing power as compared to the other method. In addition, it is also anticipated that the discrepancies between the two methods are relatively small and therefore can be regarded as negligible.

### **6. References**

460 Mechanical Engineering

"Averaged" and "Non-Average" methods. The comparison of the changes in the relevant fluid properties between the "Average" and "Non-Average" methods is shown in Fig. 16. In general, the supercooling and wetness fraction show similar trend for both methods. However, the average method seems to under-predict the droplet radius after the nucleation phase. This is probably due to the simplifications made to the governing equations and calculation procedure when numerical sensor created in the scheme automatically ignores the subsequent droplet embryos which grows for a very short period of time and later failed to survive. The cumulative error resulting from this simplification seems to increase at a very slow pace and later becomes constant. The maximum magnitude of this error is estimated to be approximately 5%. Taking into account the wetness fraction and supercooling properties predicted by both methods, and considering the amount of resources required by the Non-average method, it is anticipated that the Average method is capable of estimating the fluid properties in condensing flow. With some minor modification towards the numerical scheme, the scheme can be adopted to the real steam

turbine passage for the prediction of various fluid properties in wet steam flow.

Fig. 16. Comparison on the changes of fluid properties between the "Average" and "Non-

The mathematical model for the calculation of high speed condensing flow has been developed. This model was applied to different test cases involving dry and wet flow expansion in converging-diverging nozzle. The mathematical prediction shows excellent agreement with experimental data for pressure along the nozzle. However, the calculated

Average" methods.

**5. Conclusion** 


**20** 

*R.O.C.* 

**Experimental Study on Generation of Single** 

It has been known that the collapse of the cavitation bubbles could cause serious destruction of pressure pipes, hydraulic machineries and turbine structures. After the cavitation bubble is generated, the variation of its surrounding velocity and pressure field could result in its collapse. If the process of the collapse of a cavitation bubble appears near the solid boundary, its impact to the boundary could generate an immense water-hammer pressure effect (Plesset and Chapman, 1971). The shock wave generated in this process of bubble collapse could possibly impact or even destroy the solid boundary of structure. The bubble collapse studies include the understanding of the shock wave, the characteristics of the resultant luminescence, and the jet related fields. If the cavitation bubble is located near the solid boundary at certain suitable distance, it is more possible for the production of counterjet in the process of bubble collapse. There has not been a firm conclusion for the exact

characteristics which causes the destruction of the interface on the solid boundary.

Rayleigh (1917) studied the corrosion of high speed blade subjected to the effect of cavitation bubble. He mentioned that the bubble collapse is able to produce a high speed flow jet which damages the solid surface. Plesset (1949) further considered the influence of the physical characteristics of fluid viscosity and surface tension and derived the Rayleigh-Plesset equation. Kornfeld and Suvorov (1944) brought up the theory of bubble collapse near a solid boundary. They proposed that the bubble would be deformed to a nonspherical shape with the bubble surface tension penetrated subsequently to generate the phenomenon of flow jet. This phenomenon was proved in the experiment carried out by Naude and Ellis (1961). The numerical model in Plesset and Chapman's research (1971)also revealed this phenomenon. If the solid boundary is located on the right side of the bubble, the jet flow would be formed on the left side of the bubble and penetrates it before arriving at the right side interface of the bubble. The damage of the solid boundary might be caused by the impact of this jet flow during the bubble collapse. Benjamin and Ellis(1966)and Philipp and Lauterborn (1998) also detected the bubble collapse phenomenon and its consequent behavior of damage at the solid boundary. Recent research results reveled that the destructive power of the jet flow was not the main factor for the damage of the solid

**1. Introduction** 

**Cavitation Bubble Collapse Behavior** 

Sheng-Hsueh Yang1, Shenq-Yuh Jaw2 and Keh-Chia Yeh1

 **by a High Speed Camera Record** 

*1National Chiao Tung University Hsinchu, Taiwan, 2National Taiwan Ocean University Keelung, Taiwan,* 

