Valentina Zaccaria
email: valentina.zaccaria@netl.doe.gov

David Tucker

Alberto Traverso

Cold-air Bypass Characterization for Fuel Cell Thermal Management in Fuel Cell Turbine Hybrids

Keywords:

The effect of cathode airflow variation on the dynamics of a fuel cell gas turbine hybrid system was evaluated using a cyber-physical emulator. The coupling between cathode airflow and other parameters, such as turbine speed or pressure, was analyzed comparing the results at fixed and variable speed. In particular, attention was focused on the fuel cell temperatures, since cathode airflow is generally employed for thermal management of the stack. A significant difference was observed in the two cases in terms of turbine inlet, exhaust gas, cathode inlet, and average cell temperatures. When the turbine speed was held constant, a change in cathode airflow resulted in a strong variation in cathode inlet temperature, while average cell temperature was not significantly affected. The opposite behavior was observed at variable speed. The system dynamics were analyzed in details in order to explain this difference. Open loop response was analyzed in this work for its essential role in control systems development. But, the significant difference shown in this work proved that some trends are not adequately captured with a standard system identification procedure, because of the high coupling between turbine speed and cathode airflow. Cold air valve bypass position also showed a strong impact on surge margin in both cases.

INTRODUCTION

The hybridization of high temperature fuel cell and gas turbine facilitates very high efficiency, increased system flexibility, and extended fuel cell component lifetime [START_REF] Mueller | Synergistic integration of a gas turbine and solid oxide fuel cell for improved transient capability[END_REF][START_REF] Veyo | Tubular Solid Oxide Fuel Cell/Gas Turbine Hybrid Cycle Power Systems: Status[END_REF][START_REF] Rao | Analysis strategies for tubular solid oxide fuel cell based hybrid systems[END_REF][START_REF] Tucker | Determination of the Operating Envelope for a Direct Fired Fuel Cell Turbine Hybrid Using Hardware Based Simulation[END_REF][START_REF] Tucker | SOFC lifetime assessment in gas turbine hybrid power systems[END_REF]. Recovering the waste heat from the fuel cell with a gas turbine, it is possible to enhance the power production and use the compressor to pressurize the fuel cell at no additional cost, reaching efficiency levels that are not possible with the separate technologies [START_REF] Rao | Analysis strategies for tubular solid oxide fuel cell based hybrid systems[END_REF]. However, the interaction between the two coupled systems presents new significant challenges in terms of controls. In a hybrid system different variables, for instance turbine speed and cathode airflow, are strongly coupled, i.e. a change in one variable affects the second one and vice-versa. In addition, the time response of the two main components, fuel cell and gas turbine, are substantially different [START_REF] Liese | Technical Development Issues and Dynamic Modeling of Gas Turbine and Fuel Cell Hybrid System[END_REF]. These aspects result in more complicated control of the system.

There are only a few hardware-based facilities in the world designed to study the coupling of a gas turbine and a solid oxide fuel cell, and the associated control issues [START_REF] Ferrari | Hybrid simulation facility based on commercial 100 kWe micro gas turbine[END_REF][START_REF] Hohloch | Micro gas turbine test rig for hybrid power plant application[END_REF]. The Hybrid Performance (HyPer) facility, located in the U.S. Department of Energy, National Energy Technology Laboratory, is the only one designed to evaluate high speed dynamic phenomena in hybrid systems. The facility matches hardware and virtual components to take advantages from model flexibility and experimental accuracy [START_REF] Tucker | The Role of Solid Oxide Fuel Cells in Advanced Hybrid Power Systems of the Future[END_REF].

In order to ensure safe operability and optimize the system efficiency, the characterization of the different actuators was performed on the HyPer facility with the final goal of implementing control systems capable of facing the new challenges [START_REF] Tucker | Characterization of Bypass Control Methods in a Coal-based Fuel Cell Turbine Hybrid[END_REF][START_REF] Pezzini | Control impacts of cold-air bypass on pressurized fuel cell turbine hybrids[END_REF][START_REF] Pezzini | Decentralized Control Strategy for Fuel Cell Turbine Hybrid Systems[END_REF][START_REF] Zhou | Evaluation of cathode air flow transients in a SOFC/GT hybrid system using hardware in the loop simulation[END_REF][START_REF] Zhou | Evaluation of Compressor Bleed Air Transients in a Fuel Cell Gas Turbine Hybrid System Using Hardware Simulation[END_REF]. In particular, cathode airflow management was considered extremely important to keep cell temperature and temperature gradients in a safe range [START_REF] Inui | Analytical Investigation on Cell Temperature Control Method of Planar Solid Oxide Fuel Cell[END_REF], but its variation showed to affect several other parameters in the system, such as turbine speed, pressure, and surge margin [START_REF] Pezzini | Control impacts of cold-air bypass on pressurized fuel cell turbine hybrids[END_REF][START_REF] Zhou | Evaluation of cathode air flow transients in a SOFC/GT hybrid system using hardware in the loop simulation[END_REF][START_REF] Traverso | Turbomachinery for Air Management and Energy Recovery in Fuel Cell Gas Turbine Hybrid Systems[END_REF]. In previous works the system response to a variation in cathode airflow was evaluated in an open loop configuration, where no control action was applied to regulate turbine speed or turbine inlet temperature [START_REF] Zhou | Evaluation of cathode air flow transients in a SOFC/GT hybrid system using hardware in the loop simulation[END_REF][START_REF] Zhou | Transfer Function Development for Control of Cathode Airflow Transients in Fuel Cell Gas Turbine Hybrid Systems[END_REF]. In these works, cathode airflow was changed via manipulating a hot air bypass, and the coupling between airflow and turbine speed was shown when the fuel cell model was in the loop. The nonlinear behavior of some of the actuators was also investigated without the real-time simulation driving the system [START_REF] Pezzini | Control impacts of cold-air bypass on pressurized fuel cell turbine hybrids[END_REF][START_REF] Pezzini | Decentralized Control Strategy for Fuel Cell Turbine Hybrid Systems[END_REF]. A complete characterization of these actuators was considered very important to study possible control actions. In this work, the impact of cathode airflow variation on the system dynamics was studied comparing two different scenarios: in the first case turbine speed was free to change (open loop configuration), in the second one it was kept constant via manipulating the electric load imposed to the turbine (closed loop configuration).

Open loop tests are useful to study the transient dynamics of the system, but the system response when the speed is controlled is also important since it represents a realistic plant configuration. System identification in closed loop can give a more insight knowledge of the system dynamics. A more complete system identification helps to develop accurate control strategies.

FACILITY DESCRIPTION

Two experiments were carried out using a cyberphysical simulation approach. In this approach the hardware components of the HyPer facility were coupled with a numerical real-time model of a solid oxide fuel cell (SOFC), which drove the only heat source of the system. The hardware components of the fuel cell emulator included a natural gas combustor to simulate the heat from the fuel cell and two vessels to emulate the volumes of the fuel cell system, and they were physically coupled to a recuperated gas turbine.

A schematic representation of the plant is presented in Figure 1, where all the dash lines represent the numerical model. The model also includes a dynamic module of a biomass gasifier, not included in this work [START_REF] Traverso | Preliminary Experimental Results of Integrated Gasification Fuel Cell Operation using Hardware Simulation[END_REF].

The compressed air was preheated by the turbine exhaust into the two counter-flow heat exchangers, before feeding the cathode and air manifold volumes emulator, which consisted of a pressurized vessel. Here the pressure dynamics of the fuel cell were simulated. After the volume, a natural gas combustor generated the amount of heat calculated in the fuel cell model, which was used to drive a 120 kW turbine. A second vessel was located after the combustor to simulate the volume of the post-combustor in the fuel cell system. Three bypass valves were used for control purposes: a bleed air valve, which blew compressed air into the atmosphere, a cold air bypass, which diverted air from the compressor outlet to the post-combustor volume, and a hot air bypass, which was located at the heat exchangers outlet and bypassed the fuel cell emulator. Cold air valve, which was employed in this work, was observed to have a strong impact on cathode airflow, turbine inlet temperature, turbine speed, system pressures and surge margin. It was considered a critical actuator, because it is very effective in varying fuel cell temperature distribution. Hence, its complete characterization was considered fundamental for system control.

MODEL DESCRIPTION

The one-dimensional model simulates a planar, co-flow, anode-supported SOFC. It employs a finite difference approach to solve the thermal equations and a finite volume approach for the electrochemistry. Details on the numerical model can be found in previous work [START_REF] Hughes | A Real-Time Spatial SOFC Model for Hardware-Based Simulation of Hybrid Systems[END_REF]. Data are fed every 80 ms into the model from sensors embedded in the physical system. Distributed profiles of all the cell parameters (composition, temperatures, current density, overpotentials, and others) are calculated at each sample time. The model calculates in real-time the waste heat transferred to the turbine, as shown in Eq. 1, where hin is the specific enthalpy change of the air at the cathode inlet, multiplied by cathode inlet airflow min, and hout is the specific enthalpy change of the exhaust at the post-combustor outlet, where the unutilized fuel is burned, times the exhaust gas flow rate mout.

in in out out h m h m Q    - = [kW] ( 1
)
This value is then fed to a feedforward controller that commands a fuel valve. A fuel valve position is calculated in order to match the simulated heat transfer and the heat generated in the real combustor upstream the turbine.

During the test, the fuel valve was initially controlled by a PID to start up the turbine without the model in the loop. In the meantime, the model received real-time data from the sensors located at the volume inlet. When both the physical system and the numerical model reached the steady state and the fuel valve position calculated in the model matched the actual position, the model was connected switching the fuel valve controller from the PID to the feedforward model control.

When the model is in the loop, any change in the hardware system affects the fuel cell model, and any change in the model affects the hardware system. In this way it is possible to study the dynamics of the coupled system taking advantages from the model flexibility and ensuring high fidelity.

INSTRUMENTATION Rotational Speed Measurement

Rotational speed is measured by an optical sensor, which picks up laser light reflected from a rotating target on the end of the generator shaft and transmits the pulse train to the frequency input of the control system. The optical sensor provides a 1,200 Hz signal at the nominal 40,500 rpm turbine speed with observed standard deviation in measurements of 50 rpm or 0.12% relative error in the precision of the measurement. The dynamic range of the speed variable is 1,000 to 50,000 rpm.

Compressor Inlet Flow

Compressor inlet flow is measured using an annubar flow element, which provides a mechanical average of the difference between stagnation pressure and static pressure in the inlet pipe to determine flow. Compressor inlet temperature and pressure reflect ambient conditions in the facility test cell during operation, and are used for calculating referred or corrected airflow and compressor pressure ratio. The time response is 300 ms.

Fuel Cell Simulator Flow

Primary airflow through the fuel cell simulator is measured at the entrance of the vessel using an annubar flow meter similar to the one at the compressor inlet. The combined temperature of the mixed streams from the heat exchangers is measured just upstream of the hot-air bypass valve. Pressure is also measured inside the air plenum of the fuel cell simulator.

RESULTS AND DISCUSSION

Two experiments were carried out moving the cold air bypass valve with a 15% step up from its nominal position. This value was chosen as the maximum step change allowed when the turbine speed is not controlled, in order to avoid damages to the turbine. The cold air valve bypasses the heat exchanger and the fuel cell volume, diverting air to the turbine inlet. In the first case (open loop) no control system was applied to the turbine speed, which was free to change, while in the second case (closed loop) the speed was kept constant by varying the turbine load. Each experiment was conducted three times to ensure reproducibility of the results. In both the experiments, the fuel valve was driven by the fuel cell model, simulating the heat transfer from the fuel cell system to the turbine. In order to keep constant turbine speed in the closed loop experiment, a load based speed controller was employed, which implementation was described in a previous work [START_REF] Pezzini | Decentralized Control Strategy for Fuel Cell Turbine Hybrid Systems[END_REF].

Nominal conditions of the system are illustrated in Table 1. The results showed the strong coupling between turbine speed, shown in Figure 2, and several other system parameters, such as cathode airflow and cell temperatures. After the bypass valve was opened, cathode airflow showed a significant reduction, as illustrated in Figure 3. When the turbine speed was free to change, the variation in cathode airflow was higher due to a self-propagating effect. The reduced turbine speed resulted in a lower compressor inlet airflow, as Figure 4 shows, causing a further reduction in cathode airflow. A 2% decrement in the turbine speed at steady state caused a 4% difference in cathode airflow compared with the closed loop scenario, when the speed was constant.

Turbine speed decreased in the first scenario because of two effects. First, the cold air diverted from the compressor discharge to the turbine inlet, bypassing heat exchangers and fuel cell emulator, caused a reduction in turbine inlet temperature (Fig. 5). In addition, a decrease in transferred heat from the fuel cell to the turbine was observed, as a consequence of a lower amount of cooling flow through the cell that caused more thermal energy to be stored in the cell, as explained later. This thermal effluent, shown in Fig. 6, is the only heating source of the system, driving the fuel valve in the combustor that simulates the fuel cell heat power. In closed loop, the decrement in turbine inlet temperature (TIT) and transferred heat was compensated by a PI controller, which imposed a lower electric load to the turbine in order to keep the speed constant. 5. In open loop, the variation in TIT and TOT was about 3.5% of the nominal value, and the new steady state was reached after few minutes from the valve step change. The 2% reduction in compressed airflow, due to the lower turbine speed, resulted in a lower amount of bypassed air to the turbine compared with the closed loop case, and globally in a lower amount of airflow in the combustor. This was believed to mitigate, to some extent, the decrement in TIT. In closed loop a significant difference was observed in temperature trends. The immediate reduction in TIT was about 4%, and then the temperature kept decreasing gradually during time. Turbine outlet temperature decreased around 6%, following the same trend of TIT. The higher variation in closed loop compared with open loop was mainly due to the different compressor inlet mass flow, which increased of 2% in closed loop (Fig. 4).

The thermal energy transferred to the turbine was very similar in both cases, as Figure 6 shows. In open loop, the decreased total airflow in the system after the speed reduction resulted in a higher temperature at the combustor exit. In addition, the amount of bypass flow increased around 26%, while in contrast, in closed loop the amount of airflow through the compressor increased slightly, and the increment in bypass flow was 32%. In both cases the bypass flow changed from 32% to 41% of the compressor inlet flow.

A larger variation of TOT than TIT in closed loop indicated that the efficiency of the turbine expander increased. The main cause was considered the reduction in turbine load to maintain constant speed. An increment in turbine inlet pressure was also believed to contribute, which was not observed in open loop where, in contrast, the inlet pressure decreased (Fig. 7). The heat transferred from the fuel cell to the turbine showed an initial reduction due to the lower cathode airflow, which caused a higher amount of generated heat to be stored in the fuel cell. Then it gradually recovered because the cathode inlet temperature decreased, as shown later in Fig. 9. As such, a reduction in air inlet temperature increased the temperature difference between airstream and fuel cell solid material, and consequently the heat transfer to the airflow after the initial decrement. The pressure trends, presented in Figure 7, were significantly different in open loop and closed loop. In open loop both compressor discharge pressure and turbine inlet pressure decreased considerably, following the turbine speed trend. Turbine inlet pressure reduced about 2% less than compressor outlet pressure, because bypassing heat exchangers and additional volumes, the system pressure drop decreased significantly. In closed loop, compressor discharge pressure showed a small reduction because of the slight increment in compressor mass flow at constant speed. Also in this case the system pressure drop reduced due to the lower airflow through the recuperators and the volumes.

Consequently, turbine inlet pressure increased. Pressure drops are presented in Figure 8. The air exiting the compressor was pre-heated with the turbine exhaust through two heat exchangers before feeding the cathode side of the cell. The significant decrement in TOT reduced the amount of heat transferred to the cold side of the recuperators, and this resulted in a lower cathode inlet temperature in both cases. In open loop, since the variation in TOT was smaller, cathode inlet temperature decreased about 12 °C, or 1.7% of the nominal value. In closed loop the variation was higher, and cathode inlet temperature reduced of 26 °C, or 3.5%. The trends are shown in Figure 9. The difference in cathode inlet temperature mainly drove the different trends in cell average temperature, which was calculated as the mean value of the temperatures in the 20 nodes of the cell. In open loop the reduction in cathode airflow caused the average solid temperature in the cell to rise of 17 °C, around 2% of the initial temperature, as Figure 10 shows. In closed loop the temperature showed a different trend. The initial increment was significantly lower, less than 0.5%, and after 1000 seconds the temperature started decreasing again slowly.

After 3000 seconds, a scaling factor of 0.001 was applied to the cell heat capacity in order to reduce the fuel cell thermal capacity, accelerate the simulation, and evaluate the final steady state value. In open loop, the final temperature value was pretty close to the one at 3000 seconds, while in closed loop, the temperature showed a further decrement of 1% (8 °C) with respect to the initial steady state value. The closed loop behavior was mainly caused by the considerable reduction in cathode inlet temperature, which compensated the effect of the cooling flow decrease.

The reduced inlet temperature and cathode airflow caused in both cases a substantial increment in the temperature difference between cathode outlet and inlet, as Figure 11 shows. This difference can give an indication of the temperature gradients along the cell, which are not measurable but can induce detrimental stresses on the ceramic electrolyte if their value is too high. Hence, air temperature difference represents an important parameter for the control of the cell performance. As illustrated in Figure 11, the impact of cold air on ΔT was the same in closed loop and open loop by the time the new steady state was reached. Even though in closed loop both cathode inlet and outlet temperature were observed to be lower than in open loop, the increment in ΔT was in both cases around 20% (from approximately 170 °C to 205 °C).

This aspect is very important for the thermal management of the system. Normally, for large size systems, the turbine speed needs to be held constant if the system is connected to the grid, and this means that cold air bypass is not very effective in changing the average cell temperature. On the other hand, the impact on cathode inlet temperature and air ΔT across the cell was very high in both cases. This suggests that cold air bypass could be used in closed loop to mitigate ΔT variations without affecting the average temperature significantly.

In addition, ΔT response to airflow change was faster in closed loop, reaching the 63% of the new steady state value in half of the time compared with the open loop response. This was mainly due to the greater magnitude in cathode inlet temperature drop. The 63% criterion was considered for system identification and control research to identify the response time of different system parameters. The conclusion was that temperature difference could be controlled faster in closed loop if the cold air bypass was employed as actuator. When the cold air valve was opened, surge margin increased in both cases. In open loop, a reduction in both compressor inlet mass flow and pressure ratio were observed due to the decrement in turbine speed. In closed loop, the operative point moved along the isovelocity curve while compressor inlet mass flow increased and the pressure ratio slightly decreased. The surge margin variation is presented in Figure 12, and it was comparable in the two cases. In both cases the reduction in system pressure drop due to the bypass of heat exchangers, fuel cell volume, and post combustor volume, was considered to contribute significantly to the surge margin increase. While the bleed air valve can provide a strong impact on surge margin, penalizing the efficiency of the system significantly, the cold air valve was considered an important actuator for long-term regulation of surge margin, since the bypassed air was still expanded into the turbine and the global efficiency was not affected as much.

CONCLUSIONS

The impact of cold air bypass variation on a fuel cell gas turbine hybrid system was analyzed comparing two different scenarios: open loop, where no control was applied to the turbine speed or the turbine inlet temperature, and closed loop, where the turbine speed was kept constant by varying the turbine load. Air was diverted from the compressor discharge to the turbine inlet, bypassing the heat exchangers and the fuel cell emulator.

Significant differences in the system parameter behavior was observed in the two cases, proving the strong coupling between cathode airflow and turbine speed in these systems. A variation in speed affected all the other variables in the open loop configuration, with a selfpropagating effect. In particular, cathode airflow, system pressures, and average cell temperature showed stronger variations in open loop than in closed loop. In contrast, in closed loop cold air valve position affected cathode inlet temperature more significantly and ΔT across the fuel cell quicker than open loop cold air changes. Hence, the cold air valve could be very effective in regulating cell temperature difference without impacting the average cell temperature in a constant speed scenario.

Cold air valve also proved to be very effective in controlling the surge margin, in both cases. Thus, it is considered an important actuator for the safety of the system.

Figure 1 .

 1 Figure 1. Hyper facility schematic diagram

Figure 2 .Figure 3 .

 23 Figure 2. Turbine speed trend in open and closed loops

Figure 4 .

 4 Figure 4. Compressor inlet mass flow trends in open and closed loops

Figure 5 .

 5 Figure 5. Turbine inlet and outlet temperature trends in open and closed loops

Figure 6 .

 6 Figure 6. Heat transfer trends from the fuel cell system in open and closed loops

Figure 7 .Figure 8 .

 78 Figure 7. Pressure trends in open and closed loops

Figure 9 .

 9 Figure 9. Cathode inlet temperature trends in open and closed loops

Figure 10 .Figure 11 .

 1011 Figure 10. Average solid temperature trends in open and closed loops

Figure 12 .

 12 Figure 12. Surge margin trends in loop and closed loops

Table 1 .

 1 Nominal system operating conditions

	Turbine load	40 kW
	Turbine speed	40,500 rpm
	Pressure ratio	3.6
	Compressor inlet flow 2 kg/s
	Bleed air	1% of compressor inlet flow
	Cold air	32% of compressor inlet flow
	Hot air	2% of compressor inlet flow
	Fuel cell load	220 A
	Fuel utilization	70%
	Anode fuel flow	0.1 kg/s
	Cathode inlet flow	1 kg/s

ACKNOWLEDGMENTS

This work was funded through the Crosscutting Research Program of the U.S. Department of Energy, National Energy Technology Laboratory (NETL) in Morgantown, WV. The work was supported in part by the NETL Research Participation Program, administrated by Oak Ridge Institute for Science and Education. The authors would also like to thank Dr. Paolo Pezzini from Ames Laboratory and Dr. Nor Farida Harun from McMaster University (Canada) for their valuable support.


Inez Von Deschwanden
email: inez.deschwanden@uni-due.de

Friedrich-Karl Benra

Dieter Brillert

Hans Josef Dohmen

Droplet evaporation in the context of interstage injection

Keywords:

Wet compression is a widely used approach to maximize the power rate of stationary gas turbines. This technique is practically realized by liquid water injection into the air main flow. Water is injected via a nozzle rack at the air inlet duct of the compressor. Only minor parts of the injected water evaporate on their way to the compressor inlet. Therefore, a multi-phase flow of humid air and water droplets enters a compressor operating in wet compression mode. Water injection between compressor stages is called interstage injection. This technique is challenging and rarely investigated yet.

. The compression process is calculated using an axial compressor model. Physical limits referring to evaporation process and internal compressor cooling are discussed. A thermodynamic analysis of interstage injection is presented with focus on the amount of water required for optimized cooling.

Introduction

Inlet air cooling is a commonly used technique to increase gas turbine power output. The increase of ambient air temperature decreases simultaneously the gas turbine power output of about 0.54 %-0.9 % per degree, Chaker, et al. [START_REF] Chaker | Inlet Fogging of Gas Turbine Engines-Part I: Fog Droplet Thermodynamics, Heat Transfer, and Practical Considerations[END_REF]. There are four common evaporative inlet cooling systems:

• evaporation cooling • inlet fogging • wet compression • interstage injection
The installation of the four water injection systems is schematically pictured in Fig. 1. For evaporation cooling the air mass flow is cooled while the air goes through wet gauze, placed at the entrance of the air inlet duct. In this application evaporative cooling of the air is achieved through the air contact with the wet gauze. No water is injected into the air mass flow. For inlet fogging the air mass flow is cooled by water injection into the air inlet duct until the wet bulb temperature is attained. Within this process only the water amount needed for air saturation is injected. Hence saturated air at wet bulb condition but no water droplets enters the compressor.

The third technique is wet compression. In literature this technique is also called overspray fogging. The technique of wet compression is similar to the one used for inlet fogging. The air mass flow is similarly to the inlet fogging technique cooled down by water injection into the air inlet duct of the compressor. However, the amount of injected water is higher than the amount which is injected for inlet fogging. In contrast to inlet fogging, during the wet compression process a mixture of saturated air and water droplets enters the compressor. The major advantage of the wet compression system compared to the inlet fogging is the evaporation of water droplets during the compression process.

Figure 1 Evaporative cooling systems for gas turbines

The continuously cooled compression process leads to an increase of the gas turbine power output. Thermodynamic analyses of the wet compression process are given by Zheng, et al. [START_REF] Zheng | Thermodynamic Analyses of Wet Compression Process in the Compressor of Gas Turbine[END_REF], Zheng, et al. [START_REF] Zheng | Thermodynamic Performance of Wet Compression and Regenerative (WCR) Gas Turbine[END_REF] and Bianchi, et al. [START_REF] Bianchi | Influence of Water Droplet Size and Temperature on Compression[END_REF]. Aerodynamic analyses of the wet compression process e.g. by Brun, et al. [START_REF] Brun | Aerodynamic Instability and Life-Limiting Effects of Inlet and Interstage Water Injection Into Gas Turbines[END_REF] and Sun [START_REF] Sun | Numerical Simulation of a Complete Gas Turbine Engine With Wet Compression[END_REF] point out that there is the risk of aerodynamic instabilities for wet compression processes due to changed compressor operating conditions.

Water injection between compressor stages is called interstage injection. For wet compression systems just one or two injection points exist in the air inlet duct. Small droplets are injected because of following the compressor main flow best. The aim is to prevent that droplets hit the walls. If droplets hit walls they form a water film. The evaporation of water films performs slower than droplet evaporation process. Additionally, the evaporation time of small droplets is faster than the

Evaporation cooling Inlet fogging / Wet compression

Interstage injection

Droplet evaporation in the context of interstage injection -2 evaporation time of bigger droplets. A shorter evaporation time leads to a cooling effect mainly in the first compressor stages.

Interstage injection enables a continuous water injection during the compression process. With this technique further cooling of the last compressor stages can be achieved.

Interstage injection offers the possibility for water injection at specific positions. Therefore optimized cooling can be realized during the compression process. The amount of injected water can be adapted to operating conditions. For specific operating conditions water can be injected at individual injection positions in order to improve compressor cooling. Additionally, the risk of blade icing at the first compressor stages can be avoided if water is injected between the stages and not in front of the compressor.

Injecting water between the compressor stages leads also to changed compressor operation. Comparable to wet compression there is also the risk of aerodynamic instabilities; see e.g. Brun, et al. [START_REF] Brun | Aerodynamic Instability and Life-Limiting Effects of Inlet and Interstage Water Injection Into Gas Turbines[END_REF]. Different methods for interstage injection exist. Wang, et al. [START_REF] Wang | The Effective Positions to Inject Water Into the Cascade of Compressor[END_REF] present a model for water injection between high-pressure and low-pressure compressor parts. Bagnoli, et al. [START_REF] Bagnoli | A Parametric Study of Interstage Injection on GE Frame 7EA Gas Turbine[END_REF] present an interstage injection system with eight water injection points. The first injection point is within the inlet duct to saturate the air in the inlet duct. The seven additional injection points are located between the first eight compressor stages. Further water injection between the stages leads to a supplemental evaporation during the compression process and therefore cools the air down. Other examples demonstrate water injection just in between the compressor stages; see e.g. Brun, et al. [START_REF] Brun | Aerodynamic Instability and Life-Limiting Effects of Inlet and Interstage Water Injection Into Gas Turbines[END_REF].

Interstage injection systems are rarely investigated yet. Bagnoli, et al. [START_REF] Bagnoli | A Parametric Study of Interstage Injection on GE Frame 7EA Gas Turbine[END_REF] and Roumeliotis, et al. [START_REF] Roumeliotis | Evaluation of Interstage Water Injection Effect on Compressor and Engine Performance[END_REF] calculate the benefit of the interstage injection process for some operation conditions. Ingistov [START_REF] Ingistov | Interstage Injection System for Heavy Duty Industrial Gas Turbine Model 7EA[END_REF] describes a first nozzle test of an interstage injection system for online compressor washing due to water injection between the compressor stages.

In this paper a study of the evaporation of droplets in the context of interstage injection for axial compressors is presented. The evaporation performance is calculated for different initial spray conditions.

Nomenclature

𝐵

Transfer number 𝐶 Compression rate, s -1 𝑐 𝑝 Specific heat capacity, J (kg K)

Methods

Water injection between compressor stages is technically challenging. Nozzles are required to inject the water between the compressor stages. The influence of the water injection on the compressor flow should be minimized. Therefore is it essential to have nozzles producing a fine widely dispersed water spray. The spray should be homogeneous and consisting of tiny droplets with a diameter of less than 10 µm, see e.g. Schnitzler, et al. [START_REF] Schnitzler | Experimental Determination of a Four Stage Axial Compressor Map Operating in Wet Compression[END_REF]. Tiny droplets are required because they follow the compressor flow best. The short relaxing time of those tiny droplets reduces blade erosion due to avoiding the impact of droplets on the blades.

The residence time of water droplets inside the compressor is very short, in the range of 15 ms-30 ms. Thus, high evaporation rates are needed for the evaporative cooling process. The evaporation rates increase significantly when the droplet diameter is decreased. Nevertheless might the evaporation process of droplets with a diameter of 10 µm be even too slow for their overall residence time in a compressor.

An increase of the temperature of the injected water has the effect that water surface tension is reduced. This reduction leads to a higher Weber-number. Spray generation at higher Weber-numbers results in reduced droplet diameter for the same nozzle hardware. A reduced droplet diameter enables the injection of higher water mass fractions due to the faster evaporation of smaller droplets. Therefore the injection even of higher water mass fractions with its consequence on the evaporation performance during the compression process is analyzed. This analyzation enables the estimation of the potential of interstage injection in axial compressors. The droplet evaporation model by Abramzon, et al. [START_REF] Abramzon | Approximate theory of a single droplet vaporization in a convective field: effects of variable properties, Stefan flow and transient liquid heating[END_REF] is used to calculate the droplet evaporation. An influencing factor for the evaporation performance is the relative velocity between droplet and gas phase. The strength of this factor for the actual calculations is therefore firstly determined. The dependence of the droplets diameter on its ability to follow the main flow is investigated. In the second step is the dependence of the evaporation on the droplet temperature established. Finally the calculation of the evaporation ability at compressor conditions needs to be modeled.

Calculation model

The evaluation of the droplet evaporation in this paper consists of a heat-and mass-transfer model for the droplet evaporation determination and a model for the compression process. There are different methods for calculating the effect of evaporative cooling. White, et al. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF] uses a 1D-approach to calculate the evaporation effect on the compressor performance. He assumes that entropie change due to aerodynamic effects can be neglected

𝑇 𝑑𝑠 = 𝑑ℎ -𝑣 𝑑𝑝 (1)
For ideal gas and ideal mixture leads Eq. 1 to

𝑐 𝑝 * 𝑑𝑇 𝑇 = 𝑅 * 𝑑𝑝 𝑝 (2)
with 𝑐 𝑝 * and 𝑅 * as the specific heat and gas constant for the air-vapor and water mixture. For further information see White, et al. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF]. This assumption is also used by Young [START_REF] Young | The fundamental equations of gas-droplet multiphase flow[END_REF]. Härtel, et al. [START_REF] Härtel | Model Analysis of High-Fogging Effects on the Work of Compression[END_REF] calculates the effect of the wet compression process on compressor performance with two models. The first model is an ideal model, which predicts fast evaporation with saturated air at any time. The second model is a droplet evaporation model which also assumes permanently saturated air. Their results fit well for a droplet diameter of 1 µm. But for droplet diameters bigger than 1 µm the evaporation process performance slows down. Due to the slower evaporation process it is impossible to get a continuous saturation of the air. This leads to a minor temperature decrease due to the evaporation process. Härtel, et al. [START_REF] Härtel | Model Analysis of High-Fogging Effects on the Work of Compression[END_REF] concludes that just a spray consisting of droplets with an diameter of 1 µm can saturate the air at any time liquid water is present.

Producing a fine and well distributed spray with mean droplet radii smaller than 5 µm for fast evaporation is technically challenging. To produce a spray with mean droplet diameter of 1 µm is technically rather impossible. Commonly used spray nozzles provide a spray with a mean diameter of about 30 µm.

For the presented calculation a combination of a heat-and mass-transfer model for droplet evaporation and a thermodynamic model for the compression process calculation are investigated.

Compressor performance model

The calculation of compressor performance is realized by a simplified thermodynamic model for an adiabatic compressor. Acceleration and deceleration of the gas flow inside the compressor is neglected. Polytropic efficiency of the compressor is assumed to be

𝜂 𝑝𝑜𝑙𝑦 = 𝑣 𝑑𝑝 𝑑ℎ = 0.9 (3)
The compression rate 𝐶 is set to

𝐶 = 1 𝑝 𝑑𝑝 𝑑𝑡 = 200 𝑠 -1 (4)
according to White, et al. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF]. This means that the pressure increases exponentially. Figure 2 shows calculated pressure and temperature trends of air according to the formula for dry compression.

Figure 2 Pressure and air temperature trend for dry compression

For the calculations the same pressure increase is assumed for the dry and the wet compression process. Experimental and numerical studies reveal a stage mismatch and changed mass flow rate due to water injection for wet compression process see White, et al. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF] and Eisfeld, et al. [START_REF] Eisfeld | Experimental Investigation of the Aerodynamic Performance of a Linear Axial Compressor Cascade With Water Droplet Loading[END_REF]. These changed operating conditions lead to a minor pressure increase in the first compressor stages and a major pressure increase in the last compressor stages. The changed pressure increase influences the temperature rise due to the compression process. Irrespective of these changed operating conditions due to water injection is the pressure increase and dry air mass flow in the present study kept constant. This consideration affords to focus only on the water evaporation at identical conditions.

The temperature increase for the wet compression is determined in two steps. First the change of air temperature in (5)

𝑚̇𝑣 𝑎𝑝 𝑖 * = 𝑚̇𝑣 𝑎𝑝 𝑖 + 𝑑𝑚 𝑖 (6)
𝑚̇𝑤 𝑖 * = 𝑚̇𝑤 𝑖 -𝑑𝑚 𝑖 (7)

𝑇 𝑤 𝑖 * = 𝑇 𝑤 𝑖 + 𝑑𝑇 𝑖 (8)
𝑑𝑚 𝑖 denotes the evaporated mass flow which is estimated with the evaporation model. The water temperature change 𝑑𝑇 𝑖 is also estimated using the evaporation model. Subsequent is the temperature increase due to compression process evaluated using the polytropic equation. The polytropic equation is formulated using the Gibbs-equation

𝑇𝑑𝑠 = 𝑑ℎ -𝑣𝑑𝑝 (9)
Inserting the polytropic efficiency

1 𝜂 𝑝𝑜𝑙𝑦 = 𝑑ℎ 𝑣 𝑑𝑝 (10)
leads to

𝑇𝑑𝑠 = 𝑑ℎ (1 - 1 𝜂 𝑝𝑜𝑙𝑦 ) (11)
The real gas factor z of air for the regarded pressure and temperature range can be stated with 𝑧 ≈ 1 (0.996 ≤ z ≤ 1.004). As a result it is reasonable to use the equations for ideal gas. Therefore, the specific volume is calculated with the formula 𝑣 = 𝑅𝑇 𝑝 . The enthalpy is expressed with the equation 𝑑ℎ = 𝑐 𝑝 𝑑𝑇. This results in the known polytropic equation

𝑇 𝑎𝑖𝑟 𝑖+1 = 𝑇 𝑎𝑖𝑟 𝑖 * • ( 𝑝 𝑎𝑖𝑟 𝑖+1 𝑝 𝑎𝑖𝑟 𝑖 ) 𝑛-1 𝑛 (12)
The integration of Eq. 11 requires a further iteration step. Solving the problem by using the polytropic equation (Eq. 12) instead of the integration of Eq. 11 speeds the computation time up. The polytropic exponent 𝑛 changes with vapor mass fraction in the air according to the change of 𝜅.

𝑛 -1 𝑛 = 1 𝜂 𝑝𝑜𝑙𝑦 𝜅 -1 𝜅 (13)
𝜅 depends on pressure, air temperature and vapor mass fraction. The polytropic efficiency 𝜂 𝑝𝑜𝑙𝑦 is kept constant. Figure 3 illustrates the scheme of the temperature calculation within the two steps.

Droplet evaporation

The droplet evaporation model by Abramzon, et al. [START_REF] Abramzon | Approximate theory of a single droplet vaporization in a convective field: effects of variable properties, Stefan flow and transient liquid heating[END_REF] (Abramzon-Sirignano model) is used to compute the evaporation process with a one-dimensional calculation tool. The following assumptions are used:

 Spherical droplets  Monodisperse spray with mean diameter 𝑟 𝑑  Droplet breakup or conglomeration is neglected  Wall interaction is neglected
The one-dimensional tool is well suited for a fast computational performance. The evaporation of one droplet with its characteristic diameter for the monodisperse spray is calculated. The droplet number of the spray is determined according to:

𝑧 𝑑 = 𝑥 𝑤 • 𝑚 𝑎𝑖𝑟,𝑑𝑟𝑦 4 3 𝜋 𝑟 𝑑 3 𝜌 𝑤 (14)
with 𝑟 𝑑 as mean droplet diameter. For humid air, gas properties are taken from the property library by Kretzschmar, et al. [START_REF] Kretzschmar | Property Library for Humid Air Calculated as Ideal Mixture of Real Fluids[END_REF]. Water properties according to IAPWS-IF97 tables are used from the property library by Kretzschmar, et al. [START_REF] Kretzschmar | Property Library for the Industrial Formulation IAPWS-IF97 for Water and Steam[END_REF]. The Abramzon-Sirignano model uses the extended film model to describe the droplet evaporation. The droplet vaporization rate is estimated with the formula

𝑑𝑚 𝑖 = 2𝜋 𝜌 𝑔 ̅ 𝐷 𝑔 ̅̅̅ 𝑟 𝑑 𝑆ℎ * ln(1 + 𝐵 𝑀 ) (15)
𝜌̅ 𝑔 and D ̅ 𝑔 are the average values of the gas film vapor density and of the diffusion coefficient. The average temperature and mass concentration of the gas film 𝑇 ̅ 𝑔 and 𝑌 ̅ 𝑔 can be determined using the "1/3-rule" introduced by Yuen, et al. [START_REF] Yuen | On Drag of Evaporating Liquid Droplets[END_REF].

𝑇 ̅ 𝑔 = 𝑇 𝑤 + 1 3 (𝑇 𝑎𝑖𝑟 -𝑇 𝑤 ) (16)
𝑌 ̅ 𝑔 = 𝑌 𝑤,𝑠 + 1 3 (𝑌 ∞ -𝑌 𝑤,𝑠 ) (17)
𝑌 𝑤,𝑠 is the vapor mass concentration at the film surface and 𝑌 ∞ is the vapor mass concentration far from the droplet. Figure 4 illustrates a droplet with film layer to clarify the definition of the vapor mass concentrations. The diffusion coefficient D ̅ 𝑔 in Eq. 15 is determined using the method given by Fuller, et al. [START_REF] Fuller | New Method for Prediction of Binary Gas-Phase Diffusion Coefficients[END_REF]. Sherwood number 𝑆ℎ is the dimensionless mass transfer gradient at a surface. Ranz, et al. [START_REF] Ranz | Evaporation from drops Part II[END_REF] gives a correlation for estimating the Sherwood number of a non-vaporizing droplet 𝑆ℎ 0 , which is used here

𝑆ℎ 0 = 2 + 0.552 𝑅𝑒 1/2 𝑆𝑐 1/3 (18)
According to Abramzon, et al. [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] has the Sherwoodnumber of the non-vaporizing droplet 𝑆ℎ 0 to be modified to consider the convective mass transfer between droplet surface and gas environment using the Spalding mass transfer number

𝐵 𝑀 𝑆ℎ * = 2 + (𝑆ℎ 0 -2) ln(1 + 𝐵 𝑀 ) 𝐵 𝑀 (19)
The Spalding mass transfer number 𝐵 𝑀 , an amount for the mass transfer between gas and liquid can be determined from the vapor mass fraction concentrations 𝑌

𝐵 𝑀 = 𝑌 𝑤,𝑠 -𝑌 ∞ 1 -𝑌 𝑤,𝑠 (20)
The heat 𝑄 ̇𝐿 which is transferred into the liquid droplet is estimated using the formula by Abramzon, et al. [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] 𝑄 ̇𝐿 = 𝑚̇{ 𝑐̅ 𝑝 𝑤,𝑠 (𝑇 𝑎𝑖𝑟 -𝑇 𝑤 ) 𝐵 𝑇 -𝐿(𝑇 𝑤 )}

The latent heat of evaporation 𝐿(𝑇 𝑤 ) is calculated for the surface temperature of the droplet 𝑇 𝑤 . The radii of droplets used for interstage injection should be less than 5 µm. For such tiny droplets a uniform droplet temperature can be assumed. This leads to equilibrium of surface temperature of the droplet and the temperature at any point inside the droplet. With this assumption the droplet temperature can be exposed with the formula

𝑑𝑇 𝑤 𝑑𝑡 = 𝑄 ̇𝐿 𝑚 𝑑 𝑐 𝑝,𝑤 (22)
𝐵 𝑇 is the Spalding heat transfer number according to the definition by Abramzon, et al. [START_REF] Abramzon | Approximate theory of a single droplet vaporization in a convective field: effects of variable properties, Stefan flow and transient liquid heating[END_REF] 𝐵 𝑇 = (1 + 𝐵 𝑀 ) 𝜙 -1 [START_REF] Fuller | New Method for Prediction of Binary Gas-Phase Diffusion Coefficients[END_REF] with the non-dimensional correction parameter 𝜙 given by Abramzon, et al. [START_REF] Abramzon | Approximate theory of a single droplet vaporization in a convective field: effects of variable properties, Stefan flow and transient liquid heating[END_REF]

𝜙 = ( 𝑐̅ 𝑝𝑤 𝑐̅ 𝑝𝑔 ) ( 𝑆ℎ * 𝑁𝑢 * ) 1 𝐿𝑒 (24)
Correspondent to the Sherwood-number for mass transfer is the Nusselt-number the dimensionless temperature gradient at the droplet surface. The modified Nusselt-number is similarly to the modified Sherwood-number defined by Abramzon, et al. [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF]:

𝑁𝑢 * = 2 + (𝑁𝑢 0 -2) ln(1 + 𝐵 𝑇 ) 𝐵 𝑇 (25)
with 𝑁𝑢 0 , the correlation for Nusselt-number of a nonvaporizing droplet, given by Ranz, et al. [START_REF] Ranz | Evaporation from drops Part II[END_REF]:

𝑁𝑢 0 = 2 + 0.552 𝑅𝑒 1/2 𝑃𝑟 1/3 ( 26
)
The assumption is made that droplets are injected into the gas mass flow with the droplet velocity u d = 160 ms -1 . The air velocity during the compression process is set constant to u ∞ = 175 ms -1 corresponding to typical axial-compressor specifications. According to Abramzon, et al. [START_REF] Abramzon | Droplet vaporization model for spray combustion calculations[END_REF] can the change of the droplet velocity be evaluated by the formula

𝑑𝑢 𝑑 𝑑𝑡 = 3 𝐶 𝐷 2 𝑟 𝑑 ( 𝜌 ∞ 𝜌 𝑤 ) |𝑢 ∞ -𝑢 𝑑 |(𝑢 ∞ -𝑢 𝑑 ) (27)
The drag coefficient 𝐶 𝐷 is defined by Faeth [START_REF] Faeth | Current status of droplet and liquid combustion[END_REF]

𝐶 𝐷 = 24 𝑅𝑒 [1 + 𝑅𝑒 2/3 6 ] (28)
The relative velocity between gas phase and droplet influences the evaporation performance. The relative velocity 𝑢 𝑟𝑒𝑙 = |𝑢 𝑑 -𝑢 ∞ | between gas and droplet is used to determine the Reynolds-number 𝑅𝑒 of the droplet. This Reynolds-number is used for the droplets Sherwood-number 𝑆ℎ 0 (Eq.18) and Nusselt-number 𝑁𝑢 0 (Eq. 26). A one-dimensional approach incorporates of cause a one dimensional relative velocity. Considering the droplets acceleration and deceleration in relation to the air the relative velocity vector transfers this into a one-dimensional scheme and enables the usage of a onedimensional evaporation model. Inside the compressor, a uniform air velocity of 175 ms -1 is assumed. To validate the small droplets ability to follow the main gas flow a previous study is performed. Droplets with initial radii of 2 µm and 5 µm are selected for the pre-study. The relative velocity between gas phase and droplets is set to 15 ms -1 . In this case amounts the relaxing time for droplets with an initial radius of 2 µm to 0.05 ms. 5 µm droplets reach 0.4 ms after injection the gas velocity, see Fig. 5.

Figure 5 Velocity of droplets injected into compressor air mass flow, relative velocity 𝒖 𝒓𝒆𝒍 = 𝟏𝟓 𝒎𝒔 -𝟏

The size of droplets used in the present calculations is so small that the three dimensional character of the compressor flow is neglected due to the fact that the acceleration process of these small droplets is very fast. No slip velocity condition is used for the presented study.

Comparison of droplet evaporation models

Kim, et al. [START_REF] Kim | An Assessment of High-Fogging Potential for Enhanced Compressor Performance[END_REF] and Kim, et al. [START_REF] Kim | Analytical modeling of wet compression of gas turbine systems[END_REF] estimate the droplet evaporation for a technical analysis of the wet compression process with two different evaporation models. In the first publication Kim, et al. [START_REF] Kim | An Assessment of High-Fogging Potential for Enhanced Compressor Performance[END_REF] uses the droplet evaporation model by Spalding. Within the second publication (Kim, et al. [START_REF] Kim | Analytical modeling of wet compression of gas turbine systems[END_REF]) the analytical non-equilibrium solution is used to compute the wet compression process. For both calculations the pressure rise inside the compressor is determined with Eq. (4) according to White, et al. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF]. The compressor specifications are:

 compression rate 𝐶 = 200 1 𝑠  pressure ratio 𝜋 = 15
 polytropic compressor efficiency 𝜂 𝑝𝑜𝑙𝑦 = 90%  ISO-conditions for the air at compressor inlet In the presented calculation the same compressor model and the same ISO conditions of the air are used. As mentioned in chapter 2.3 no slip velocity condition is assumed. All three models show different evaporation performances for the same compression process, see Fig. 6. The Abramzon-Sirignano model calculates a slower evaporation performance compared to the two other models. The difference in the evaporation time between the Spalding model and the Abramzon-Sirignano model is about 0.002 s for a 2 µm droplet. The same evaporation level of the 4 µm droplet is calculated with a time delay of about 0.0023 s with the Abramzon-Sirignano model compared to the Spalding model [START_REF] Faeth | Current status of droplet and liquid combustion[END_REF]. The spray consisting of 4 µm droplets is not completely evaporated when the compression process is finished. The evaporation process calculated with the Abramzon-Sirignano model for a spray consisting of droplets with an initial diameter of 5 µm is still not finished when the compression process is finished. The non-equilibrium assumption shows a complete evaporation of the 5 µm droplets 0.005 s before the end of the compression process.

The evaporation trend of the 2 µm and the 4 µm droplets by the Abramzon-Sirignano model and by the Spalding model are similar. In contrast to the Abramzon-Sirignano model the evaporation trend of the non-equilibrium model differs significantly, see Fig. 6. It has to be considered that different evaporation models result in different evaporation performances at compressor conditions. Because of the lack of experimental data no thorough validation of the evaporation models is available. In the presented calculations the Abramzon-Sirignano model is used because of its good acceptance in the research community.

Results and discussion

The evaporation performance in a compressor depends mainly on three factors

• Relative velocity between gas and liquid • Droplet radius • Temperature difference between gas and liquid

The relevance of the relative velocity between gas and droplet is not apprenticed if the droplet radius goes below the target value of 5 µm. Consequently this examination disregards the impact of the relative velocity on the evaporation performance. The analysis how the droplet radius and the temperature difference between gas and liquid impacts the evaporation performance can give an indication how the evaporation process for compressor cooling can be optimized. For all calculations it is postulated that air enters the compressor at ISO conditions. Basic conditions for the calculations are given in Tab. 2. Spalding model r i = 4 µm [START_REF] Faeth | Current status of droplet and liquid combustion[END_REF] non-eq. model r i = 5 µm [START_REF] Roumeliotis | Evaluation of Interstage Water Injection Effect on Compressor and Engine Performance[END_REF] compression process finished Droplet evaporation in the context of interstage injection -7

Droplet radii and water mass fraction

The droplet radius is one of the main factors influencing the evaporation performance of droplets. Spalding [START_REF] Spalding | The combustion of liquid fuels[END_REF] gives the correlation, that the square of the droplet diameter decreases linear with time. This assumption fits well for droplets in resting air and at ambient conditions. In the present study, the droplets are injected in an air mass flow with changing conditions due to compression and evaporation process. The relation between droplet volume and droplet surface is important for the present study. Decreasing the droplet diameter increases the ratio of droplet surface to droplet volume.

In Tab. 3 is the evaporation time in dependence of droplet radius and water mass fraction listed. The basic conditions from Tab. 2 are used for the calculations. The initial droplet radii are 1 µm, 2 µm, 3 µm, 4 µm and 5 µm respectively. A decreasing droplet radius decreases the evaporation time of the droplet. Consequently decreases the evaporation time with decreasing water mass ratio. Also a higher water mass ratio leads to a higher vapor mass fraction in the air during the evaporation process. The Spalding mass transfer number 𝐵 𝑀 is an indicator for the mass transfer between the droplets and the surrounding gas. In Fig. 7 is the Spalding mass transfer number 𝐵 𝑀 in dependence of the evaporation time pictured. The calculations are made for a spray, consisting of droplets with an initial diameter of 1 µm but different water mass fractions. The mass transfer number refers to the ratio between vapor mass fraction in the air and at the droplet surface. The difference between the vapor mass fraction at the droplet surface and in the gas mass flow away from the droplet increases with decreasing water mass ratio. Hence 𝐵 𝑀 gets higher if less water is injected and this leads to a faster evaporation process. For the same droplet radius results an increased water mass ratio in an augmented number of water droplets. The augmented number of droplets causes a slower evaporation process of a single droplet. The slower evaporation follows from the enlarged vapor mass in surrounding according to the numerous evaporating droplets.

Figure 8 shows the air temperature degradation due to the water evaporation within 0.004 s during the compression process. Different water mass fractions as well as different spray droplet radii are regarded. As demonstrated before a water mass fraction of 1 % consisting of 1 µm droplets evaporates within 0.004 s. The depicted temperature drop of 35.39 K is the maximum temperature drop which can be reached with this configuration. Besides, the temperature drop raises in consequence of the spray evaporation with an increasing water mass fraction but unchanged droplet radius. The maximum temperature decrease (∆𝑇 = 52.94 𝐾) can be reached with an injected water mass fraction of 5 % and a spray consisting of droplets with an initial radius of 1 µm. As expected, leads a smaller water mass fraction with same droplet radii in the regarded time to a minor evaporative air cooling effect than a higher amount of injected water.

Furthermore, the cooling effect diminishes if the droplet radius increases but the water mass fraction is kept constant. The trend is irrespective of the analyzed water mass fractions of 1 % to 5 %. Thus, the strength of the temperature drop depends on the droplet radius. The evaporation rate is substantial for the cooling effect within the short period of 0.004 s. A water mass fraction of 1 % and a spray consisting of 1 µm droplets leads to an air temperature drop of 35.39 K whereas a water mass fraction of 3 % und a spray consisting of 2 µm droplets leads to an air temperature drop of 36.11 K. The difference is only 0.72 K. It is worth mentioning that in this case a higher water mass fraction with increased droplet radius leads to a slightly better cooling effect though the evaporation of the single droplets performs slower. Even within one diameter class enhances the cooling effect with an enlarged water mass fraction. The decelerated evaporation performance of a single droplet due to the exceeded vapor mass fraction in the air is less significant than the cooling effect introduced by the evaporation of the huge number of droplets in the spray. Therefore, the cooling effect of a higher amount of injected water with the same droplet radius is more substantial than the evaporation effect of a small water mass ratio. Though a substantial augmented droplet radius but same water mass fraction leads to a declining evaporation performance and therefore a declining air temperature drop. Air is heated up at T air = 371.45 K due to the compression process if no water is injected. The injection of 5 % water mass fraction consisting of droplets with an initial radius of 1 µm decreases the air temperature by about ΔT = 52.9 𝐾. For the same mass fraction but droplets with an initial radius of 5 µm is the air temperature decreased by about ΔT = 17.5 K. It can be pointed out that both, water mass fraction and droplet radii, have a significant influence on the cooling performance of compressed air.

Temperature difference between gas and liquid

Increasing the water temperature leads to a reduction of water surface tension which results in an enlarged Webernumber. Spray generation at enlarged Weber-numbers results in reduced droplet diameter. The evaporation performance for two exemplary water temperatures is presented in this study. The lower water temperature is set to 293.15 K, the higher water temperature is set to 353.15 K. This temperature increase leads to a decrease of the surface tension about 16 %. Weber-number for atomizers is defined as

𝑊𝑒 = 𝑑 𝑢² 𝜌 𝜎 (29)
Fluid velocity 𝑢 and the nozzle diameter 𝑑 are kept constant for both cases. The ratio of the Weber-number at 293.15 K to the Weber-number at 353. [START_REF] White | An Evaluation of the Effects of Water Injection on Compressor Performance[END_REF] This means a rise of Weber-number about 11.5 % for a water temperature increase of ∆T w = 60K. Also a slight augmentation of water temperature has a positive effect on the generation of small droplets. Furthermore a spray consisting of small droplets leads to an improved air cooling effect.

Besides the effect of increased water temperature on the evaporation performance has to be considered. In Fig. 9 the water temperature reduction for initial droplet radii of 2 µm, 5 µm and 10 µm, injected with an initial water temperature T w = 353.15 K is pictured. Basic conditions, mentioned in Tab. 2 are used for the determination of the evaporation performance.

The droplet temperature decreases faster with diminishing droplet diameter. The air temperature at the beginning of the calculation is set to 288.15 K. The temperature of a droplet with a radius of 2 µm declines within 0.0003 s from 353.15 K to air temperature level. For a droplet with an initial radius of 5 µm the water temperature decreases due to convection and latent heat of vaporization within 0.001 s. Consequential drops the temperature of the 10 µm droplet slower than the temperature of the droplets with 2 µm and 5 µm. The initial water temperature for the first case is set to 293.15 K. For the second case is the initial water temperature set to 353.15 K. As mentioned before, the droplets injected with an initial temperature of 353.15 K reach air temperature level within 0.0003 s. It can be shown that within these 0.0003 seconds the evaporation rate is slightly higher than the evaporation rate of the spray with minor initial water temperature. The droplet radius for an initial water temperature of 353.15 K decreases by 4 %. The radius of a droplet injected at 293.15 K decreases in the same time by 0.5 %. After that period the evaporation performance of both sprays is comparable. The initial offset due to faster evaporation of the spray injected at higher initial water temperature is almost constant for the whole evaporation process. Just a small advantage in the evaporation performance due to a rise of water temperature can be observed.

Multiple water injection during the compression process

The last section of this paper deals with the evaporative cooling of the compressed air by multiple water injection during the compression process. Due to the fact that droplets with a small initial radius evaporate faster than droplets with a bigger radius, computations are realized for a spray consisting of droplets with an initial radius of 1 µm. Three case studies are performed. The sequenced injection of 1 % water mass fraction at six positions is analyzed in case 1. In case 2 the injection of 2 % water mass fraction injected at three sequenced positions is examined. The third case focuses on the injection of 6 % water mass fraction at once at the compressor inlet. The sum of the injected water mass fraction in all three cases amounts to 6 % of the dry air mass flow. The results of the different water injection strategies are shown in Fig. 11.

Figure 11 Temperature air for compression process with multiple water injection

The injection positions are marked with circles. The black line pictures the dry compression process without any water injection as reference. The blue line illustrates the air tempera-ture for case 1 (1 % water injection at six positions); the red line depicts the air temperature for the second case (2 % water injection at tree positions) and the green line depicts the air temperature for case 3 (6 % water injection at one single position). In all three cases the whole amount of injected water evaporates before the compression process is finished (the particular positions are marked with arrows). The amount of 6 % water mass fraction evaporates slightly faster, if the entire water is injected at once (case 3). For the slower evaporation performance of the cases with multiple injections (case 1 and case 2) one effect is responsible: In every calculation and at every injection position the water injected is at 288.15 K. The later the water is injected the higher the temperature difference is between the injected droplets and the gas phase. Additionally the vapor mass fraction in the air is higher if the water is injected in further stages. Vapor from the heated gas phase condenses at the cold droplet surface until the water temperature exceeds the specific dew point temperature. The droplet radius increases due to the added mass. The droplet radius augmentation increases with a higher vapor mass fraction in the gas phase and a greater temperature difference between liquid and gas. The radius of the droplets injected at position six for case 3 for example augments about 4 %. The increased droplet radius results in a slower evaporation performance. The condensation results in a reduced cooling effect.

The air temperature at the compressor outlet decreases in all three cases about 260-270 K due to the injection of 6 % water mass fraction. Numerous injections with smaller water amounts each (case 1) lead to a slightly higher outlet temperature of 7.5 K than fewer injections (case 2) with the same overall water amount but higher water amounts at each injection position. The difference between six continuous water injections and one primary water injection (case 1) is 13.3 K. The difference of the temperature rise depends also on the variation of the polytropic coefficient due to the evaporation rate. A higher amount of water vapor in the gas phase leads to decreasing polytropic coefficients. For smaller polytropic coefficients the air temperature increases more significantly due to the compression process (see Eq. 9).

In the presented investigation droplet deposition due to droplet-wall interactions is neglected. With this assumption the calculated evaporation rate surpasses the technically feasible rate because of the smaller evaporation rate of water films compared to droplet evaporation.

Although small droplets follow the main flow best, it is not possible to avoid a contact between droplets and walls completely. The risk of droplet deposition on walls increases with higher amounts of injected water. Smaller amounts of water during a single injection decrease the droplet deposition. Simultaneously the risk of blade erosion due to droplet impact decreases. Hence the injection of smaller amounts of water at numerous positons is preferable to higher amounts injected at one single position. The distinct evaporative cooling leads to reduced power consumption of the compression process. The higher mass amount due to water injection increases the mass flow and therefore the turbine workload in a gas turbine cycle. One disadvantage of a decreased compressor outlet temperature is the need of higher fuel mass to reach the requested turbine inlet temperature. An analysis of the whole gas turbine cycle is necessary for a clear statement as to which effect predominates.

Conclusion

This contribution deals with spray evaporation during a compression process in an axial compressor. The influence of the injected water mass and the droplet radius is demonstrated. As expected, a smaller droplet radius results in faster evaporation. This leads to a stronger decrease of air temperature when the same water amount with smaller droplets is injected. Increasing the water mass for a spray consisting of droplets of the same diameter decelerates the evaporation process. An elevated initial droplet temperature results in a minor positive effect on the evaporation performance but in higher Weber-numbers and therefore smaller droplets for the same nozzle configuration.

Water injection during the compression process decreases the air temperature continuously. Injection of a spray consisting of tiny droplets with a radius r d = 1 µm at different positions into the compressor allows a strong decrease of the compressor outlet temperature. In the investigated cases a water mass fraction of 6 % is injected. This leads to a compressor outlet temperature decrease about 65 %. The injection of higher water amounts at fewer positions leads to a slightly stronger temperature drop at the compressor outlet than the water injection at numerous positions.

The application of different evaporation models results in varying evaporation performances. For further investigation of the evaporation during the compression process a detailed comparison of the evaporation model performances should be executed. Correspondent to the comparison the model which reproduces the influence of high temperature and high pressure conditions on the evaporation process in the best way should be chosen. The present approach neglects changes in the compressor performance due to the water injection. Future precise calculations should take into consideration that water injection may lead to stage mismatching and therefore to a changed compressor performance.

Figure 3

 3 Figure 3 Scheme of temperature calculation

5 Figure 4

 54 Figure 4 Scheme of water droplet with surrounding film layer and mass concentrations

  Droplet evaporation in the context of interstage injection -6

Figure 6

 6 Figure 6 Droplet evaporation performance calculated with C=200 s -1 , π=15, ηpoly=90%, air ISO conditions at compressor inlet, water mass fraction xw= 5%

  mass ratio [%] Abramzon model r i = 2 µm Abramzon model r i =4 µm Abramzon model r i = 5 µm Spalding model r i = 2 µm [25]

Figure 7

 7 Figure 7 Dependency of Spalding number of mass transfer BM on water mass ratio xw and evaporation time for a spray with an initial droplet radius of 1 µm

Droplet evaporation in the context of interstage injection - 8 Figure 8

 88 Figure 8 Temperature decrease for different droplet radii and water mass fractions, evaporation time = 0.004 s

Figure 9

 9 Figure 9 Decrease of water temperature for different initial droplet radii Figure 10 depicts the influence of the water temperature on the evaporation time of the generated droplets. The spray mass flow is set to 2 %. The initial droplet radius is set to r d = 2 µm.The initial water temperature for the first case is set to 293.15 K. For the second case is the initial water temperature set to 353.15 K. As mentioned before, the droplets injected with an initial temperature of 353.15 K reach air temperature level within 0.0003 s. It can be shown that within these 0.0003 seconds the evaporation rate is slightly higher than the evaporation rate of the spray with minor initial water temperature. The droplet radius for an initial water temperature of 353.15 K decreases by 4 %. The radius of a droplet injected at 293.15 K decreases in the same time by 0.5 %. After that period the evaporation performance of both sprays is comparable. The initial offset due to faster evaporation of the spray injected at higher initial water temperature is almost constant for the whole evaporation process. Just a small advantage in the evaporation performance due to a rise of water temperature can be observed.

Droplet evaporation in the context of interstage injection - 9 Figure 10

 910 Figure 10 Droplet evaporation and water temperature change for a spray with initial radius r=2 µm and initial water temperature of 293.15 K and 353.15 K

  the context of interstage injection -10

Table 1 Basic conditions for determination of velocity change for water injected into compressor flow

 1

	Property
	Air velocity (constant)	175 ms -1
	Droplet velocity at injection point	160 ms -1
	Air temperature	288.15 K
	Water temperature	288.15 K
	Air pressure	1 bar
	Droplet radii	2 µm, 5 µm
	Droplet Reynolds-number	26325, 65711
	Droplet Weber-number	0.006, 0.015

Table 2 Basic conditions for calculations

 2

	Property
	Air temperature compressor inlet	288.15 K
	Air pressure compressor inlet	1 bar
	Air humidity (relative)	60 %
	Water temperature injection point	288.15 K
	Pressure ratio 𝜋 Compression rate C	15 200𝑠 -1

Table 3 Evaporation time in dependence on droplet radius and water mass fraction

 3

		xw=5%	xw=4%	xw=3%	xw=2%	xw=1%
	Radius	evap.	evap.	evap.	evap.	evap.
	[µm]	time [s]	time [s]	time [s]	time [s]	time [s]
	5	n.evap.	n.evap.	n.evap.	0,013	0,012
	4	n.evap.	n.evap.	0,012	0,011	0,010
	3	0,013	0,012	0,010	0,009	0,008
	2	0,012	0,010	0,009	0,007	0,006
	1	0,011	0,009	0,007	0,005	0,004

Table 4 Temperature decrease due to multiple water injection

 4

	Water mass fraction xw [%]	Temperature at compressor outlet [K]	Temperature decrease due to evaporative cooling [K]
	0	673.2	0
	6 times 1 %	416.0	257.2
	3 times 2 %	408.5	264.7
	1 time 6 %	402.7	270.5