Numerical study of the heterogeneous condensation effect on the steam turbine performance

The investigation of the loss and efficiency of steam turbine holds immense significance in improving the production of electric energy as a pivotal power conversion device in the electric power industry. However, during the expansion of steam in the steam turbine, the existence of heterogeneous particles leads to the heterogeneous condensation, resulting in a significant reduction in the turbine efficiency and safety of its operation. This study investigates the impact of heterogeneous condensation flow on the performance of steam turbines. First, a condensation model is developed, and numerical calculations are performed using the Bakhtar stator blade cascade. The validity of the proposed model is verified by comparing its results with existing experimental data. Then, the adiabatic flow (non-condensing), the homogeneous condensation flow, and the heterogeneous condensation flow on solid particles with a radius of 10-8[m] and particle concentration of 1015 and 1016[1/kg] are employed to investigate the effect of each flow type on steam turbine performance, and the loss, power, and efficiency in the turbine are detailedly and quantitatively calculated. The results show that in the presence of heterogeneous particles, increasing particle concentration appropriately can effectively reduce the loss caused by condensation and improve thermal efficiency.


Introduction
The steam turbine is a crucial energy conversion device that is widely applied in power generation, industrial production, and aerospace industries [1].The efficiency and loss issues of steam turbines have always been a focal point of research due to their direct impact on energy utilization efficiency and system performance improvement [2].The efficiency of a steam turbine refers to the ratio of the useful work generated by the system to the given input energy.Enhancing the efficiency of steam turbines is of great significance in terms of energy conservation, emission reduction, and operational cost reduction.
However, various loss mechanisms exist within steam turbines, including thermal losses, friction losses, leakage losses, and condensation losses [3].These losses result in energy wastage and a decrease in system performance, ultimately affecting the efficiency of the steam turbine.Among these mechanisms, condensation loss is a significant factor in the overall losses of steam turbines.During turbine operation, the energy in the steam is transferred to the turbine as the steam expands, pushing the turbine to work while rapidly lowering its temperature and pressure, resulting in the formation of non-equilibrium condensation on the turbine blade.Once the non-equilibrium condensation flow has formed, the wetness in the back of the stage will gradually increase, forming larger diameter droplets [4][5] [6].High-speed droplets will not only erode the blade but also result in the efficiency of the turbine decreasing, increasing its loss [7].Condensation losses negatively affect steam turbine performance, as widely acknowledged.A 1% rise in the liquid phase content leads to a corresponding 1% reduction in efficiency [8] [9].Hence, it becomes imperative to investigate the influence of condensation-related losses on steam turbine efficiency.Zhang et al. [10] conducted a study to examine the impact of NaCl particle presence on the condensation process and losses in turbine cascades, their findings indicate that the condensation process and losses are significantly influenced by vapor impurities.Wen et al. [11] developed a wet steam model to investigate the phase change process in turbine blade cascades.Their research revealed that the condensation loss resulting from the steam condensation process in the blade cascade amounted to 0.118 [MW].Aliabadi et al. [12] explored the utilization of water droplet injection for controlling two-phase heat transfer and condensation losses in turbine blade cascades.Their study highlighted the effect of droplet radius and quantity on condensation losses.Heiler et al. [13] conducted a study on homogeneous/heterogeneous condensation phenomena in supersonic nozzle flow, and the results indicate that the occurrence of heterogeneous condensation can prevent shock and reduce nonequilibrium losses associated with homogeneous condensation.However, it is worth noting that the aforementioned studies lack a classification of loss types and specific loss magnitudes, which could provide more detailed insights into the efficiency of the system.The primary objective of this study is to bridge the gaps in current research by presenting a comprehensive classification and calculation method for losses attributed to heterogeneous condensation on solid particles in a steam turbine.And a comparison between the heterogeneous condensation process and adiabatic process as well as homogeneous condensation process will be conducted to investigate the impact of heterogeneous condensation on the performance of the steam turbine.These results contribute to a better understanding of the impact of heterogeneous condensation on loss distributions in the examined flows, which is crucial for optimizing the design and performance of related systems.

Governing equations
During the numerical calculations conducted using the ANSYS FLUENT software, several assumptions are made to simplify the model.These assumptions include: 1.The droplet is considered to be surrounded by an infinite number of gases.2. The droplet is assumed to have a spherical shape.3.Both the gas and liquid phases are assumed to experience the same pressure, denoted as  =   =   .4. There is no relative motion between the droplet and the vapor.Under these assumptions, the conservation equations for mass, momentum, and energy can be expressed as follows: The symbol  denotes the source term, where equation ( 5) corresponds to the mass source term due to homogeneous condensation, equation ( 6) represents the droplet source term, and equation ( 7) signifies the mass source term related to heterogeneous condensation.The source terms are defined by the following equations: ℎ =  (9) Where  is the nucleation rate,  is the droplet radius.

Nucleation rate and droplet growth model
The classic expression for the homogeneous nucleation rate, which describes the rate at which droplets form below the critical radius from the supercooled vapor, can be expressed as follows: The Kantrowitz [14] correction  is expressed as: The surface tension  is a crucial parameter in the expression for the nucleation rate.The common value of  for vapor is calculated using the relationship proposed by IAPWS-IF97 [15].It is defined as follows: In IAPWS-IF97, only the variation of  with temperature was considered.Nevertheless, in this study, the Kirkwood-Buff equation is employed to establish the relationship between surface tension and the radius of curvature, given by the following equation: Therefore, the modified nucleation rate equation can be expressed as: * is the critical size, defined as: The growth rate of the droplet is determined by the heat transfer between the droplets and the surrounding vapor, as well as the mass transfer involved in the condensation process.The theory proposed by Young [16] [17], can be defined as: (20) Where ( 20) describes the droplet radius of insoluble particles, and (21) describes the droplet radius of soluble particles.

Turbulence model
The  −  SST turbulence model is a type of Reynolds-Averaged Navier-Stokes (RANS) model used to describe the distribution of velocity and turbulent kinetic energy, as well as the turbulent energy dissipation rate, in turbulent flow fields.It employs two equations, one for modeling turbulent kinetic energy () and another for modeling turbulent energy dissipation rate ().Due to its effectiveness in capturing the distribution of velocity and turbulent energy in turbulent flow fields, the  −  SST model has certain advantages when simulating non-equilibrium condensation processes.This model is based on physical principles and can account for the influence of velocity and turbulent energy in turbulent flow fields on the condensation process, making it more accurate for simulating nonequilibrium condensation.

Model validation
In order to assess the reliability of the modified condensation model in predicting the behavior of nonequilibrium condensation flow in a two-dimensional blade cascade, the Bakhtar [19] blade cascade is employed as a validation case for the condensing steam flow model.As can be seen in Figure 2, the modified model enables more accurate prediction of the pressure distribution on the blade surface, particularly in the region of rapid condensation.Both models exhibit pressure distributions on the pressure side that are reasonably consistent with the experimental data.The original model tends to overestimate the pressure on the pressure side, while the modified model underestimates the pressure values compared to the experimental data.Furthermore, both models successfully capture a small pressure increase near the trailing edge of the pressure side.However, there are two instances of pressure rise on the suction side.The pressure rise near the trailing edge can be attributed to the decrease in flow velocity resulting from the release of latent heat during the rapid condensation of droplets in the condensation region.The pressure surge on the suction side, away from the trailing edge, may be linked to the structural characteristics of the blade.

Estimation of the losses and power in the steam flow with heterogeneous condensation
The loss coefficient and thermal efficiency are first defined before the calculation, as shown in the Figure 3:   The inlet and outlet boundary conditions are determined using the streamline curvature method [20].The specific boundary conditions are detailed in Table 3 and visually represented in Figure 6.These conditions correspond to the total unit power of 216 [MW].
Table 3.The boundary condition for the Baumann rotor.Figure 6.Boundary conditions at the inlet and outlet along the blade length.As the leaf height increases, the nucleation rate of both homogeneous and heterogeneous condensation processes decrease gradually.Furthermore, the heterogeneous condensation process exerts an inhibitory effect on the nucleation rate.The reason behind this phenomenon is that the expansion rate has a significant impact on non-equilibrium condensation flow.An excessively high expansion rate leads to a large resumption time difference, resulting in a high nucleation rate.However, as the blade height increases, the expansion rate decreases gradually, leading to a gradual decrease in the nucleation rate.As can be seen in Figure 8, it can be observed that the heterogenous condensation flow process with solid particles reduces the total liquid mass fraction generated by the turbine.As the concentration of solid particles increases from 10 15 to 10 16 [1/], the total liquid mass fraction also decreases.This indicates that appropriately increasing particle concentration can effectively reduce humidity and minimize losses.Furthermore, with the span increasing, the total liquid mass fraction gradually decreases.The observed phenomenon can be attributed to the gradual decrease in expansion rate as the span increases.As a result, the nucleation rate decreases, resulting in a gradual reduction in the homogeneous liquid mass fraction produced.Ultimately, this leads to an overall decrease in humidity.As depicted in Figure 9, noticeable distinctions are observed in the potential energy loss coefficient (   ) and the entropy loss coefficient (   ) near the rotor tip.Specifically, the flow featuring heterogeneous condensation experiences a significant decreases in the entropy loss coefficient (  ).It is caused by the increase of the entropy at the inlet due to heterogeneous condensation.It decreases the entropy differences between the outlet and inlet.The entropy loss coefficient (  ) and energy loss coefficient (  ) increase close to the hub of the rotor for the flow with heterogeneous condensation.However, there is almost no difference in the kinetic energy loss coefficient (  ) among these four types of flows.     Heterogeneous n het =10 16   (c) Entropy loss coefficient (  ) Figure 9. Distribution of the loss coefficients along the blade length.As depicted in Figure 10, when comparing to adiabatic flow, both homogeneous condensation and heterogeneous condensation flows exhibit a decrease in thermal efficiency and power of 1.9%, 2.7%, and 2.2%, respectively.These findings indicate that the non-equilibrium condensation flow process leads to an unfavorable reduction in the turbine thermal efficiency and power output.Moreover, when compared to homogeneous condensation, the occurrence of heterogeneous condensation on solid particles results in more significant decrease in efficiency and power.However, upon examining flow processes with different particle concentrations, it is observed that the generated power gradually decreases with increasing particle concentration, while the thermal efficiency experiences a slight increase.This phenomenon can be attributed to the release of latent heat during heterogeneous condensation, which weakens the homogeneous condensation process.While heterogeneous condensation itself leads to a decrease in thermal efficiency, the overall improvement in thermal efficiency resulting from the weakening of the homogeneous condensation process outweighs the decrease caused by the heterogeneous condensation process itself, ultimately resulting in an overall increase in thermal efficiency.Therefore, by appropriately adjusting the particle concentration, it is possible to increase the thermal efficiency and reduce losses.

Figure 1 .
Figure 1.The heterogeneous droplet.Such droplets can grow according to the droplet growth equation (17)  ℎ =  +∆ ℎ .The given concentration of these particles,  ℎ , is a constant.For heterogeneous condensation on soluble particles, according to Pruppacher and Klett[18], the chemical activity coefficient of water   can be obtained:

Figure 2 .
Figure 2. The prediction of the blade surface pressure by different models.

Figure 3 .
Figure 3. Definitions of the loss coefficients and efficiency in the expansion flow.To estimate the losses and power in the steam flow with heterogeneous condensation, numerical calculations are conducted for the lower portion geometry of the Baumann stage rotor in the lowpressure (LP) section of a 200 [MW] steam turbine.The specific geometry used in this study is shown in Figure 4.The rotor consists of 92 blades, and the rotational speed is set at 3000 revolutions per minute.

Figure 4 .Figure 5 .
Figure 4. Schematic representation of the LP steam turbine featuring the Baumann stage and 3-D view of Baumann stage rotor.For discretization of the computational domain for the Baumann stage rotor a multi-block structured grid is used.The numerical grid consists of 8 blocks with 240,000 nodes (Figure.5)after the grid independence is verified.

Figure 7 .
Figure 7. Nucleation rate distribution of four different flow processes at various leaf heights.

Figure 8 .
Figure 8.Total liquid mass fraction distribution of four different flow processes at various spans.Four types of flows are considered in this section, which are flow without condensation (adiabatic flow), flow with homogeneous condensation and flow with heterogeneous condensation on an insoluble particles with two values of impurities concentration  ℎ =10 15 [ −3 ] and  ℎ =10 16 [ −3 ].For the heterogeneous flow the particle radius of  ℎ =10 −8 [] is assumed.As depicted in Figure9, noticeable distinctions are observed in the potential energy loss coefficient (   ) and the entropy loss coefficient (   ) near the rotor tip.Specifically, the flow featuring heterogeneous condensation experiences a significant decreases in the entropy loss coefficient (  ).It is caused by the increase of the entropy at the inlet due to heterogeneous condensation.It decreases the entropy differences between the outlet and inlet.The entropy loss coefficient (  ) and energy loss coefficient (  ) increase close to the hub of the rotor for the flow with heterogeneous condensation.However, there is almost no difference in the kinetic energy loss coefficient (  ) among these four types of flows.

Figure 10 .
Figure 10.Values of efficiencies for the numerical calculations of the flow through Baumann stage rotor.
,  ,  and  are the time, velocity vector, density and pressure, respectively.is temperature,  is energy,  and  are Kronocker delta and stress tensor.is effective,   is source term due to condensation and it has following form:  = (  ℎ +   ℎ ) •  (4) Where  = ℎ  − ℎ  is the latent heat.An additional transport equation is employed to characterize the phase transition phenomena occurring during steam condensation, as represented by the following equation:

Table 1 .
Table 1 provides the geometric parameters of the Bakhtar stator blade cascade, while Table 2 outlines the boundary conditions for the blade.Blade geometry parameters.