Numerical Investigation of the Influence of Air Contaminants on the Interfacial Heat Transfer in Transonic Flow in a Compressor Rotor

Abstract

Atmospheric air is a commonly used working fluid in turbomachinery. The air typically contains a certain amount of suspended solid particles, as well as water in the form of vapor or droplets. In the current paper, we focus on the numerical modeling of humid air transonic flow in turbomachinery. In this paper we demonstrate a rarely considered, but as presented herein important influence of air humidity, pollution and liquid water content on the performance of the first stage of the gas turbine compressor and turbofan engine fan (NASA rotors 37 and 67). We also discuss the impact of the interfacial heat transfer associated with steam condensation or water evaporation on the distribution of stagnation parameters at the rotor outlet, the rotor performance, and flow conditions, as well as losses. Results demonstrate the impact of the number of pollution particles and water droplets on the compression process in the analyzed rotors, especially on the Mach number distribution in the blade-to-blade channel. In this paper we highlight that the air pollution and liquid water content, together with such physical phenomena as steam condensation or water droplets evaporation, exert a significant influence on work parameters, losses and efficiency, and thus should be considered in high-velocity airflow simulations.

1. Introduction

Air humidity and its pollution has an impact on the performance of turbomachinery (e.g., compressors, fans, blowers or even wind turbines). Atmospheric air always contains a certain amount of water in the form of vapor (steam) or liquid droplets (fog). The air used as a working fluid in many technical applications is usually not filtered or dried. So far, much research has been conducted on the influence of air humidity on the loss generation in transonic flows in nozzles or around airfoils. However, thorough studies on the impact of air pollution and liquid water content on the flow conditions in compressors or fans are limited.

Steam condensation can be divided into two processes based on the triggering phenomena in transonic flows. Homogeneous condensation is triggered by the spontaneous nucleation occurring when the condensation nuclei reach the critical radius and grow further to form water droplets. In sonic flow conditions, the pressure drop is significant and gas supercooling is favorable for homogeneous condensation. Therefore, condensation may rapidly occur, forming a so-called condensation wave on which, due to the rapid release of the condensation latent heat, significant flow losses arise [1]. Heterogeneous condensation occurs on droplets or solid particles already present in the air (e.g., water droplets or dust particles). The interfacial heat transfer is followed by the release or absorption of latent heat, depending on whether, e.g., a drop is growing or evaporating. In the case of condensation, the latent heat is released to the carrier gas, and on the contrary when liquid droplets evaporate. In both cases, we may observe a significant change of the flow structure, affecting the performance of turbomachinery [2,3]. In high-velocity flows with shock waves, for instance in the compressor rotor flows, the water droplets may evaporate on the shock wave and absorb the latent heat from the surrounding gaseous phase [4]. The sudden heat release tends to occur in homogeneous condensation and the phase change process via the heterogeneous condensation is more gradual and therefore lower thermodynamic losses are expected. However, this may result in droplets of relatively big diameters. The impact of condensation and evaporation on the performance of turbomachinery is strictly connected to the air parameters and the suspended particles, namely, the total pressure, total temperature, humidity, and number and diameter of the particles. The heterogeneous condensation is of major importance if the air contains many suspended foreign nuclei. And additionally, when the air humidity is high the heterogeneous condensation may be dominated, weakening the conditions for the homogeneous condensation by changing the air supercooling value. Humidity is a crucial parameter and is generally given as a nondimensional number, referring to the actual water vapor mass to the maximum vapor mass held by the air. Humidity values typically range from 40% to 50%, however, in certain conditions they can reach 100%. Fog is formed when the water mass exceeds the mass capacity of the air. We stress that the aforementioned factors (i.e., air parameters, humidity, pollution or existence of water droplets) should not be neglected in the study of transonic airflow.

The condensation phenomena in internal and external transonic flows have been the focus of much research, with studies initiated over 30 years ago and continuing to present times [5]. Schnerr et al. conducted pioneering experimental and numerical studies on condensation in transonic internal flows through Laval nozzles [1,6,7] and became a reference for researchers, particularly the kinetic nucleation theorem model [8,9,10]. The non-stationarity of the flow was investigated by Adam, both experimentally and numerically, to determine the influence of relative humidity on the interaction between the shock and condensation waves, as well as the oscillation frequency of condensation in a nozzle with parallel walls [10]. Dykas et al. developed a 3D academic code that was validated against in-house experimental studies, in which the air–water mixture is treated as a single fluid that allows for the reliable computation of external and internal flow with phase change [8,11,12]. The condensation models available there were then recently implemented into commercial software and validated against experimental data, with extensions considering the difference in velocity between humid air and water droplets, i.e., between gaseous and dispersed liquid phase [13,14]. Yamamoto produced valuable contributions regarding simulation condensation techniques in complex 3D structures [15,16]. The computational costs of numerical condensation studies are extremely high, however, the increase in computation power available have led to attempts in simulating the entire rotor with nonuniform circumferential distributions of wetness at the compressor inlet [17]. Moriguchi et al. highlight that the presence of liquid water at the compressor inlet might lead to significant change of static parameters at the rotor outlet [4,18]. Studies on condensation in transonic flows are generally conducted via a single-fluid approach, where the mixture of air and water is treated as a continuous fluid. This approach can be justified if the droplet inertia is negligible. However, if the droplets reach a relatively large size or the flow swirl is significant, a two-fluid approach, in which water droplets are treated as a separate phase, should be considered. Recent work has attempted to model multi-phase humid air and wet steam [17,19,20]. The consideration of steam condensation in the flow modeling in turbomachinery plays an important role. It allows for estimating the flow losses and thus the efficiency correctly. Correct estimation of turbomachinery efficiency has a key influence on the power installations efficiency as well as on the final energy consumption. Kermani et al. provided insight into two types of losses induced by wet-steam condensation: (i) thermodynamic losses due to the nucleation; and (ii) losses related to the perturbation of temperature between phases on a normal shock [21]. The condensation process not only occurs in humid airflow, and hence similar models have been employed to consider the phase change in wet steam for nozzle shape optimization or to investigate dehumidification techniques [22,23,24,25,26]. Analogous models have also been employed to investigate complex phenomena such as the supersonic separation of CO₂ [27].

In the current paper, we employed ANSYS Fluent (ver.2020R1, ANSYS Inc., Canonsburg, PA, USA), a commercial computational fluid dynamics (CFD) software, for the numerical analysis. The phase change phenomenon was implemented via User Defined Functions (UDFs). We consider the homogeneous and heterogeneous condensation influence on the first-stage rotor performance of the gas turbine compressor and a turbofan engine fan. In this paper we present the influence of air pollution in form of solid and liquid particles on the turbine’s compressor rotor and turbofan’s fan performance, which is an open, not-investigated scientific issue that needs further study, especially as the performance of the first stages of a turbine affects the work of a whole turbine. Conducted experiments demonstrate the impact of air humidity, liquid droplets, and solid particles on the work of typical turbomachinery. The appearance of solid particles (HS) and liquid droplets (HW) drives the heterogeneous condensation. The liquid droplets input the additional amount of latent heat that must be absorbed when the liquid evaporates on the shock wave. Results presented in this study provide an overall view of the air relative humidity influence as well as the phase change phenomena on the transonic flow structure, while focusing on the losses and performance of the first-stage rotor of the gas turbine compressor and a fan of a turbofan. On the basis of the results presented herein, it can be clearly stated that the influence of the above-mentioned phenomena on turbomachinery performance is significant, it should not be neglected, and further detailed studies are required.

2. Physical and Numerical Model

We employed the single-fluid approach, whereby the mixture of air and water is treated as a continuous fluid and the slip velocity between the phases is neglected. This is a common simplification and can be justified as the droplets are small, thus their inertia is negligible [18]. The conservation equations were formulated for compressible flow and the phase change phenomena were introduced via the ANSYS Fluent UDFs. Moreover, the Reynolds-Averaged Navier-Stokes Equations with Eddy-Viscosity Turbulence Models were adopted to model the time-averaged flow quantities. The k-ω Shear-Stress-Transport (SST) model proposed by Menter was used to model the turbulence effects [28]. Additional governing equations were required to model the phase change due to homogeneous and heterogeneous condensation. Thus, we included PDEs describing the number of droplets, n, in homogeneous condensation, and the liquid mass fraction, y, due to the homogeneous and heterogeneous condensation. The number of solid particles in a heterogeneous process is constant and is based on the literature [29]. The additional governing equations take following form:

$$\frac{\partial \rho y_{hom}}{\partial t}+\frac{\partial }{\partial x_j}\left(\rho v_j{y}_{hom}\right)=S_{y_{hom}}$$

(1)

$$\frac{\partial \rho n_{hom}}{\partial t}+\frac{\partial }{\partial x_j}\left(\rho v_j{n}_{hom}\right)=S_{n_{hom}}$$

(2)

$$\frac{\partial \rho y_{het}}{\partial t}+\frac{\partial }{\partial x_j}\left(\rho v_j{y}_{het}\right)=S_{y_{het}}$$

(3)

where $\rho$ is the density, t is time, $v$ is the velocity vector, S represents source term, respectively (Equation (1)) the source of mass of the condensed liquid due to the homogenous condensation, (Equation (2)) the source of droplets per kilogram of air generated due to the nucleation process, (Equation (3)) the source of mass of water owing heterogeneous condensation.

The source terms in Equations (1)–(3) have following form:

$$S_{y_{hom}}=\frac{4}{3}\pi \rho {\rho }_l{r}^{*}{}^3{J}_{hom}+4\pi \rho {\rho }_l{n}_{hom}{r}_{hom}^2\frac{d{r}_{hom}}{dt}$$

(4)

$$S_{n_{hom}}=\rho J_{hom}$$

(5)

$$S_{y_{het}}=4\pi {\rho }_l\rho n_{het}{r}_{het}^2\frac{d{r}_{het}}{dt}$$

(6)

where r* is critical radius, J is nucleation process, r is the droplet radius, which is calculated based on the water mass and number of droplets.

The homogeneous condensation process is triggered by nucleation, whereby the nuclei spontaneously form in the fluid and continue to grow if the critical radius is reached. In our model, the critical radius was based on the Kelvin equation [30]:

$$r^{*}=\frac{2\sigma }{{\rho }_{l}R{T}_v\ln \left(\frac{p_v}{p_s}\right)}$$

(7)

where $\sigma$ is the surface tension, $R$ is the individual gas constant, T is the temperature and $p_v/p_s$ is the ratio of vapor pressure to saturation pressure. The Classical Nucleation and Molecular-Kinetic Droplet Growth theories (Equations (8) and (9)) were integrated to describe the nucleation process and the growth of the droplets:

$$J_{hom}=C\sqrt{\frac{2\sigma }{\pi m_v^3}}\frac{{\rho }_v}{{\rho }_l}{e}^{\left(-\frac{4\pi r^{*2}\sigma }{3k{T}_v}\right)}$$

(8)

$$\frac{dr}{dt}=\frac{1}{{\rho }_l}\frac{p_v-p_s}{\sqrt{2\pi RT}}$$

(9)

where $m_v$ is water molecular mass, $k$ is Boltzman constant and C is the Kantrowitz correction factor. The Kantrowitz correction was considered to account the temperature difference between continuous and dispersed phases [31,32,33]. The Molecular-Kinetic Droplet Growth model, known as the Hertz–Knudsen model is recently the most commonly used model for condensation study in humid air transonic flows [34,35,36,37]. This model is especially recommended for the flows with high Knudsen number, i.e., when the droplet radius is sufficiently small in reference to the mean-free path of the vapor molecules, what is a case in homogeneous condensation in humid air [18]. All equations were implemented via the UDFs. For more information regarding the solver settings, numerical techniques, and the implementation of the condensation, we refer the reader to our previous work [5,11,12,13,14].

It should be mentioned, that User Defined Real Gas Model was used for computations of the fluid properties, i.e., density, specific heat, viscosity etc. [38]. Using this approach allows to consider the change of fluid properties due to the change of state of water in the flow.

As the Spatial Discretization Scheme, the pressure-based solver available in ANSYS Fluent was used. The MUSCL method was employed to achieve the third-order accuracy in space. For all presented in the paper results, the steady-state solution was obtained by means of the implicit time integration method. As a convergence criterion of the iteration process, it was assumed that the mass flow rate difference between rotor inlet and outlet was less than 0.1%. The absolute total parameters at the inlet, total pressure and total temperature, as well as static pressure at the rotor outlet were assumed as the boundary conditions.

3. Results and Discussion

3.1. Validation

The proposed model was validated for internal and external 2D flows with in-house experiments and data from the literature. The validation procedure is described in detail in our previous work, however, for clarity we present the boundary conditions in

Table 1. Boundary conditions for validation purposes.

	${\mathit{p}}_{0,\mathit{i}\mathit{n}}, \mathbf{Pa}$	${\mathit{T}}_{0,\mathit{i}\mathit{n}}, \mathbf{K}$	f, rad/s	${\mathit{p}}_{\mathit{o}\mathit{u}\mathit{t}}, \mathbf{Pa}$	Blade Length, m	Number of Blades
Rotor 37	101,325	288.15	1800	120,000	0.0749	36
Rotor 67	101,325	288.15	1680	105,000	0.1320	22

[13,14,39].

The current paper focuses on numerical investigations of homogeneous and heterogeneous condensation influences on the first-stage performance of the gas turbine compressor rotor and the turbofan engine fan. We base our methods and the computational domains for NASA rotors 37 and 67 on previous work available in literature [40,41]. The inflow and outflow for NASA rotor 37 are 5 cm and 8 cm, respectively, whereas the inflow and outflow for NASA rotor 67 are 3 cm and 4 cm, respectively. Note that the tip clearance was neglected as the focus of the study is the condensation phenomena. We generated three numerical meshes for each rotor in accordance with literature guidelines and conduced mesh-dependence study [42]. The meshes named Mesh 1, Mesh 2 and Mesh 3 have, respectively, 0.25, 0.60 and 1.30 million elements for rotor 37 and 0.50, 1.00 and 1.70 million of elements for rotor 67. Despite the fact that Mesh 2 already provides a reasonable solution, and a further increase of mesh density does not improve the convergence of the numerical results with available experiment data, Mesh 3 for both rotor 37 and rotor 67 were selected for target studies due to the better discretization near to the wall. It is of major importance to keep the good discretization of the boundary layer regions, very crucial by modeling the interaction of condensation shock or shock wave with solid wall. It has to be mentioned, owing to the use of the k-ω SST turbulence model, the boundary layer was introduced to ensure y⁺ $\approx$ 1. Figure 1 presents the computational domain, mesh and the compression process line in an adiabatic compressor. Figure 2 compares for meshes of different density the ratio of the pitch-wise averaged absolute total pressure at the control plane downstream of the blade, p₀₂ to the absolute total pressure at the inlet, p₀₁, obtained numerically and experimentally [43]. We may conclude that even for the coarse numerical mesh, Mesh 1, the comparison of the CFD results with the experiment is satisfactory. However, the condensing flow modeling requires fine mesh to capture the condensation wave properly. The outlet absolute total pressure of rotor 37 was measured at the section located 10.67 cm downstream of the blade hub leading edge. The inlet control plane was located 4.19 cm upstream of the blade hub leading edge. The outlet total pressure for rotor 67 was measured at the section placed 11.01 cm downstream of the blade hub leading edge, while the inlet control plane was located 2.47 cm upstream of the blade hub leading edge. These comparisons demonstrate the effectiveness of the proposed CFD method in modelling the flow in a transonic compressor rotor.

Table 2. Experimental (EXP) and numerical (CFD) mass flow rate, total pressure and total temperature ratio.

	Rotor 37				Rotor 67
	$\dot{\mathit{m}}$, kg/s	p02/p01, -	T02/T01, -	$\mathit{\eta }$, -	$\dot{\mathit{m}}$, kg/s	p02/p01, -	T02/T01, -	$\mathit{\eta }$, -
EXP	20.19	2.11	1.27	0.88	32.31	1.73	1.19	0.89
CFD	20.41	2.09	1.27	0.87	31.84	1.71	1.19	0.87
Error, %	1.09	0.95	0	1.14	1.45	1.16	0	2.25

reports the overall mass flow rate, total temperature and total pressure ratios determined from the CFD simulations and experiments. The <1.5% discrepancy between the numerical and experimental results may be attributed to the omission of the tip clearance in the numerical analysis. The good agreement between the numerical and experimental results lays solid foundations for further studies.

3.2. Comparative Study

Numerical research on the heterogeneous condensation influence in moist air transonic flow was performed following the two approaches described in our previous work [8,11,12,13,14]. The HS and HW contamination are assumed to be solid particles and liquid, respectively. Investigations were conducted for varying droplet numbers of the same size, followed by varying mass fractions. Cases HS1, HS2 and HS3 employ suspended solid particle numbers, $n_{het}$, equal to ${10}^{13}$, ${10}^{14}$ and ${10}^{15}$ per kg of air, respectively, based on the data of PM10 air pollution available in the literature [29], where HS1, HS2 and HS3 can be considered as air of good, average and bad quality respectively. It has to be mentioned that the particle radius is constant and equal to ${10}^{-7}$ m, therefore smaller than standard PM10, however it can be justified as the PM10 consists of particles with radius up 10⁻⁵ m. In the HW approach, cases HW1, HW2 and HW3 employ suspended droplets numbers, $n_{het}$, equal to ${10}^{11}$, $2.5\times {10}^{11}$ and ${10}^{12}$ per kg of air, respectively, which result in standard mass values of water suspended in the air across different weather conditions (e.g., cloud and fog) [44]. The diameter of the droplets at the inlet is equal to ${10}^{-6}$ m. The air is assumed to be humid, with a relative humidity, φ, equal to 70% and temperature set at 303 K.

Table 3. Boundary conditions.

	${\mathit{p}}_{0,\mathit{i}\mathit{n}}, \mathbf{Pa}$	${\mathit{T}}_{0,\mathit{i}\mathit{n}}, \mathbf{K}$	Φ, %	f, rad/s	${\mathit{p}}_{\mathit{o}\mathit{u}\mathit{t}}, \mathbf{Pa}$		${\mathit{n}}_{\mathit{h}\mathit{e}\mathit{t}},\frac{1}{\mathbf{k}\mathbf{g}}$	r, m
Rotor 37	101,325	303.15	70	1800	120,000	HS1	${10}^{13}$	${10}^{-7}$
						HS2	${10}^{14}$	${10}^{-7}$
						HS3	${10}^{15}$	${10}^{-7}$
Rotor 67	101,325	303.15	70	1680	105,000	HW1	${10}^{11}$	${10}^{-6}$
						HW2	2.5 × 1011	${10}^{-6}$
						HW3	${10}^{12}$	${10}^{-6}$

presents summarized boundary conditions for comparative study.

3.3. Rotor Performance Assessment

For assessing the influence of the different flow conditions on the rotor performance the value of the specific work was used. The specific work includes in its definition information of the thermodynamic state of the compression process. Additionally, the total pressure at the rotor outlet in the absolute frame was compared for analyzed test cases. Figure 3 and Figure 4 present the influence of steam condensation on the specific work and distribution of absolute total pressure at the domain outlet. The influence of HS and HW on the total pressure profile at the outlet is observed to be significant. The total pressure increases in the region close to the hub and decreases in the upper blade component. This is attributed to the higher velocity at the blade tip, which facilitates condensation and a stronger shock wave. Variations can be observed between specific work cases, and the heterogeneous condensation resulting from the solid particles in the air influences the specific work value along the blade span. For low particle numbers, the specific work is greater than for the adiabatic case. This is attributed to the higher specific heat of the humid air. Due to condensation-induced losses, the specific work decreases with the increasing number of suspended particles in the flow. The influence of fluid droplets at the rotor inlet is even more significant. In case of low liquid water mass fraction at the inlet to the rotor, the effect is not significant, however, it increases with the number of suspended droplets, which causes the specific work to decline with increasing liquid content at the inlet. This is induced by the temperature decrement due to the latent heat absorption in the evaporation process.

Figure 5 depicts the mass averaged values of specific work, l, and the ratio of the outlet to the inlet relative total pressure. The specific work is a difference between total specific enthalpies at the outlet and at the inlet to the rotor:

$$l=h_{02}-h_{01}$$

(10)

whereas the total specific enthalpy is computed as follows:

$$h_0=h+\frac{1}{2}{c}^2$$

(11)

It has to be pointed out, that the specific enthalpies of steam as well as water are calculated using the IAPWS formulations, and that the specific enthalpy is being computed for the humid air and water mixture:

$$h=h_a\left(1-y_{max}\right)+h_v\left(y_{max}-y\right)+h_{l}y$$

(12)

The presence of solid (liquid) particles increases (decreases) the specific work. The enhanced specific work in the flow with suspended solid particles is attributed to the air humidity, and consequently, the increased air specific heat. The reduced specific work for the liquid droplets at the inlet is caused by the absorption of the latent heat via the evaporation on the shock wave.

3.4. Flow Field Analysis

The steam condensation phenomenon significantly alters the flow field structures. Figure 6 shows a 3D quantitative presentation of mass fraction of water in the rotor of a compressor, it presents the HW3 condensation case in rotor 67. It is noticeable that the condensation occurs mainly in the mid and tip parts of the rotor; therefore, to ensure this article is easy to read, we focus only on these areas.

Figure 7 and Figure 8 present the mass fraction of the liquid phase and Mach number contours for HS and HW, respectively. The liquid water mass fraction maps in Figure 7 indicate condensation occurs on the solid particles and evaporation on the shock wave. As the number of solid particles in the air increases, the condensed water content also increases, which subsequently releases the latent heat and reduces the Mach number. The enhanced condensed water mass induced by the increasing number of suspended particles is attributed to the more favorable condensation conditions. More specifically, condensation occurs on the solid particles suspended in the air.

Figure 8 demonstrates that condensation occurs on the suspended droplets entering the computation domain. The water then evaporates on the shock wave. For the flow with liquid droplets at the inlet, the lower number of droplets results in less favorable condensation conditions compared to the flow with suspended solid particles. Thus, the condensed water mass upstream of the shock wave is lower compared to the case with suspended solid particles. The impact of the evaporation is principally observed in the region of the shock wave. The evaporating fluid absorbs the latent heat, which reduces the temperature and the shock wave moves downstream of the blade-to-blade channel.

Figure 9 and Figure 10 show the comparison of the pressure distribution along the blade surface at 10%, 50% and 90% span for rotor 37 and rotor 67, respectively, and considered are adiabatic, HS and HW cases. The distribution of the static pressure on the blade surface at 90% span slightly differs in comparison to the data available in the literature, especially for rotor 67. It is owing to neglect the tip gap. Regardless of the discrepancies near to the shroud region for the adiabatic case, the impact of heterogeneous condensation is clear and that is an aspect we focus on in this study. Figure 9 shows the influence of HS and HW condensation on the blade loading of rotor 37, it is visible that for HS cases, the shock wave moves upstream at the pressure side of the blade. If the hub region of the suction side of the blade is considered the shock wave moves downstream, while on the other hand, for the shroud region the shock wave moves upstream. If the liquid droplets are present at the inlet, HW cases, then the shock wave on both sides of the blade moves downstream the blade-to-blade channel; this is caused by the heat release due to the evaporation, which drives the drop of fluid temperature.

Figure 10 shows the influence of HS and HW condensation on the blade loading for rotor 67. It is visible that for HS cases, the shock position on the pressure side is affected only in the near shroud region and it moves downstream, on the suction side the condensation delays the shock wave. The influence of liquid droplet presence at the inlet on the blade loading for rotor 67 is the same as in rotor 37. The character of pressure distribution along the blade at span 0.5 (Figure 9 and Figure 10) fits the Mach number contours (Figure 7 and Figure 8).

Figure 11 presents the water mass fraction and Mach number for HW3 and HS3 condensation case at 0.9 span of rotor 37 and rotor 67. As it can be seen, the condensation process near the tip of the rotor is more gradual than at 0.5 span and the condensation starts further upwind the rotor, however, the droplet growth is slower, and therefore, the total amount of condensed water is lower in comparison to mid-span area.

3.5. Losses Assessment

In the condensing flow of moist air, we deal with both aerodynamic and thermodynamic losses. Aerodynamic losses such as those caused by turbulence or shock waves can be intensified or weakened in presence of the condensation phenomenon. Thermodynamic losses caused by phase change processes can be dominant and they may affect significantly the entropy change. With the increment of air humidity at the inlet to the rotor, the efficiency of the compression process in first stage of compressor or fan of turbofan engine decreases what is connected with the increasing specific entropy of the fluid due to the condensation. The homogeneous condensation occurring in the region near to the blade affects the local properties of the fluid and induces additional losses. It has to be pointed out that the specific entropies of steam as well as water are calculated using the IAPWS formulations. The fluid specific entropy is computed as the sum of dry air, vapor and liquid-specific entropies from following equation:

$$s=s_a\left(1-y_{max}\right)+s_v\left(y_{max}-y\right)+s_{l}y$$

(13)

where $s_a$, $s_v$, $s_l$ are the specific entropies of dry air, of water vapor, and of condensate, respectively.

The isentropic efficiency is being computed as follows:

$$\eta =\left({\left(\frac{p_{02}}{p_{01}}\right)}^{\left(\frac{\gamma -1}{\gamma }\right) }-1\right)/\left(\frac{T_{02}}{T_{01}}-1\right)$$

(14)

where $T_{01}$ and $T_{02}$ are the inlet and outlet total temperature, $p_{01}$ and $p_{02}$ are the inlet and outlet total pressure and $\gamma$ is the specific heats ratio.

Figure 12 shows the influence of the air humidity on the specific entropy increment in the blade-to-blade channel and its efficiency. With increasing relative humidity, the entropy increment in the blade-to-blade channel is augmented, it is due to the condensation losses, influencing flow parameters, and increasing importance of vapor entropy in mixture entropy. The influence of homogeneous condensation on the efficiency of rotor 37 is negligible, while the efficiency decrement, due to the presence of humidity in the working fluid, in rotor 67 is significant. The influence of homogeneous condensation on rotor 67 efficiency is higher in comparison to the influence of homogeneous condensation on rotor 37 efficiency, as the conditions for condensation in rotor 67 are more favorable. Moreover, the compression work realized by the rotor 67 is lower compared to the compression work realized by rotor 37, therefore the effect of release or absorption of comparable amount of latent heat, due to the condensation and evaporation, has a greater impact on the flow parameters, so on the efficiency of the machinery.

Figure 13 shows the influence of the solid particle and water droplet presence at the inlet to the compressor on the specific entropy increment and rotors efficiency. With the increasing number of suspended solid particles, the heterogeneous condensation becomes a major condensation process. The heterogenous condensation triggered by the presence of foreign nuclei in the fluid accelerates the condensation upstream the rotor what decreases the importance and speed of nucleation and droplets growth due to homogeneous process. The same phenomena occur if the air contains suspended liquid droplets, and moreover, with the increasing water content at the inlet to the rotor, more latent heat is being absorbed during its evaporation om the shock wave, what leads to significant drop of fluid total temperature. The adopted definition of isentropic efficiency (Equation (14)) may result in values above 1 when dealing with evaporation of a large amount of water, the latent heat evaporation should be taken into account here. As a result, a significant increment of the efficiency occurs, in reference to the adiabatic case. This is of major importance, especially in fans, where the total temperature increment is relatively low in comparison to the compressor, and the absorption of latent heat during evaporation exerts a strong effect on the flow structure and turbomachinery efficiency.

Figure 14 shows the influence of the liquid water content on the performance curve of rotor 37. The performance curve for adiabatic and for HW3 cases are confronted with experimental results. The adiabatic simulation overestimates the compression in the rotor stage, due to neglecting tip losses. If the tip losses are considered, the adiabatic curve would move to a lower pressure ratio, thus would fit better to the experimental results. However, in this study, the focus is on the shift of the performance curve due to the humidity and phase change phenomena. If the liquid water is present at the inlet to the compressor, the compression ratio drops due to the latent heat absorption on the shock wave, which leads to the decrement of fluid pressure and temperature. Moreover, it has to be mentioned that assuming the air humidity and presence of water at the rotor inlet, we have to be aware of the fact of the fluid density change, which consequently must affect the mass flow rate as well. In considered HW3 case, the drop of fluid density due to the presence of steam is much higher than fluid density rise due to the presence of water droplets at the inlet. It leads to the fluid density decrease and finally to the drop of mass flow rate. The choking mass flow rate ${\dot{m}}^{*},$ based on the experimental study, is equal to 20.93 kg/s, ${\dot{m}}^{*}$ obtained numerically for adiabatic flow equals 20.83 kg/s, and ${\dot{m}}^{*}$ for HW3 case is equal to 19.87 kg/s.

4. Conclusions

In the current paper, we investigate the influence of suspended particles and liquid water content on the flow in the first compressor rotor and fan of a turbofan engine. Numerical experiments were performed via ANSYS Fluent and a corresponding condensation model extension with the implementation of UDFs. The homogeneous and heterogeneous condensation models, triggered by the presence of suspended solid particles and liquid water, respectively, are based on the classical nucleation theorem and kinetic droplet growth. The models were validated against data available in the literature.

Results demonstrate the significance of humidity, condensation, and evaporation on the flow in turbomachinery. If humidity is considered, the fluid properties are altered and the specific heat of the air increases. This enhances the specific work in the rotor. The presence of the solid particles triggers the condensation, which consequently affects the pitch-wise-averaged total pressure profile and the total temperature. This has a negative impact on the rotor-specific work. The liquid water content at the inlet proves to be problematic as the liquid tends to evaporate on the shock wave in the blade-to-blade channel. During evaporation, the latent heat is absorbed by the fluid and the temperature of the air decreases, which is linked to the reduced total temperature and entropy at the rotor outlet. Our results are of major importance for cases with a large number of solid particles or relatively high liquid water content. Suspended solid or liquid particles in the air exert a strong impact on the flow field structure and the performance of turbo engines. The condensation triggered by the solid particles decreases the Mach number in the flow, while the presence of liquid droplets in the air shifts the shock wave downstream of the blade-to-blade channel. Moreover, it has to be mentioned that the condensation phenomenon, for both HS and HW cases affects the blade loading. Finally, it has to be mentioned that when the humidity and suspended particles in the air are considered, the rotor performance curve is affected. This is due to the change of flow rate owing to the influence of humidity on the fluid density, and the phase change affects the compression process, so the compression ratio decreases.

In summary, both homogeneous and heterogeneous condensation exhibit non-negligible impacts on the performance of the first stage of compressor and turbo engine fans within the transonic flow regime. The influence of air pollution and water content exert a significant impact on the flow. The presence of suspended particles in the air impacts the specific work of the rotor, the total parameter distribution and the flow field. Considering the importance of the aforementioned problems, we recommend taking into account the humidity and pollution of air during numerical experiments on the transonic flow of air and the turbomachinery design process.