A first principles based approach for dynamic modeling of turbomachinery.
As the cost and complexity of modern aircraft systems increases, emphasis has been placed on model-based design as a means for reducing development cost and optimizing performance. To facilitate this, an appropriate modeling environment is required that allows developers to rapidly explore a wider design space than can cost effectively be considered through hardware construction and testing. This wide design space can then yield solutions that are far more energy efficient than previous generation designs. In addition, non-intuitive cross-coupled subsystem behavior can also be explored to ensure integrated system stability prior to hardware fabrication and testing. In recent years, optimization of control strategies between coupled subsystems has necessitated the understanding of the integrated system dynamics. To this end, a dynamic vapor cycle modeling toolset known as the AFRL Transient Thermal Management and Optimization (ATTMO) toolset was developed to address two-phase flow systems. This toolset has been further expanded to include components typical of air cycle topologies. Current air-cycle modeling tools rely heavily on user supplied performance maps to predict turbomachinery output. This approach has a variety of limitations. First, at a conceptual design level, modifications to a specific component design requires the generation of a new map; thereby, limiting one's ability to rapidly evaluate and optimize across a wide design space. Second, interpolation routines in map-based approaches fail to yield precise solutions near critical operating points, such as compressor stall lines, due to unknown data outside of the stall point. As a result, development of control strategies around the stall margin of a machine is difficult due to the model induced instabilities or incorrect predictions from interpolation around those operating points. Lastly, modeling startup and shutdown requires torque prediction near or at zero speed, which for map-based approaches is ill-defined. To address these limitations, first principles models of traditional turbomachinery components have been developed and will be discussed in this paper. Conservation of mass, energy, and momentum are applied to capture appropriate volume dynamics relevant to plant and controls engineers. Enthalpy based calculations derived from machine geometry and fluid flow conditions allow for design optimization of machine parameters while still maintaining accurate performance predictions. These approaches have been implemented in the open-source Simulink toolset, ATTMO. Based on user defined system architecture and associated design parameters, a time-domain simulation for an integrated system analyses can be formed.
CITATION: McCarthy, P., Niedbalski, N., McCarthy, K., Walters, E. et al., "A First Principles Based Approach for Dynamic Modeling of Turbomachinery," SAE Int. J. Aerosp. 9(1):2016, doi:10.4271/2016-01-1995.
Turbomachinery modeling has been studied and performed for decades. Currently, there are many software toolsets available ranging in complexity, capability, and applicability. In particular, the National Aeronautics and Space Administration (NASA) has worked extensively in the development of design and sizing of turbomachinery components. Specifically, the NASA Glenn Research Center in Cleveland, OH provides extensive software available for download related to sizing and design of compressors, turbines, compressible flows, fluid dynamics, and additional thermal, structural, acoustic, and electromagnetic modeling [1, 2, 3, 4]. In addition to NASA's software, companies like Concepts NREC provide wide-ranging software capabilities for preliminary sizing and design of turbomachinery which, like the NASA software, is largely developed from first principles based calculations . First principle implies physics and enthalpy-based calculations and correlations derived from machine geometry and fluid flow conditions, with conservation of mass, energy, and momentum applied. This method allows for design optimization of machine parameters while still maintaining accurate performance predictions over a wide range of turbomachinery designs. Another widely utilized software based on first principle derivations is numerical propulsion system simulation (NPSS), an often utilized toolset for steady-state calculations  via map-based turbomachinery calculations. It consists of vapor cycle and air cycle components for developing integrated system studies. Other software companies, such as Simerics Inc. , SoftinWay Inc. , and PCA Engineers Limited , provide detailed CFD calculations for modeling turbomachinery components. All of these software toolsets are capable of generating accurate performance maps of turbomachinery that can then be used in steady-state or transient modeling toolsets.
Despite the extensive capabilities available for estimating turbomachinery performance data, map-based transient studies suffer from several limitations. Firstly, modifications to a specific component design frequently require the generation of a new map; thereby, limiting one's ability to rapidly evaluate and optimize across a wide design space. Secondly, within a specific study, interpolation routines in map-based approaches fail to yield precise solutions near critical operating points, such as compressor stall lines due to unknown data outside of the stall point. In turn, control strategies around the stall margin of a machine are difficult due to model induced instabilities or incorrect predictions from interpolation around those operating points. Finally, modeling startup and shutdown requires torque prediction near or at zero speed, which for map-based approaches is ill-defined.
Based on the motivations described above, work has been explored regarding a first principles based transient toolset for air cycle machine (ACM) and turbomachinery based studies as an alternative to the traditional map-based performance calculations. The toolset is being developed as an extension of the AFRL Transient Thermal Modeling and Optimization (ATTMO) toolset [10-11]. Given the extensive capabilities of the ATTMO toolset in transient modeling of vapor cycle systems (VCSs) and additional heat transfer specific components, development of a first principles, transient ACM toolset for direct integration with ATTMO based components will allow for highly complex, integrated system studies that could make significant contributions in modeling and simulation of aerospace environments. Details of the initial first principles design calculations, and supporting transient calculations are provided below. As a preliminary verification study, steady-state performance maps were developed and correlated to the NASA based toolsets. As a representation of future capabilities of the ACM components developed, a final verification study was then performed via a system level transient model, the specifics of which are provided below.
In this effort, four main ACM components were developed: radial turbine, centrifugal compressor, axial turbine, and axial compressor. Additionally, for integrated dynamic studies, shaft and bearing loss components were developed and utilized within a Simulink environment. A first principles based design code was developed for each component based on relevant publications. In a first principles based code, ideal gas law and isentropic pressure rise equations are modified to account for losses within the ACM component. Examples of losses that result in inefficiencies in ACM performance include clearance between turbine/compressor blade and housing, incidence and deviation flow angles on rotor/stator blades, and friction losses. Detailed equations related to each ACM component are provided later in the report. Once design codes were developed for each component, Simulink blocks were established with dynamic calculations applied to the design equations for accurate transient performance.
While many references were employed in developing these components, the NASA publications are extensive and well detailed warranting a distinctive note [1, 2, 3, 4]. Additionally, books by Gorla  and Dixon  provided wide-ranging information regarding the design and performance of the turbomachinery established in this report, along with the relevant fluid mechanics required for thermodynamic analysis.
Figure 1 provides simplified two-dimensional schematics of radial turbine and centrifugal compressor components. In a radial turbine, air is input at the volute and forced through stationary nozzle blades, and then through the rotational impeller blades. The work performed by the fluid on the impeller blades, generates torque on the supporting shaft and results in a reduction of pressure and temperature of the fluid. In a nearly opposite performance, a centrifugal compressor is rotated by its supporting shaft forcing air in at the impeller and subsequently through the stationary diffuser (nozzle) blades and out through the volute. The work performed by the compressor on the fluid generates a rise in pressure and temperature of the air. Figure 2 provides example schematics for the axial turbine and axial compressor components, where stator blades are stationary and rotor blades rotate with the supporting shaft. The same general work generation of the radial turbine applies to the axial turbine, while the same work generation of the centrifugal compressor applies to the axial compressor.
Prior to providing detailed calculations for the various components, velocity diagrams (or velocity triangles) should be defined. Example velocity triangles are shown in Figure 3 for the radial turbine and centrifugal compressor, and velocity triangles for the axial components are provided in Figure 4. The absolute velocity (C), rotational velocity (W), and turbine/compressor velocity (U) are plotted. The absolute flow angles ([alpha]) will approximately match the blade angles of the stationary nozzle and stator blades, while the rotational flow angles ([beta]) will approximately match the blade angles of the rotational rotor blades, as demonstrated in Figures 3 and 4. For the equations provided within this report, flow velocities are defined as being positive if they match the direction shown in the schematic below, which depends on the direction of rotation of the ACM component. As shown in Figure 3, absolute velocities are positive in the direction of rotor rotation and rotational velocities are positive in the opposite direction of the rotor rotation. As shown in Figure 4, axial turbine absolute and rotational velocities are positive in the direction of rotor rotation at the inlet of the rotor, but are positive in the opposite direction of rotor rotation at the rotor outlet. For axial compressors, absolute velocities are always positive in the direction of rotor rotation while rotational velocities are always positive in the opposite direction of rotor rotation.
Turbomachinery can operate at high rotational speeds, generating large velocities within the air flow that can achieve supersonic conditions. As such it is important to differentiate between static, total, and rotational fluid properties. Static fluid properties contain no contributions due to velocity, while total properties contain contributions due to absolute velocities and rotational properties contain contributions due to rotational velocities. The static, total, and rotational temperatures are related by the following equations:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where t refers to total, s refers to static, and r refers to rotational, T is temperature, [gamma] is the heat capacity ratio, and M is the fluid Mach number. The Mach numbers utilized above are calculated as:
[M.sub.t] = C/c (3)
[M.sub.r] = W/c (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [R.sub.sp] is the specific gas constant of air, and c is the speed of sound. Pressure relations are then defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
The enthalpy of the fluid is a function of the temperature, and is utilized in the impeller/rotor work calculations. Recognizing that for high Mach numbers, total and/or rotational temperatures and pressures can be much larger than the static properties, it is critical to establish that only static properties are used for calculating fluid properties such as density, viscosity, and specific heat. Total and rotational properties will still play crucial roles in the ACM calculations, as is shown later in this section.
Radial Turbine Design
The first ACM component developed in this effort was the radial turbine. The radial turbine consists of three main segments: the volute, the nozzle (stator blades), and the impeller (rotor blades). Within and between each segment, loses can occur. Volute losses were established based on work performed by Vilim at Argonne National Laboratory . Nozzle losses were taken from work by Benson  and expansion and impeller losses were derived from NASA software . Across the volute and nozzle, it is assumed that no total enthalpy losses occur. Total pressure losses will occur, though they are typically small. Velocities are calculated at volute inlet/outlet and nozzle inlet/outlet in order to ensure accurate calculation of fluid properties which contribute significantly in loss calculations.
In the radial turbine, the fluid performs work on the impeller blades resulting in a drop in enthalpy of:
[dh.sub.imp] = [U.sub.2][C.sub.[theta]2] - [U.sub.1][C.sub.[theta]1] (8)
where [U.sub.2] is the impeller velocity at its outer diameter (inlet), [C.sub.[theta]2] is the rotational component of fluid absolute velocity at the impeller inlet, [U.sub.1] is the impeller velocity at its inner diameter (outlet), and [C.sub.[theta]1] is the rotational component of fluid absolute velocity at the impeller outlet. With the impeller work known, the outlet to inlet temperature ratio can be calculated as:
[T.sub.t1]/ [T.sub.tv] = 1 - [dh.sub.imp]/ [C.sub.p] [T.sub.tv] (9)
where [T.sub.tv] is the volute inlet total temperature, [T.sub.t1] is the impeller exit total temperature, and [c.sub.p] is the specific heat of the fluid. In turn the pressure ratio is calculated as:
[P.sub.t1]/[P.sub.tv] = (1 - (1 - [T.sub.t1]/[T.sub.tv])/[[eta]).sup.[gamma]/([gamma]-1)] (10)
where [P.sub.tv] is the volute inlet total pressure, [P.sub.t1] is the impeller exit total pressure, and [eta] is the total efficiency of the radial turbine. Finally, the efficiency of the turbine is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [dh.sub.Vf] are volute losses due to friction, [dh.sub.Vex] are volute expansion losses due to the change in area of the flow path along the volute, [dh.sub.N] are the nozzle losses, [dh.sub.Nexp] are nozzle exit expansion losses, [dh.sub.VS] are friction losses within the vaneless space between the nozzle and impeller, [dh.sub.inc] are impeller incidence and deviation flow losses, and [dh.sub.iexp] are impeller exit expansion losses.
In order to maintain a concise report, details of the loss equations are provided in the Appendix section, and many more specific calculations are detailed in the references provided.
Centrifugal Compressor Design
The majority of calculations utilized for the centrifugal compressor were adopted from work performed by Gravdahl [16, 17, 18, 19]. Gravdahl has worked extensively on dynamic modeling of centrifugal compressors with a particularly emphasis on control strategies around the stall region. His work, along with that executed by Jiang , provided the foundation for the compressor modeling performed in this report. In turn, the turbomachinery loss calculations provided by NASA  were studied and implemented. In this report, the NASA/Gravdahl calculations were utilized for impeller and diffuser loss calculations, while volute losses were developed from the turbine loss equations provided by Vilim .
The centrifugal compressor can be treated as an inversion of the radial turbine with fluid entering the inner diameter of the impeller, and exiting out of the volute. In a compressor, the impeller performs the work on the fluid generating a rise in enthalpy and pressure. Equation 8 is again utilized to describe the change in enthalpy across the compressor, with the additional assumption that [C.sub.[theta]1] is zero as the fluid entering the impeller will have no rotational component. The temperature and pressure rise across the impeller are then described by:
[T.sub.tv]/[T.sub.t1] = 1 + [dh.sub.imp]/[c.sub.p] [T.sub.t1] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
The efficiency of the compressor is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [dh.sub.inc] are impeller and diffuser incidence flow losses, [dh.sub.Nf] are the diffuser friction losses, [dh.sub.if] are the impeller friction losses, [dh.sub.bld] are impeller blade loading losses, and [dh.sub.mix] are impeller fluid mixing losses.
Total enthalpy is assumed to be constant from the impeller exit to the volute exit, while total pressure is allowed to rise. Velocities are again evaluated at impeller inlet/outlet, diffuser inlet/outlet, and volute inlet/outlet in order to accurately calculate static fluid properties and loss equations. For both the radial turbine and centrifugal compressor, the velocity between the impeller and diffuser (nozzle) is typically most susceptible to choked flow conditions (Mach number greater than 1), and therefore must be monitored closely when designing these components. Further details regarding centrifugal loss correlations are provided in the Appendix.
Axial Turbine Design
Axial components consists of alternating stator and rotor blade segments in which stator blades are placed along the stationary outer casing and rotor blades are placed along the diameter of the rotating shaft. The axial component design calculations vary significantly from the radial components. Axial turbine losses have been well-studied and are accurately depicted via correlations developed by Ainley and Mathieson . Sawyer's book on turbine theory and design provides loss correlations for stator and rotor blades that have precisely predicted axial turbine losses for decades . As such, they were utilized in this effort for axial turbine design considerations. Additionally, incidence loss calculations for off-design conditions were adopted from work by Horlock .
For the axial turbine stator blades, total enthalpy is assumed constant across blade length. The change in total pressure is calculated via isentropic assumptions/losses where the static enthalpy of air at the stator exit under isentropic expansion ([h.sub.1i]) is calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
where [h.sub.1s] is the actual stator outlet static enthalpy, [C.sub.1] is the absolute fluid velocity at the stator exit, [[epsilon].sub.si] is the stator loss coefficient and, [[alpha].sub.0] is the stator inlet absolute flow angle, and [[beta].sub.sb_0] is the stator inlet blade angle. The isentropic outlet temperature of the fluid can be calculated as a function of the isentropic outlet enthalpy, and in turn the static pressure at the outlet ([P.sub.1s]) can be calculated from the isentropic relation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
where [P.sub.0s] is the static inlet pressure, [T.sub.1i] is the isentropic stator outlet temperature, and [T.sub.0s] is the inlet static temperature. With the static pressure established, total pressure can be calculated via Equation 6.
The stator outlet conditions correspond to the rotor inlet conditions. For the rotor blade section, the work performed by the fluid on the rotor blades is calculated as:
[dh.sub.imp] = [U.sub.2][C.sub.[theta]2] + [U.sub.1][C.sub.[theta]1] (17)
where 2 refers to the rotor exit, and 1 refers to the rotor inlet. There is a change in sign in Equation 17 compared to Equation 8. This is due to the change in sign that was established with the flow triangles for axial turbines in Figure 4. The outlet total enthalpy of the rotor blade section is then simply:
[h.sub.2t] = [h.sub.1t] [dh.sub.imp] (18)
where [h.sub.1t] is the rotor inlet total enthalpy and [h.sub.2t] is the rotor outlet total enthalpy. The rotor outlet static pressure is calculated in a manner similar to the stator blade section where:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
where [h.sub.2] is the static enthalpy at the exit of the rotor under isentropic expansion, [h.sub.2s] is the actual static enthalpy at the rotor exit, [W.sub.2] is the rotational velocity at the rotor exit, [[epsilon].sub.rt] is the rotor loss coefficient, [[beta].sub.1] is the rotational velocity angle at the rotor inlet, and [[beta].sub.rb_1] is the rotor blade angle at the rotor inlet. Once again, rotor outlet isentropic temperature is calculated as a function of isentropic enthalpy and the static outlet pressure of the rotor is calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
where [T.sub.2i] is the isentropic rotor outlet temperature. If another stage of the turbine exists, the outlet conditions of the rotor are the same as the inlet conditions of the next stator segment. This process of calculating exit enthalpy/pressure along each turbine segment is continued for each stage of the turbine.
Once each segment is evaluated, total turbine pressure ratio ([PR.sub.T]) and exit total temperature under isentropic expansion ([T.sub.es]) are calculated as:
[PR.sub.T] = [P.sub.0t]/[P.sub.et] (21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
where [T.sub.0t] is the turbine inlet total temperature, [P.sub.0t] is the turbine inlet total pressure, and [P.sub.et] is the turbine exit total pressure. Turbine exit total isentropic enthalpy is calculated as a function of exit total isentropic temperature and in turn the total turbine efficiency ([[eta].sub.T]) is calculated as:
[[eta].sub.T] = ([h.sub.ot] - [h.sub.et])/([h.sub.ot] - [h.sub.ets]) (23)
where [h.sub.0t] is the turbine inlet total enthalpy, [h.sub.et] is the turbine outlet total enthalpy, and [h.sub.ets] is the turbine exit total isentropic enthalpy.
Further details regarding the stator and rotor loss coefficients can be found in Chapter 4 of Sawyer's book, written by Carmichael .
Axial Compressor Design
Though the axial compressor can principally be evaluated similarly to the axial turbine, a different strategy was utilized for calculating axial compressor performance. Extensive details regarding available loss correlations from Tuncer  and Falck  were researched for this effort. Ultimately, the design calculations were adopted from work developed by Tournier . Similar to the axial turbine, it is assumed that total enthalpy is constant along the stator blade. Loss calculations are handled differently in the compressor though. The change in total pressure along the stator blade is calculated as:
[P.sub.1t] = [P.sub.0t] - [Y.sub.st] ([P.sub.0t] - [P.sub.0s]) (24)
Where [P.sub.0s] is the stator static inlet pressure, [P.sub.0t] is the stator total inlet pressure, [P.sub.1t] is the stator total outlet pressure, and [Y.sub.st] is the stator loss coefficient. For calculating the work performed by the stator blades on the fluid, a 'rothalpy' term (I) is introduced and for the stator outlet it is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
where [h.sub.1s] is the static enthalpy at stator exit, [W.sub.1] is the rotational velocity at the stator exit, and [U.sub.1] is the turbine speed at the stator exit
The rothalpy is assumed to be constant along the rotor blade length. From this assumption, the rotor outlet enthalpy is calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
where [h.sub.ss] is the static enthalpy at rotor exit, [W.sub.2] is the rotational velocity at the rotor exit, and [U.sub.2] is the turbine speed at the rotor exit. The pressure rise across the rotor is calculated as a function of rotational pressure:
[P.sub.2r] = [P.sub.1r] - [Y.sub.rt]([P.sub.1r] - [P.sub.1s]) (27)
where [P.sub.2r] is the rotor rotational outlet pressure, [P.sub.1r] is the rotor rotational inlet pressure, [P.sub.1s] is the rotor static inlet pressure, and [Y.sub.rt] is the rotor loss coefficient. All forms of static and total properties are then calculated via the relations established by Equations 1, 2, 3, 4, 5_, 6, 7.
Once each segment is evaluated, total compressor pressure ratio ([PR.sub.C]) and exit total temperature under isentropic compression ([T.sub.ets]) are calculated as:
[PR.sub.C] = [P.sub.et]/[P.sub.0t] (28)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
The total compressor efficiency ([[eta].sub.C]) is then calculated as:
[[eta].sub.C] = ([h.sub.ets] - [h.sub.0t])/([h.sub.et] - [h.sub.0t]) (30)
Further details regarding the stator and rotor loss coefficients can be found in Tournier's publication .
With radial, centrifugal, and axial ACM component thermodynamic and loss calculations established, dynamic Simulink blocks were developed. Following the established ATTMO technique, inlet total pressure, outlet total enthalpy, and outlet flow rate were calculated as transient states. Transient equations were established for the radial turbine and centrifugal compressor based on work by Gravdahl [16, 17, 18, 19]. The loss equations described prior were utilized to estimate pressure ratios, efficiencies, and compressor/turbine work and torque values that drive the state equations. The inlet pressure state was calculated as:
[dP.sub.0t]/dt = [T.sub.0t][R.sub.sp]([[??].sub.in] - [[??].sub.out])/V (31)
where V is the turbine/compressor volume, [??] is the fluid flow rate, and t is time. With inlet flow rate known, the outlet flow rate of the compressor is calculated as:
[d[??].sub.out]/dt = [V.sup.2]([PR.sub.C][P.sub.0t] - [P.sub.2t])/(2D) (32)
where D is the nozzle/diffuser outer diameter, inlet and outlet total pressure ([P.sub.0t] and [P.sub.2] respectively) are known state variables at a given time step, and [PR.sub.C] is the pressure ratio resulting from the design equations. [PR.sub.C]*[P.sub.0t] equates to [P.sub.2t] calculated from the design code and as such, Equation 32 drives the outlet flow rate based on the difference between the outlet total pressure value at the previous time step, and the outlet total pressure value calculated from the design code. Similarly, the radial turbine outlet flow rate is calculated as:
[d[??].sub.out]/dt = [V.sup.2]([P.sub.0t]/[PR.sub.T] - [P.sub.2t])/(2D) (33)
The outlet enthalpy state equation for the centrifugal compressor is:
[dh.sub.2t]/dt = ([[??].sub.in][h.sub.0t] - [[??].sub.out][h.sub.2t] + [[??].sub.out][dh.sub.imp])/[[rho].sub.0]V (34)
where [[rho].sub.0] is the inlet density of the fluid. Similarly for the radial turbine, the outlet enthalpy is calculated as:
[dh.sub.2t]/dt = ([[??].sub.in][h.sub.0t] - [[??].sub.out][h.sub.2t] - [[??].sub.out][dh.sub.imp])/[[rho].sub.0]V (35)
For the Simulink component models, density values are calculated at the inlet and exit of the turbine/compressor with an additional value calculated between the nozzle and impeller. Each density value is utilized appropriately for calculating velocity and loss calculations.
Different transient state calculation techniques were required for the axial components. State variables were only required at the inlet and outlet of the radial turbine and centrifugal compressor blocks where velocities are relatively small and therefore the difference between total and static conditions are not very large. For axial components, state variables are required at the inlet and outlet of each stator/rotor segment. The states within the axial components contain high velocities where the differences between total and static conditions are very large. Additionally, instabilities in flow rate and pressure calculations can arise when a large number of segments are implemented. As such, the transient calculations were developed for static density states at volumes between segments. This technique provided higher numerical stability in part because large enthalpy and pressure changes are not required in the dynamic equations.
The volume dynamics utilized for the axial components were once again developed by work performed at NASA . In this strategy, total temperature, static density, and outlet flow rate were calculated in a small volume in between each segment. The outlet flow rate is calculated by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)
where A is the flow cross-sectional area at the segment entrance, and L is the flow length of the segment. Total temperature is calculated via the equation:
[dT.sub.t_out]/dt = ([[??].sub.in][T.sub.t_in] - [[??].sub.out][T.sub.t+out])[gamma]/[rho]V (37)
Finally, the static density is calculated as:
d[rho]/dt = ([[??].sub.in] - [[??].sub.out])/V (38)
While the inlet pressure is not calculated directly from these equations, it can be calculated from the density and temperature state equations at a given time:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)
With the appropriate dynamics for fluid and ACM component models developed, the ACM shaft dynamics must also be implemented. Shaft speed was established within ATTMO as being a function of internal ACM component torques and externally applied torques. Torque and moment of inertia calculations are implemented into each ACM component and feed into the dynamic shaft calculations. Additionally, bearing losses can be implemented into the dynamics that restrict the torque applied to the shaft. The overriding transient equation for shaft speed is adopted from Gravdahl :
dN/dt = (30 [summation][TAU])/([pi][summation]J) (40)
where N is the shaft speed, [tau] is the torque of an ATTMO component, and J is the moment of inertia of an ATTMO component . Every component attached to a given shaft contributes to the total torque and moment of inertia, hence the summation terms in Equation 40.
For map-based calculations, torque must be calculated as a function of speed in the denominator, meaning that torque at zero speed cannot be defined. This limits any zero speed startup models that would want to be modeled. In the first principle models, with component geometries known, the Euler turbomachinery equations can be utilized to calculate torque allowing for zero speed calculations. The torque equation for the radial turbine is:
[[TAU].sub.T] = [[??].sub.in][r.sub.2][C.sub.[theta]2] - [[??].sub.out][r.sub.1][C.sub.[theta]1] (41)
where [r.sub.2] is the outer impeller radius, and [r.sub.1] is the impeller inner radius. For an axial turbine rotor stage, the torque is calculated as:
[[TAU].sub.T] = [[??].sub.in][r.sub.2][C[theta].sub.2] + [[??].sub.out][r.sub.1][C.sub.[theta]1] (42)
The torques of every rotor stage must be summed together to provide the total torque of the axial turbine. The equation for the centrifugal compressor and axial compressor stage torque is the same:
[[TAU].sub.T] = [[??].sub.out][r.sub.1][C.sub.[theta]1] - [[??].sub.in][r.sub.2][C.sub.[theta]2] (43)
As with the axial turbine, the torques of every rotor stage of the axial compressor must be summed together for the total torque. The turbine torque equations will always result in a positive value while the compressor torque equation will always result in a negative value. Therefore, compressor torques work to reduce shaft speed while turbine torques increase shaft speed. The moment of inertia for each component is approximated using either conical or cylindrical shapes, or if known more precisely can be entered as a user input.
RESULTS AND DISCUSSION
As an initial verification study, performance maps were generated within ATTMO and NASA based ACM software and respective results were compared. Turbine and compressor performance and geometries at the chosen design point were input to both sets of software and results are provided below.
The corrected mass flow rate is specified for both turbines and compressors but they are defined differently. For the turbines, corrected flow is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)
where [T.sub.in] is the fluid inlet temperature and [P.sub.in] is the fluid inlet pressure. For the compressors, corrected flow is calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)
where [T.sub.std] is the fluid standard temperature (288.2 K) and [P.sub.std] is the fluid standard pressure (101.3 psi).
The NASA sizing and design codes are complex and often rely on assumptions regarding where choked and stalled flow conditions occur. Many additional loss values and reductions in efficiencies can also be input. As such, some variation in the outputted NASA performance maps is expected depending on how input fles are manipulated by the user. In this study though, the default values were utilized almost exclusively with the notable exception of stall and choke flow conditions in the axial compressor which were extended in order to acquire more data than was provided by the default NASA code.
Radial Turbine Results
For the radial turbine, NASA software was utilized to size a turbine with air inlet temperature of 130 [degrees]C, inlet pressure of 1440 kPa, and speed of 50000 rpm. Additionally, the design flow rate was chosen to be 1.7 kg/s and a turbine power was calculated as 230 kW for an efficiency of 90%. A variety of assumptions not specified here regarding ratios of various impeller/nozzle radii and blade angles were also input. The resulting impeller consists of 10 blades and a radius of 7.6 cm. More details regarding the turbine geometry are provided in the Appendix. Figure 5 provides corrected flow versus pressure ratio for the ATTMO and NASA based results. The choked flow of the ATTMO design is slightly higher than that of the NASA design. Otherwise, the corrected flow rates and pressure ratios matched well.
Figure 6 provides efficiency versus pressure ratio for the ATTMO and NASA based results. The ATTMO and NASA efficiency curves are largely in agreement at pressure ratios above 2.0, but the NASA code provides a more rapid reduction in efficiency as pressure ratio approaches 1.0. The ATTMO code currently provides slightly higher efficiencies than the NASA code at higher speeds potentially suggesting that incidence losses calculated within the ATTMO code are too small. Future efforts in experimental validation of the ATTMO software could motivate improvements in loss calculation accuracy.
Centrifugal Compressor Results
NASA software was utilized to size a 50000 rpm centrifugal compressor with fluid inlet temperature of 55 [degrees]C and inlet pressure of 270 kPa. The design flow rate was selected as 1.5 kg/s with a total compressor power of 700 kW at an efficiency of 80 %. As with the radial turbine, a number of assumptions were made regarding the anticipated Mach numbers, blade angle and geometries. An 11.2 cm impeller with 15 blades was sized by the NASA software. Further details regarding the geometry of the compressor are provided in the Appendix.
Results from the centrifugal compressor comparison study are provided in Figure 7. Overall the ATTMO design code tends to have a slightly higher efficiency and reduced pressure ratio while extending out to higher choked flows. The stall line is calculated and plotted in the ATTMO plot (dashed blue line), while the lowest flow point on a given speed line of the NASA performance map is the approximate stall point. The stall lines of the two maps are reasonable close. As with the radial turbine, the ATTMO code provides calculations under low flow / stalled flow conditions which NASA does not perform by default.
While the stall region is not functional to real-world centrifugal compressors, and therefore does not strictly provide an accurate representation of a compressor operating in this region, the capability to model within a stalled region is critical from a model stability and controls perspective. Within a transient Simulink model, an inaccurate interpolation/extrapolation of the NASA contour map shown above may suggest that an increase in pressure ratio and or efficiency is achievable in a stall region which could fool control strategies into forcing the compressor into an inoperable range. With the ATTMO software, direct calculation of the stall line in real time provides better controls to the compressor block by utilizing knowledge of where the compressor could not operate. Decreases in pressure ratio and efficiency in the stall region provide an approximate representation of what would occur were a centrifugal compressor to slip into stalled flow conditions.
Axial Turbine Results
The axial component comparisons were performed slightly differently from the radial turbine and centrifugal compressor studies. Rather than utilizing the NASA software to size the components, geometries and blade angles were established within the ATTMO design codes and then translated to the NASA off-design software for calculations. For the axial turbine off-design software, blade radii and angles were required for the NASA software along with approximate efficiency at the design point. The selected geometries are provided in the Appendix. A 50000 rpm, single-stage axial turbine was calculated for a fluid inlet temperature of 130 [degrees]C and inlet pressure of 1440 kPa. The turbine stage operates at a design power of 115 kW and flow rate of 2.0 kg/s.
Corrected flow rate versus pressure ratio for the axial turbine study are provided in Figure 8. In this example, the corrected flow rate curves overlap closely, though the ATTMO code provides a choked flow value slightly higher than the NASA code. Figure 9 provides efficiency versus pressure ratio curves for the ATTMO and NASA results. Good agreement is demonstrated between the two sets of performance curves at higher turbine speeds. The NASA code provides lower efficiencies at low speed and small pressure ratios than what is provided within the ATTMO code. The two sets of curves do tend towards similar efficiencies at higher pressure ratios (near choked conditions) regardless of turbine speed.
Axial Compressor Results
A final comparison study was performed with the axial compressor design code. Design point performance was selected within the ATTMO design code, and the given set of parameter inputs was utilized within the NASA software. The NASA software required blade radii and angles, as well as stage by stage pressure ratio and efficiencies at the design point as inputs to the off-design software, which were easily extracted from the ATTMO design code. A two-stage 50000 rpm axial compressor was developed with inlet temperature of 55 [degrees]C and inlet pressure of 270 kPa. The design flow rate was approximately 2.0 kg/s with a resulting compressor power of 670 kW. Precise geometries are provided in the Appendix section.
Performance contour maps generated by the ATTMO and NASA design codes are plotted in Figure 10. Currently, the ATTMO axial compressor design code does not calculate a stall line and as such is not provided. The NASA developed performance data provides efficiency values between 49 - 57 % and as such the resulting contour plot is relatively limited in the information provided. The ATTMO design code calculates slightly higher efficiencies and pressure ratios at low speeds than what the NASA code predicts. In the NASA contour, the pressure ratios tend to decrease much more rapidly away from the maximum efficiency and pressure ratio for a given speed line than what is demonstrated in the ATTMO design code. Finally, the peak efficiencies and pressure ratios for a given speed line tend to occur at higher flow rates within the NASA code than what was demonstrated in the ATTMO code.
As with the centrifugal compressor design code, a much larger range of performance is calculated relative to the NASA software, which will in turn enhance performance in a transient simulation relative to an interpolated performance map. Future implementation of real-time stall calculations will need to be implemented in order to further enhance control strategies for the axial compressor Simulink component model.
Integrated Model Results
While comparisons of the performance calculations were readily available through the NASA public domain code, such comparisons were not easily obtainable for dynamic simulations. Future studies may allow for model to hardware comparisons, but for this paper the models were presented in their integrated form solely to better illustrate how the software toolset utilizes the dynamic models, rather than to validate against known results.
To test the integration of the dynamic code, a simple open loop ACM topology was selected. This approach, commonly known as a bootstrap configuration, utilizes high pressure, high temperature air taken from the engine that is first cooled, then compressed, then cooled again before delivery to an expansion turbine through which cold air is generated. The expansion turbine is used to drive the compressor, with the higher pressure ratio observed at the expansion turbine providing the necessary power to overcome the inefficiencies of the relatively lower pressure ratio compressor. An image of the integrated model is shown in Figure 11.
The models are connected as one might see in a schematic. To satisfy the component dynamic equations, information must be passed from upstream components to downstream components. The ATTMO modeling framework implements intelligent connection logic to pass information, eliminating additional overhead from end-users in signal line creation while maintaining a cleaner model appearance and minimizing user error.
In the proposed demonstration, a control was built to regulate the flow valve at the inlet to the primary heat exchanger. The control objective is the expansion turbine's exit temperature, which was targeted to 10 [degrees]C. The ACM is initialized at zero speed. During the simulation, a step change is imposed to the secondary side of the heat exchangers in the form of a sudden drop in air flow at 100 s. At 200 s, a sudden drop in air flow temperature also occurs. The control must therefore respond to these external forcing functions as their impacts drive the expansion turbine outlet temperature. One might expect that a sudden drop in air flow would result in higher temperatures at the inlet to the expansion turbine, forcing the controller to drive more flow through the cycle. The inverse would occur if there was a sudden drop in secondary side temperature, allowing the flow to decrease through the expansion turbine to maintain the same temperature. Results for the calculated ACM shaft speed and target outlet temperature are shown in Figures 12 and 13, respectively
The results are as anticipated with a speed increase observed when the secondary flow decreases, and a speed decrease when the secondary flow temperature suddenly drops. The impacts are more substantial for the temperature drop than the flow drop. This is likely due to the fact that the heat transfer in the heat exchangers is limited by the much lower flow rate of the ACM, whereas a temperature drop on the secondary side can substantially increase the heat transfer to the secondary side. Both the ACM speed response and the target temperature controlled response reflect these relative impacts.
At startup there is a much larger undershoot in target temperature while the machine gets up to speed and the control can stabilize. The machine rapidly accelerates as there is very little inertia on the shaft. The secondary flow decrease forces the system to work marginally harder with the control increasing flow and subsequent ACM speed. A large undershoot is observed when the secondary flow temperature decreases as the system is operating at too high of a capacity for those boundary conditions. The dynamics are not purely plant based but a combination of the plant, controls, and forcing functions. The dynamic models serve to facilitate these complex interactions to form a total system dynamic response representative of actual hardware.
While many excellent turbomachinery sizing and design software toolsets are readily available, they often consists of CFD based calculations and/or high fidelity calculations meant to develop steady-state performance maps. Use of these performance maps in transient studies can be limited, depending on the application. While first principles based calculations are often utilized in preliminary sizing and design software for developing performance maps, the application of first principles based transient turbomachinery toolsets has not been widely developed and utilized.
An initial verification study of first principles based turbomachinery design codes for utilization within the ATTMO toolset was presented in this paper. Comparisons of the ATTMO performance maps were made to existing NASA software with results being largely comparable in terms of pressure ratios and efficiencies of the ACM components. The complexity of the NASA software allows for the possibility of extending ranges of the resulting performance maps presented here, and allows for some variation in pressure ratio and efficiency, suggesting that notable uncertainty in the comparisons provided may be present. However, given that NASA software has been well documented and verified and validated in literature [1, 2, 3, 4], initial results are promising.
Dynamic Simulink component models have also been developed for transient calculations utilizing the calculation techniques described in this report allowing for first principles based, dynamic calculations of turbomachinery that can easily be integrated into system level transient. Initial results provided in an integrated model study verify the ability to model zero speed conditions for shaft, compressor, and turbine dynamics while demonstrating proper control implementation. Transient responses/delays were evident in shaft speed and turbine temperature responses. Future work will include the development of an ACM sizing toolset in addition to the design calculations presented here. Optimization of the component loss calculations and control strategies will continue to be investigated.
[1.] Meitner, P.L., and Glassman, A.J., "Computer Code for Off-Design Performance Analysis of Radial-Inflow Turbines With Rotor Blade Sweep," NASA Technical Paper 2199, AVRADCOM Technical Report 83-C-4, Cleveland, 1983
[2.] Galvas, M.R., "Fortran Program for Predicting Off-Design Performance of Centrifugal Compressors," NASA Technical Note: NASA TN D-7487, Washington D.C., 1973
[3.] Chen, S.C.S., "Preliminary Axial Flow Turbine Design and Off-Design Performance Analysis Methods for Rotary Wing Aircraft Engines; I-Validation," Prepared for the 65th Annual Forum and Technology Display, USA, May 27-29, 2009
[4.] Steinke, R.J., "A Computer Code for Predicting Multistage Axial-Flow Compressor Performance by a Meanline Stage-Stacking Method," NASA Technical Paper 2020, Cleveland, May 1982
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[10.] Kania, M., Koeln, J., Alleyne, A., McCarthy, K., et al., "A Dynamic Modeling Toolbox for Air Vehicle Vapor Cycle Systems," SAE Technical Paper 2012-01-2172, 2012, doi: 10.4271/2012-01-2172
[11.] McCarthy, K., McCarthy, P., Wu, N., Alleyne, A., et al., "Model Accuracy of Variable Fidelity Vapor Cycle: System Simulations," SAE Technical Paper 2014-01-2140, 2014, doi: 10.4271/2014-01-2140
[12.] Gorla, R.S.R., and Khan, A.A., "Turbomachinery: Design and Theory" (New York, Marcel Dekker, Inc., 2003), ISBN: 0824709802
[13.] Dixon, S.L., and Hall, C.A., "Fluid Mechanics, Thermodynamics of Turbomachinery, Sixth Edition,"(Boston, Elsevier, 2010), 93-136, doi:10.1016/B978-1-85617-793-1.00024-9Contact Information
[14.] Vilim, R.B., "Development and Validation of a Radial Inflow Turbine Model for Simulation of the SNL S-C[O.sub.2] Split-Flow Loop," Report Prepared for Argonne National Laboratory, Nuclear Engineering Division, ANL-ARC-195, Aug. 15, 2011
[15.] Benson, R.S., "A Review of Methods For Assessing Loss Coefficients in Radial Gas Turbines," International Journal of Mechanical Sciences, 12, 905-932, 1970, doi:10.1016/0020-7403(70)90027-5
[16.] Gravdahl, J.T., and Egeland, O., "Centrifugal Compressor Surge and Speed Control," IEEE Transactions on Control Systems Technology, 7(5), 567-579, 1999, doi: 10.1109/87.784420
[17.] Gravdahl, J.T., Willems, F., de Jager, B., and Egeland, O., "Modeling of Surge in Free-Spool Centrifugal Compressors: Experimental Validation," Journal of Propulsion and Power, 20(5), 849-857, 2004, doi: 10.2514/1.10052
[18.] Gravdahl, J.T., "Modeling and Control of Surge and Rotating Stall in Compressors," PhD Thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology, Norway, 1998
[19.] Gravdahl, J.T., Egeland, O., and Vatland, S.O., "Drive Torque Actuation in Active Surge Control of Centrifugal Compressors," Automatica, 38(11), 1881-1893, 2002, doi: 10.1016/S0005-1098(02)00113-9
[20.] Jiang, W., Khan, J., and Dougal, R.A., "Dynamic Centrifugal Compressor Model for System Simulation,", Journal of Power Sources, 158(2), 1333-1343, 2006, doi: 10.1016/j.jpowsour.2005.10.093
[21.] Ainley, D.G., and Mathieson, G.C.R., "A Method of performance estimation for axial-flow turbines," Ministry of Supply: Aeronautical Research Council Reports and Memoranda, No. 2974, London, England, 1951
[22.] Sawyer, J.W., and Japikse, D., "Sawyer's Gas Turbine Engineering Handbook, Volume 1: Theory & Design, Third Edition" (Norwalk, Turbomachinery International Publications, 1985), 62-78, ISBN: 0937506141
[23.] Horlock, J.H., "Losses and Efficiencies in Axial-Flow Turbines," International Journal of Mechanical Sciences, 2(2), 48-75, 1960, doi: 10.1016/0020-7403(60)90013-8
[24.] Tuncer, O., "Axial Compressor Design: Part II," Istanbul Technical University Presentation, Dec. 2011
[25.] Falck, N., "Axial Flow Compressor Mean Line Design," Master Thesis, Department of Energy Sciences, Lund University, Sweden, 2008
[26.] Tournier, J.M., and El-Genk, F.S., "Axial Flow, Multi-Stage Turbine and Compressor Models," Energy Conversion and Management 51: 16-29, 2010, doi: 10.1016/j.enconman.2009.08.005
[27.] Kopasakis, G., Connolly, J.W., Paxson, D.E., and Ma, P., "Volume Dynamics Propulsion System Modeling of Supersonics Vehicle Research," Journal of Turbomachinery, 132(4), 041003-041011, 2010, doi: 10.11510.1115/1.3192148
PC Krause & Associates Inc.
Air Force Research Laboratory
Kevin McCarthy and Eric Walters
PC Krause & Associates Inc.
US Air Force Research Laboratory
PC Krause and Associates
West Lafayette, IN, USA
This work was supported by the Air Force Research Laboratory, Wright-Patterson Air Force Base under contract AF-FA8650-12-C-2231.
PC Krause and Associates, Inc. (PCKA) and the University of Illinois - Urbana-Champaign (UIUC) via Air Force Research Laboratory (AFRL) SBIR funding and in collaboration with the AFRL researchers have developed and validated ATTMO software to support the design and analysis of aircraft thermal management systems through advanced modeling and simulation of vapor-cycle and air-cycle systems. ATTMO is built as an open MATLAB/Simulink toolbox wherein the engineer can customize the modeled physics to better represent the system as necessary.
AFRL - Air Force Research Laboratory
ACM - Air Cycle Machine
ATTMO - AFRL Transient Thermal Management and Optimization
NASA - National Aeronautics and Space Administration
NPSS - Numerical Propulsion System Simulation
PCKA - PC Krause and Associates
UIUC - University of Illinois, Urbana-Champaign
VCS - Vapor Cycle System
Concise loss calculations are outlined in this Appendix section. Some loss correlations require further calculations and detailed information so appropriate references are provided where such information is needed.
RADIAL TURBINE LOSS CALCULATIONS
The radial turbine volute friction loss is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)
where f is the Darcy friction factor of the volute, [L.sub.v] is the volute flow length, [C.sub.v] is the average flow velocity in the volute, [D.sub.h_v] is the hydraulic diameter of the volute, [Re.sub.v] is the Reynolds number based on [C.sub.v] and [D.sub.h_v], and [lambda] is the surface roughness. The volute expansion loss is calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3)
where [C.sub.in] is the velocity at the inlet of the volute, [r.sub.in] is the radius of the turbine at the middle of the volute inlet, [r.sub.out] is the radius of the turbine at the volute exit, [C.sub.[theta]_out] is the rotational component of absolute velocity at the volute exit, and [C.sub.out] is the total velocity at the volute exit. Further details regarding volute losses can be found in Vilim's work .
The nozzle loss is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4)
where [[eta].sub.N] is the nozzle loss coefficient, defined by a plot provided by Benson , and [C.sub.N] is the nozzle exit velocity. The nozzle expansion term is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A5)
where [C.sub.x_N] is the axial component of the velocity at the nozzle exit, and [C.sub.x_VS] is the axial component of the velocity at the vaneless space exit. As the fluid exits the nozzle section, the flow area rapidly increases as the stator blades end. This rapid increase in area generates losses. The impeller expansion loss is similarly calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)
where [C.sub.x_iout] is the axial component of the velocity at the impeller exit, and [C.sub.x_iVS] is the axial component of the velocity at the vaneless space past the impeller exit, and [C.sub.iout] is the total impeller exit velocity. Further details regarding this nozzle and impeller expansion losses are provided by NASA documentation .
For the vaneless space loss, a different form of friction loss is utilized than what is provided in the volute:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8)
where [f.sub.VS] is the Darcy friction factor of the vaneless space, [L.sub.VS] is the vaneless space flow length, [C.sub.VS] is the flow velocity at the exit of the vaneless space, [D.sub.h_VS] is the hydraulic diameter of the vaneless space, and [Re.sub.VS] is the Reynolds number based on [C.sub.VS] and [D.sub.h_VS]. Finally, the incidence loss is calculated as a summation of losses due to incidence at the front of the impeller blades, friction along the impeller blades, and deviation at the exit of the impeller blades. Incidence occurs when the flow rotational velocity angle does not match the optimal angle (often taken as the blade angle) at the impeller inlet and deviation occurs when the rotational velocity angle does not match the optimal blade angle at the impeller outlet. Therefore, the total impeller losses are described as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)
[[beta].sub.inc] = ([beta]i_in - [[beta].sub.opt]) (A10)
[[beta].sub.dev] = ([beta]i_out - [[beta].sub.ib_out]) (A11)
where [W.sub.iout] is the rotational velocity at the outlet of the impeller, [[beta].sub.dev] is the deviation angle at the outlet of the impeller, [W.sub.i_in] is the rotational velocity at the inlet of the impeller, [[beta].sub.inc] is incidence angle at the inlet of the impeller, n equals 1.75 when [[beta].sub.inc] is greater than zero and 2.5 when [[beta].sub.inc] is less than zero, [[beta].sub.i_in] is the rotational velocity angle at the inlet of the impeller, [[beta].sub.i_out] is the rotational velocity angle at the outlet of the impeller, [[beta].sub.opt] is the optimal rotational velocity angle at the inlet of the impeller, and [[beta].sub.ib_out] is the blade angle at the outlet of the impeller. The optimal blade angle is defined within NASA documentation . The above correlation was developed specifically for this effort based on similar correlations by NASA.
CENTRIFUGAL COMPRESSOR LOSS CORRELATIONS
Centrifugal compressor volute losses are defined by the same volute loss equations of the radial turbine. However, as flow is moving in opposite directions between the two components, the ' in' and 'out ' terms must be reversed for the centrifugal compressor. Friction loss terms are similar to those outlined for the radial turbine, but are redefined for clarity. The diffuser (nozzle) friction term is defined as:
[dh.sub.Nf] = [2f.sub.D][L.sub.D][C.sub.D_in]/[D.sub.h_D] (A12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A13)
where [f.sub.D] is the Darcy friction factor of the diffuser, [L.sub.D] is the length of the diffuser, [C.sub.D_in] is the velocity at the inlet of the diffuser, and [D.sub.h_D] is the hydraulic diameter of the flow path in the diffuser. Definitions for [Re.sub.D] and [D.sub.h_D] differ from those in the radial turbine code, and can be found in the NASA documentation . The impeller friction loss term can be defined by Equations A12 and A13 as well with impeller specific friction factor ([f.sub.i]), length ([L.sub.i]), velocity ([V.sub.i_in]), Reynolds number ([Re.sub.i]), and hydraulic diameter ([D.sub.h_i]) replacing the diffuser specific terms in the equation. Once again, definitions for the specific parameters above can be found in the NASA documentation .
Compressor incidence loss terms include incidence losses at the entrance of the impeller and at the entrance to the diffuser:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A14)
where [D.sub.i_in] is the average diameter of the impeller at the impeller entrance, [omega] is the rotational speed of the compressor, [[beta].sub.ib_in] is the impeller blade angle at its entrance, [??] is the fluid flow rate, [[rho].sub.i_in] is the fluid density at the impeller entrance, [A.sub.i_in] is the flow area at the impeller entrance, [D.sub.d_in] is the diameter of the compressor at the diffuser entrance, [[beta].sub.db_in] is the diffuser blade angle at its entrance, [[rho].sub.d_in] is the fluid density at the diffuser entrance, and [A.sub.d_in] is the flow area at the diffuser entrance. Two additional loss terms are in the centrifugal compressor equations. A blade loading loss term is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A16)
where [W.sub.iout] is the rotational velocity at the impeller outlet [W.sub.itip_in] is the rotational velocity at the tip of the impeller blade at the entrance of the impeller, [U.sub.i_out] is the speed of the impeller at its exit, [D.sub.itip_in] is the diameter of the impeller tip at its entrance, [D.sub.i_out] is the diameter of the impeller blade at its exit, and Z is the number of impeller blades. Finally, a fluid mixing loss term is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A17)
where [t.sub.i] is the impeller blade thickness, [b.sub.d_out] is the width of the compressor at the diffuser exit, [b.sub.i_out] is the width of the compressor at the impeller exit, and [C.sub.i_out] is the absolute velocity at the impeller exit. Further details regarding velocity and loss calculations are provided in the NASA documentation  and Gravdahl publications [16, 17, 18, 19]. Some equations provided in the supporting documentations have been modified for this effort and therefore will not match all the equations provided here.
GRAPHICAL USER INTERFACES OF STEADY STATE CHARACTERIZATION
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|Author:||McCarthy, Patrick; Niedbalski, Nicholas; McCarthy, Kevin; Walters, Eric; Cory, Joshua; Patnaik, Soum|
|Publication:||SAE International Journal of Aerospace|
|Date:||Sep 1, 2016|
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