Due to the high steam velocity and low thermal parameters at the turbine's final stage, steam generates non-equilibrium condensation and forms a large number of small droplets during the process of pressure expansion. The wet steam mixed with droplets impinges on the turbine blades, endangering turbine operation safety and reducing turbine work efficiency. This article modifies the non-equilibrium condensation control equation and embeds it into the numerical simulation software to make the numerical calculation results more accurate. By modifying the inlet steam superheat in the classical experiments, the condensation characteristics of wet steam in Moses–Stein nozzles and Dykas cascades are studied. The results show that increasing inlet superheat can effectively suppress the generation of non-equilibrium condensation and reduce outlet liquid mass fraction. The minimum supercooling temperature of non-equilibrium condensation is only related to the working fluid characteristics (the steam model used in this article is around 20 K). When the inlet superheat of the cascade is large, the rapid condensation region is mainly near the suction surface. In contrast, when the superheat is low, the rapid condensation zone is mainly near the pressure surface. The condensation location is mainly affected by the intensity of internal condensation shock waves in the cascade. Increasing inlet superheat not only increases the shock wave intensity but also decreases the shock wave angle in the passage. When the inlet temperature increases by 20 K, the heat efficiency of the cascade increases by about 1%.

As steam expands and performs work in the steam turbine, the temperature and pressure of the final stage steam decrease to the critical point of condensation. Due to the high-speed flow of steam, it cannot be condensed immediately upon crossing the saturation line. Instead, it continues to expand as supersaturated steam until its parameters reach the Wilson point,1 at which point condensation commences. The resulting condensed droplets grow and impact the turbine blades, leading to corrosion and potential blade fractures, posing significant safety hazards. According to the Baumann criterion,3 a 1% increase in humidity in the final stage of the steam turbine can result in a 1% decrease in efficiency. Therefore, it is crucial to investigate the non-equilibrium condensation flow phenomenon of wet steam to enhance the safety and efficiency of steam turbines.

The non-equilibrium condensation flow of wet steam cannot be regarded as a simple single-phase flow, but as a two-fluid motion process with complex heat and mass transfer processes. Since the last century, people have carried out a lot of experimental and theoretical research on the non-equilibrium condensation mechanism.4–11 It is found that the nucleation rate equation and the droplet growth equation are the main controlling equations that affect the non-equilibrium condensation flow. Moses and Stein4 have carried out detailed experimental comparisons of the control equations of non-equilibrium condensation phenomena in one-dimensional Laval nozzles and analyzed the numerical differences caused by different droplet growth models. Bakhtar and Zidi8 studied pressure fluctuations caused by spontaneous condensation of superheated steam in nozzles and recorded changes in droplet size. Zhang et al.10 made empirical corrections to droplet surface tension and studied the influence of superheat on non-equilibrium condensation in IWSEP nozzles. Dykas et al.9,12 studied the non-equilibrium condensation phenomenon of wet steam on a cascade under transonic conditions through experiments, recorded pressure changes caused by spontaneous condensation of steam under different expansion rates, and obtained the distribution of shock waves at the exit of the cascade through schlieren photography, laying a foundation for studying the transonic non-equilibrium condensation phenomenon of steam turbines.

With the development of computational fluid dynamics, numerical calculations have replaced most experiments and become one of the main methods for studying non-equilibrium condensation phenomena. By analyzing the numerical calculation results and experimental results, we can obtain the impact of factors such as surface tension, condensation rate equation, and droplet growth rate equation on the key parameters of non-equilibrium condensation. The research objects have expanded from simple one-dimensional nozzle models13–16 to three-dimensional unsteady steam turbine models.2,17–20 Although non-equilibrium condensation models in commercial software21 show sufficient accuracy in predictions, they are limited by the equations and steam parameters built into the software. Nowadays, more scholars conduct numerical research on non-equilibrium condensation using user-defined functions (UDFs), which greatly improves accuracy and results, and extends research objects to wet air22 and supercritical carbon dioxide.23,24

From the above references, it can be found that the non-equilibrium condensation process involves complex phase change, heat transfer and mass transfer processes, and the non-equilibrium condensation mechanism is not yet fully understood. In particular, the influence of boundary conditions on the characteristics of non-equilibrium condensation has not been deeply studied. In addition, most of the research on non-equilibrium condensation is based on Laval nozzles, while the condensation phenomenon in steam turbines is a more complex three-dimensional flow. Therefore, further research is needed to investigate the influence of inlet boundary conditions on non-equilibrium condensation and its flow field in the blade. This article modifies the non-equilibrium condensation control equation based on the Euler–Euler method and compares it with the original model. By modifying the superheat at the inlet of the cascade, the relationship between steam parameters and non-equilibrium condensation in steam turbine blade is studied, providing theoretical basis for steam turbine dehumidification strategy.

The non-equilibrium condensation process is a rapid condensation process that occurs when the steam state crosses the saturation line due to rapid expansion of steam. As the volume and mass fraction of droplets formed by condensation are relatively small compared to the steam, the droplet motion can be regarded as carried by the steam. The liquid phase velocity is the same as that of the steam phase, and there is no slip motion between phases, i.e., the viscous force between the two phases is ignored. Therefore, the gas phase continuity equation is given by the following equations:
ρ v α v t + x j ρ v u j α v = S m ,
(1)
ρ v α v u i t + ρ v u j u i α v x j = α v p x i + x j α v μ u i x j + μ u j x i 2 3 μ δ i j u j x j + S u i ,
(2)
ρ v α v H v t + ρ v u j α v H v x j + α v p t = x j λ α v T x j + u i α v τ i j x j + S h .
(3)
Here, α v is the gas phase volume fraction, ρ v is the gas phase density, u j is the gas phase velocity, S m is the mass source term, p is steam pressure, μ is steam dynamic viscosity, S u i is the momentum source term, λ is the thermal conductivity, T is the saturation steam temperature, S h is the energy source term, and H v is the steam stagnation enthalpy.
The mass source term S m of the liquid phase can be defined as
ρ l α l t + x j ρ l α l u j = S m .
(4)
Here, α l is the volume fraction of the liquid phase, and ρ l is the density of the liquid. The volume fraction of the liquid phase can be obtained by the following equations:
α v + α l = 1 ,
(5)
ρ l α l N t + x j ρ l α l u j N = ρ l α l J d .
(6)
In non-equilibrium condensation, there are two key unknowns N and J d. N represents the number of droplets formed per unit volume and per unit time, while J d represents the rate of droplet formation per unit volume.25 Based on the physical meaning of the liquid phase mass source term, its change can be identified as the mass converted from gas phase to liquid phase. Therefore, the liquid phase mass source term, momentum source term, and energy source term can be written as
S m = m * ,
(7)
m * = 4 3 π ρ l J d r * 3 + 4 π ρ l N r 2 d r d t ,
(8)
S u i = m * u i ,
(9)
S h = m * h l .
(10)
where m * is the mass of droplets generated per unit volume. According to Eq. (8), we can get the calculation of droplet mass is composed of two parts. The first part is the droplets generated when steam begins to condense; at this time, the radius of droplets is called Kelvin–Helmholtz critical radius r *, and its equation is shown in Eq. (11), where σ is surface tension of droplets and S is the ratio of steam saturation; the second part is the mass change brought by droplet growth, and the droplet mass is controlled by growth rate equation d r / d t,
r * = 2 σ ρ l R T ln S ,
(11)
S = p p sat .
(12)
Equations (1)–(12) constitute a closed system of equations for the non-equilibrium condensation process of wet steam, in which the equations of nucleation rate J d and droplet growth rate d r / d t are unknown, and they are also the main factors that affect the non-equilibrium condensation phenomenon. The equation of nucleation rate J d has been corrected many times, and finally Kantrowitz26 proposed a non-isothermal correction method. This method adds the temperature difference to the energy transfer process in the isothermal nucleation theory. The equation can be written as
J d = q c 1 + θ ρ v 2 ρ l 2 σ π M m 3 e 4 π r * 2 σ 3 k T ,
(13)
θ = 2 γ 1 γ + 1 h v l R T h v l R T 1 2 .
(14)
q c is the coefficient of evaporation and condensation, which is usually set to 1. ρ v is the density of steam, ρ l is the density of liquid phase, M is individual droplet, k is the Boltzmann number, θ is the non-isothermal correction coefficient, γ is the ratio of steam heat capacity, and h v l is the latent heat of vaporization. The above formula is derived from experimental. The numerical changes of surface tension of droplet in the exponential term will cause a drastic change of condensation rate. Benson and Shuttleworth27 corrected the droplet surface tension equation by introducing temperature term, which makes the non-isothermal correction model more consistent with the actual physical phenomenon, which is shown in Eq. (15). Subsequently, Zhang et al.2,10,15,28 introduced the influence of droplet radius on droplet surface tension on the basis of Benson and obtained a more accurate numerical model, shown as Eq. (16),
σ 0 T = 0.2358 × 1 T 647.3 1.256 × 1 0.625 1 T 647.3 ,
(15)
σ = σ 0 ( T ) × ( 1 ρ l / m 3 4.836 r ) .
(16)
The growth rate model of droplet has always been one of the key issues in the study of non-equilibrium condensation. Based on the thermodynamic growth rate model proposed by Gyarmathy, Young made a correction, making the calculated value more consistent with the experimental results. The droplet growth rate model proposed by Young is given in Eqs. (17) and (18), in which λ v is the steam thermal conductivity, T d T is the supercooling degree of steam, K n is the dimensionless Knudsen number, which is represented as K n = l ¯ / 2 r, P r v is the steam Prandtl number, and α and β are adjustment coefficients, which are used to improve the accuracy of numerical calculations,
d r d t = λ v T d T h v l ρ l r 1 r * r 1 1 + 2 × β K n + 3.78 1 ψ K n P r v ,
(17)
ψ = R T s h v l α 1 2 γ + 1 2 γ C p T s h v l 2 q c 2 q c .
(18)

In this paper, ANSYS Fluent is used to calculate the above equations. The steam flow control Eqs. (1)–(3) are solved by the software, while Eqs. (4)–(18) are compiled in C language and embedded into the numerical simulation process. The calculations of the nucleation rate Eq. (13) and the droplet growth Eq. (17) are compiled using the UDF (user-defined function) Adjust macro, and the resulting mass source term, energy source term, and momentum source term of the droplet are fed back to Fluent as the UDS (user-defined scalar), the workflow is shown in Fig. 1. To ensure computational accuracy, the entire calculation process adopts double precision, uses second-order upwind scheme, and the turbulence model uses the SST k ω model. The convergence criterion is 10 4. The inlet boundary is set to total pressure and total temperature, and the wall is defined as an adiabatic no-slip wall.

FIG. 1.

Non-equilibrium condensation workflow.

FIG. 1.

Non-equilibrium condensation workflow.

Close modal

This section selects the Moses–Stein nozzle, which has rich experimental data as the research object. The physical model and mesh of the nozzle are shown in Fig. 2. The total length of the entire nozzle is 0.14 m, the nozzle contraction section is a R = 0.053 m arc, the expansion section is a R = 0.684 m arc, and the throat of the nozzle is located at x = 0.0822 m, as shown in Fig. 2(a). The throat of the nozzle has a large velocity change and is also the location where non-equilibrium condensation begins. Therefore, to ensure the accuracy of numerical calculations, local refinement was applied to the mesh in the throat region, as shown in Fig. 2(b). Before performing numerical calculations, it is necessary to verify the mesh independence. This article selects Moses–Stein nozzle experimental data No. 1934 as the verification object. The total pressure of the nozzle inlet is 43.02 kPa, the total temperature is 366.0 K, the inlet superheat Δ T = 15.5 K, and the turbulence model uses the SST k ω.

FIG. 2.

Moses–Stein nozzle model (a) geometric structure and (b) nozzle mesh.

FIG. 2.

Moses–Stein nozzle model (a) geometric structure and (b) nozzle mesh.

Close modal

In this section, the initial calculation selects 30 000 grid points. On this basis, the grid independence is verified by increasing the grid points by 20 000 each time. Finally, the pressure ratio ( P ratio = P / P 0) on the centerline of the nozzle is compared with experimental data as shown in Fig. 3. With the increase in grid points, it is found that the error between numerical calculation results and experimental results is gradually reduced. The change of numerical calculation results is mainly reflected in the rapid condensation region. The upstream and downstream of the condensation region are not sensitive to changes in grid size. When the grid size increases to 70 000, the numerical calculation results no longer change, and the grid independence verification is complete. By comparing the numerical simulation results with experimental results, it can be found that the numerical simulation results accurately capture the pressure jump region formed by non-equilibrium condensation. The position and pressure increase amplitude of these results are consistent with experimental results. It is proved that the numerical calculation method used in this article is reliable and can satisfy the predictive research of non-equilibrium condensation phenomenon.

FIG. 3.

Mesh independence verification.

FIG. 3.

Mesh independence verification.

Close modal

The parameters at the centerline of the nozzle best represent the characteristics of non-equilibrium condensation. The key parameters of non-equilibrium condensation in experiment 193 are shown in Fig. 4. The inlet superheat of the steam is 15.5 K. As the steam expands, the pressure gradually decreases, while the degree of supercooling of the steam gradually increases. When the degree of supercooling reaches about 20 K, the nucleation rate starts to have a numerical value, and condensation occurs. Since the nucleation rate is small at this time, there is no change in humidity inside the nozzle. As the steam continues to expand, the nucleation rate reaches its maximum value (Wilson point), and the non-equilibrium condensation rate inside the nozzle reaches its peak. The degree of supercooling also reaches its peak at this time. Inside the nozzle, a large number of water droplets condense rapidly and release a large amount of latent heat. The humidity at the centerline of the nozzle begins to increase, and static pressure surges in this region, which is called the rapid condensation region. When the steam passes through the rapid condensation region, the nucleation rate drops to 0, but the droplets continue to grow and release latent heat, leading to a continuous decrease in supercooling and an increase in humidity inside the nozzle.

FIG. 4.

Moses–Stein nozzle condensation characteristics (a) flow parameters at the centerline and (b) nucleation rate distribution.

FIG. 4.

Moses–Stein nozzle condensation characteristics (a) flow parameters at the centerline and (b) nucleation rate distribution.

Close modal

The superheat of the steam has a significant impact on the flow characteristics of non-equilibrium condensation, especially the position of the rapid condensation point (Wilson point), thermodynamic parameters, and key parameters such as the outlet droplets. Based on the numerical calculation results obtained from Sec. IV A, by changing the inlet temperature (temperature difference of 3 K), we obtained five different superheat conditions. The boundary conditions for these conditions are summarized in Table I.

TABLE I.

Inlet parameters for different degrees of superheat in nozzle.

No. Inlet pressure P 0(kPa) Inlet temperature T 0(K) Superheat Δ T(K)
Case 1  43.02  360  9.5 
Case 2  43.02  363  12.5 
Case 3  43.02  366  15.5 
Case 4  43.02  369  18.5 
Case 5  43.02  372  21.5 
No. Inlet pressure P 0(kPa) Inlet temperature T 0(K) Superheat Δ T(K)
Case 1  43.02  360  9.5 
Case 2  43.02  363  12.5 
Case 3  43.02  366  15.5 
Case 4  43.02  369  18.5 
Case 5  43.02  372  21.5 

When the wet steam rapidly condenses, it generates a large amount of droplets. The volume difference during the conversion process needs to be compensated by subsequent steam. Therefore, the rapid condensation of steam hinders the increase in steam velocity in the nozzle and causes local pressure increase. At the same time, a large amount of condensation latent heat is released. The location of pressure fluctuation represents the region of rapid steam condensation, and the amplitude of pressure change represents the rate of heat and mass transfer in the rapid condensation region. Figure 5(a) shows the static pressure distribution curve at the central axis and the diameter distribution of the droplets at the outlet. When the inlet superheat increases, it can be observed that the pressure surge region gradually moves toward the nozzle outlet, and the amplitude of pressure change gradually decreases. This indicates that when the inlet superheat of steam gradually increases, the position of the rapid condensation region gradually shifts toward the rear, and the rate of non-equilibrium condensation heat and mass transfer decreases. The change process of droplet radius in non-equilibrium condensation is mainly affected by droplet growth rate [Eq. (17)]. Among them, steam supercooling T d T is a phase parameter of steam. With the increase in inlet superheat, steam supercooling shows a downward trend and leads to a decrease in droplet growth rate, thereby gradually reducing the average diameter of droplets at the steam nozzle outlet. This phenomenon is consistent with the droplet diameter distribution in Fig. 5(a).

FIG. 5.

Distribution of liquid phase parameters on the centerline: (a) pressure and outlet droplet diameter, (b) nucleation rate and supercooling, and (c) mass fraction of outlet droplets.

FIG. 5.

Distribution of liquid phase parameters on the centerline: (a) pressure and outlet droplet diameter, (b) nucleation rate and supercooling, and (c) mass fraction of outlet droplets.

Close modal

Nucleation rate is crucial for non-equilibrium condensation of steam, and its magnitude and location have a significant impact on the flow of wet steam in the nozzle. Figure 5(b) shows the distribution of nucleation rate and steam supercooling at the central axis. Taking case 1 as an example, when non-equilibrium condensation does not occur, the nucleation rate is 0. When the steam pressure is greater than the saturation pressure (x = 0.075 m), non-equilibrium condensation begins, and the steam temperature is lower than the saturation steam temperature, which is one of the typical physical parameters of non-equilibrium condensation. As the steam expands within the nozzle, the nucleation rate gradually increases to the Wilson point. At this point, the supercooling caused by steam expansion still dominates, so the supercooling of steam gradually increases. After the steam passes the Wilson point, the droplet diameter rapidly increases, and the latent heat released by rapid condensation is absorbed by the steam, causing a rise in steam temperature and a decrease in supercooling (which is also the reason for the lagging peak position of supercooling). After non-equilibrium condensation disappears, the steam parameters inside the nozzle tend to be stable, and the supercooling of steam at the nozzle outlet basically remains unchanged. When the inlet superheat of steam increases, the nucleation position gradually shifts backward, and the peak value of nucleation rate decreases. This phenomenon is consistent with the static pressure curve [Fig. 5(a)] distribution. When the inlet superheat of steam increases, the supercooling of steam before the Wilson point decreases, and this change is obvious. After the steam passes the Wilson point, due to the influence of latent heat of condensation, higher nucleation rates release more latent heat. Therefore, the outlet gas supercooling increases with increasing inlet parameters. By comparing the supercooling at the condensation start point under various conditions, it can be found that the supercooling at the start of steam condensation is around 20 K, which is independent of inlet parameters.

The mass fraction of droplets at the nozzle outlet [Eq. (8)] is a comprehensive reflection of non-equilibrium condensation within the nozzle, as shown in Fig. 5(c). When steam condensation begins, due to the small nucleation rate, the mass fraction of droplets formed within the nozzle is small but not zero. After the steam passes the Wilson point, rapid condensation occurs within the nozzle, and the humidity along the central axis rapidly increases. At this time, the first term ( 4 / 3 π ρ l J d r * 3) in Eq. (8) dominates. After non-equilibrium condensation within the nozzle ends, the number of droplets in the steam no longer changes, but the droplet diameter gradually increases. At this time, the second term ( 4 π ρ l N r 2 d r / d t) in Eq. (8) dominates, and the humidity within the nozzle gradually increases, but the growth rate decreases. As the inlet steam parameters increase, the outlet humidity of the nozzle gradually decreases, which is also one of the main methods for steam turbine dehumidification.

The blade model used in this section is the straight blade made by Dykas et al.12 using the 200 MW steam turbine last stage blade type line. The blade geometric parameters are shown in Fig. 6(a), the leading edge diameter is 21.1 mm, the trailing edge diameter is 2.3 mm, the blade turning angle is 51.44°, the vertical direction blade pitch is 91.74 mm, the axial chord length is 173.97 mm, and the blade height is 110 mm. The pressure measurement point is located 75 mm behind the blade trailing edge. As shown in Fig. 6(b), the flow channel grid adopts H-type grid, and the blade is surrounded by O-type segmentation, with local encryption performed near the wall to ensure that the y + value of the blade surface is less than 10, enabling more accurate capture of the flow details near the blade wall. The boundary conditions of S1 in Ref. 12 are used, with the inlet total pressure P 0 = 103 kPa, inlet temperature T 0 = 379 K, outlet back pressure P 1 = 42 kPa, outlet temperature T 1 = 350.18 K, inlet superheat Δ T = 5.6 K, and turbulence model using SST k ω.

FIG. 6.

Dykas cascade: (a) blade parameters and (b) cascade mesh.

FIG. 6.

Dykas cascade: (a) blade parameters and (b) cascade mesh.

Close modal

Figure 7 shows the comparison between the wall static pressure distribution obtained by numerical simulation and the experimental results.12 On the suction surface of the blade, two pressure fluctuation points can be observed. The first fluctuation point is located at the middle of the blade cascade, which is caused by pressure changes due to airfoil structure. The second fluctuation point is located at the rear of the blade cascade, which is caused by the rapid condensation of a large number of small droplets and the release of a large amount of latent heat of vaporization due to the overcooling steam expansion acceleration to Wilson point. The numerical simulation results in this paper correspond well with the experimental results, especially for the prediction of Wilson point position and pressure changes caused by condensation. Combined with Fig. 7(b), the distribution of condensate nuclei rate in the blade cascade, the non-equilibrium condensation process of steam can be obtained: steam expands rapidly in the flow channel, starting from subsonic velocity at the inlet and gradually expanding to sonic velocity at the throat, and then further expanding to supersonic velocity; due to the lack of condensation nuclei, steam continues to expand without condensation when it reaches saturation temperature and becomes overcooled steam; when steam reaches maximum supercooling (Wilson point), it suddenly condenses into a large number of small droplets and releases a large amount of latent heat of condensation, and the surrounding steam pressure rapidly increases.

FIG. 7.

Numerical results and experimental results: (a) blade static pressure and (b) nucleation rate.

FIG. 7.

Numerical results and experimental results: (a) blade static pressure and (b) nucleation rate.

Close modal

In order to study the influence of superheat on the unbalanced condensation flow in the blade cascade, this section selects the S1 condition studied by Dykas et al.12 as the basis and changes the inlet temperature to vary the steam inlet superheat. The geometric structure and numerical calculation method of Dykas blade are described in detail in Sec. V A. This section selects five different inlet superheats as the research objects. Due to the complexity of the blade cascade structure, in order to better reflect the influence of superheat on unbalanced condensation, this section selects a temperature difference of 5 K, and its boundary conditions are shown in Table II.

TABLE II.

Inlet parameters for different degrees of superheat in cascade.

No. P 0(kPa) T 0(K) Superheat Δ T(K) P 1 (kPa) T 1 (kPa)
Case 1  103  369  −4.4  42  350.18 
Case 2  103  374  0.6  42  350.18 
Case 3  103  379  5.6  42  350.18 
Case 4  103  384  10.6  42  350.18 
Case 5  103  389  15.6  42  350.18 
No. P 0(kPa) T 0(K) Superheat Δ T(K) P 1 (kPa) T 1 (kPa)
Case 1  103  369  −4.4  42  350.18 
Case 2  103  374  0.6  42  350.18 
Case 3  103  379  5.6  42  350.18 
Case 4  103  384  10.6  42  350.18 
Case 5  103  389  15.6  42  350.18 

Due to the fact that the pressure parameters extracted in the experiments are wall static pressures, this section selects the wall pressure under the above five boundary conditions for analysis, as shown in Fig. 8. When the inlet superheat of the blade cascade changes, the pressure fluctuations caused by the airfoil do not change significantly, while the pressure fluctuations caused by non-equilibrium condensation have a larger change in the starting position (Wilson point) and amplitude of the fluctuations. With the increase in inlet superheat, the position of the Wilson point gradually moves toward the outlet of the blade cascade, which is consistent with the results obtained in the nozzle. The amplitude of the pressure fluctuations on the wall increases with the increase in inlet superheat, but decreases at ΔT = 15.6 K. There is no pressure surge on the pressure curve, but a period of pressure fluctuations. Since the pressure signals extracted in this section are from the blade wall, while those in the nozzle are from the central axis of the nozzle, the results obtained by the two methods are not identical.

FIG. 8.

Effect of different inlet superheat on wall static pressure of blade cascades.

FIG. 8.

Effect of different inlet superheat on wall static pressure of blade cascades.

Close modal

Figure 9 shows the influence of different blade inlet superheat on the distribution of nucleation rate. The case 3 condition is given in Fig. 8. It can be observed that the non-equilibrium condensation region of the blade mainly occurs near the throat, and the maximum value also occurs near the throat. The main reason for this phenomenon is that the steam velocity at the throat is high, and when the steam reaches the condensation pressure, there is still a certain degree of supercooling, leading to a failure of the thermal recovery of the steam to follow the inertial recovery of the steam. With the increase in inlet superheat, the non-equilibrium condensation region of the blade gradually shrinks and moves toward the outlet. In case 5, the non-equilibrium condensation region along the central axis of the blade extends to the outlet of the blade channel. The fluid in this region lacks the constraint of the pressure surface, so the non-equilibrium condensation phenomenon disappears near the suction surface, leading to a failure of the pressure in this section to continue to rise, explaining the reason for the decrease in static pressure curve in Fig. 8.

FIG. 9.

Distribution of nucleation rate in cascade: (a) case 1, (b) case 2, (c) case 4, and (d) case 5.

FIG. 9.

Distribution of nucleation rate in cascade: (a) case 1, (b) case 2, (c) case 4, and (d) case 5.

Close modal

Figure 10 shows the nucleation rate and steam supercooling at the axial positions of adjacent blades in cases 1 and 5, as extracted from Fig. 9. When the steam inlet superheat is –4.4 K, the non-equilibrium condensation starts at x = 0.119 m and ends at x = 0.167 m. The Wilson point is located before the large condensation region where the nucleation rate increases slowly. After the steam passes the Wilson point, the condensation region rapidly shrinks. When the steam inlet superheat is 15.6 K, the non-equilibrium condensation starts at x = 0.161 m and ends at x = 0.181 m. The Wilson point is located before a smaller condensation region where the nucleation rate increases rapidly. After the steam passes the Wilson point, the condensation region extends toward the outlet, which corresponds to the nucleation rate cloud map. By comparing the steam supercooling at the nucleation start position, it can be seen that the non-equilibrium condensation start steam supercooling in the blade is also around 20 K.

FIG. 10.

Nucleation rate and supercooling at the central axis of the cascade: (a) case 1 and (b) case 5.

FIG. 10.

Nucleation rate and supercooling at the central axis of the cascade: (a) case 1 and (b) case 5.

Close modal

Figure 11 shows the distribution of droplet number and humidity in cases 1 and 5. By comparing with the nucleation rate distribution in Fig. 9, it can be found that the distribution of droplet number has a high consistency with the nucleation rate distribution. With the increase in inlet steam temperature, the position of droplet generation moves toward the blade outlet. In case 5, the number of droplets generated near the suction surface of the blade is the largest, and a narrow and long region of dense droplets is formed with the flow of steam in the channel. By comparing with the nucleation rate distribution in Fig. 9, it can also be found that the nucleation rate is the largest near the suction surface wall, which is the main reason for the high number of condensed droplets here. With the decrease in inlet steam temperature in case 1, the number of droplets generated near the pressure surface of the blade is the largest, and the nucleation rate is also at a relatively large value. The outlet humidity of the blade increases with the decrease in inlet steam superheat. In the lowest temperature condition (case 1), the humidity reaches a maximum value of 3.3%. It can be found that there is no much correlation between the initial position of humidity and the initial formation position of nucleation rate and droplet number. Instead, after the nucleation rate ends, humidity begins to appear. This is because during the initial stage of non-equilibrium condensation, a large number of small droplets are generated in the channel, and at this time, the droplet radius is small, so it has little impact on humidity. The steam humidity in the blade is more affected by wet steam equilibrium condensation.

FIG. 11.

Effect of superheat on cascade condensation characteristics: (a) number of droplets and (b) mass fraction of droplets.

FIG. 11.

Effect of superheat on cascade condensation characteristics: (a) number of droplets and (b) mass fraction of droplets.

Close modal

Due to the large inlet velocity of steam, the steam accelerates through the cascade, exceeding the speed of sound at the blade outlet, and forming a unique “swallowtail wave” at the blade trailing edge. Figure 12 shows the Mach number distribution in the blade channel under cases 1 and 5 conditions. By analyzing the angle of the “swallowtail-shaped” condensation shock wave at the blade trailing edge, it can be found that the condensation shock wave angle in case 1 is 123.7°, and as the inlet steam temperature increases, the shock wave angle gradually decreases to 102.3°. In addition, the higher the inlet steam temperature, the higher the intensity of the condensation shock wave at the blade trailing edge. In case 5, a clear reflection wave can be observed after the condensation shock wave contacts the suction surface of the blade. With the decrease in inlet temperature, in case 1, the reflected wave disappears, and the higher speed region is concentrated near the pressure surface at the trailing edge. Combined with the distribution of nucleation rate and droplet number, it can be found that regions with large velocity gradients are more prone to non-equilibrium condensation, which further affects droplet generation.

FIG. 12.

Mach number distribution in cascade: (a) case 1 and (b) case 5.

FIG. 12.

Mach number distribution in cascade: (a) case 1 and (b) case 5.

Close modal
The thermodynamic loss in steam expansion is irreversible. The thermodynamic process of steam in the turbine cascade is shown in Fig. 13(a). The point 0 represents the inlet parameters of steam, the point 1 is the outlet parameters of steam in isentropic expansion process, and the outlet parameters of steam due to the existence of thermodynamic loss should be at point 2. When steam enters the cascade, it has a certain initial velocity, so the stagnation parameters at point 0 are h 0 * = h 0 + c 0 2 / 2. Under ideal conditions, the actual enthalpy drop of the cascade is h 0 * h 1, and the actual enthalpy drop of the cascade is h 0 * h 2 Therefore, the heat efficiency of the cascade is
η T = h 0 * h 2 h 0 * h 1 .
(19)
FIG. 13.

Cascade heat efficiency: (a) thermodynamic process and (b) heat efficiency.

FIG. 13.

Cascade heat efficiency: (a) thermodynamic process and (b) heat efficiency.

Close modal

Figure 13(b) shows the distribution of heat efficiency of the cascade under different inlet superheat. With the increase in inlet superheat, the heat efficiency of the cascade increases from 85.6% to 86.5%. By comparing the outlet humidity of the cascade, it can be found that the outlet humidity of case 1 decreases from 2.4% to 1.4%. The change of outlet humidity is similar to that of heat efficiency, which also verifies the Baumann criterion.

In this paper, the modified non-equilibrium condensation equation is embedded into the numerical calculation software by user-defined functions. The accuracy and reliability of the corrected equation are verified by comparing with experimental results. Based on classical experimental cases, by modifying the inlet superheat, the non-equilibrium condensation characteristics in nozzles and cascades are analyzed, and the following conclusions are obtained:

  1. The influence of steam inlet superheat on non-equilibrium condensation is universal. The inlet superheat mainly affects the position of the non-equilibrium condensation region and the size of the nucleation rate. When the inlet superheat increases, the condensation region gradually moves toward the outlet, and the numerical value of the nucleation rate decreases, leading to a decrease in outlet humidity. The minimum amount of super-cooling required for non-equilibrium condensation (about 20 K) is independent of working conditions and models, and this parameter is mainly affected by the characteristics of working fluids.

  2. The local non-equilibrium condensation phenomenon is slightly different for the more complex cascade structure compared to the nozzle. When the inlet superheat is low, the non-equilibrium condensation core is closer to the pressure surface, and when the inlet superheat is high, the condensation core is closer to the suction surface. This phenomenon also leads to non-uniformity in the number of droplets at the outlet.

  3. The change in the position of the condensation core is mainly influenced by the intensity of the shock waves at the trailing edge. When the inlet superheat of the cascade is low, regions with high velocity gradients are concentrated near the pressure surface at the trailing edge, while when the inlet superheat is high, they are mainly concentrated near the suction surface, where there is a reflection of shock waves. The change in inlet superheat alters the angle of the swallowtail wave at the trailing edge of the cascade. As the inlet superheat increases, the angle gradually decreases.

  4. The increase in inlet superheat results in a smaller droplet mass fraction at the outlet of the cascade. By analyzing the heat efficiency of the cascade, it is found that when the inlet temperature increases by 20 K, the heat efficiency of the cascade improves by approximately 1%.

This work was supported by the National Natural Science Foundation of China (Grant No. 51776217) and the Fundamental Research Funds for the Central Universities (Grant No. 2024QN11075).

The authors have no conflicts to disclose.

Di Liang: Conceptualization (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Zining Ji: Software (equal). Yimin Li: Funding acquisition (lead); Writing – original draft (equal); Writing – review & editing (equal). Zhongning Zhou: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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