Patent Publication Number: US-2023141059-A1

Title: Thermal bias control in turbomachines

Description:
FIELD 
     The present disclosure generally pertains to managing residual heat within various regions of a turbomachine after a turbomachine has been shut down and/or prior to starting a turbomachine. 
     BACKGROUND 
     When a turbomachine is shut down after a period of operation, hot air within the turbomachine tends to rise, which may lead to uneven cooling within the turbomachine. As a result, for a period of time after a turbomachine has been shut down, various regions of the turbomachine may exhibit different amounts of thermal expansion, which is sometimes referred to as “rotor bow” or a “bowed rotor.” For example, a rotor shaft of the turbomachine may exhibit more thermal expansion at an upward portion of the rotor shaft relative to a downward portion of the rotor shaft. 
     Over time, thermal expansion within the turbomachine may generally approach an equilibrium. However, if a start-up method for a turbomachine is initiated under a bowed rotor condition, the turbomachine may exhibit vibration or cause airfoil blades to approach contact with a surrounding casing. 
     Accordingly, it would be welcomed in the art to provide turbomachines with improved capabilities for managing residual heat, including improved engine control system, and improved methods of operating a turbomachine, such as on-ground cooling methods, engine shut-down methods, and engine start-up methods. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A full and enabling disclosure, including the best mode thereof, directed to one of ordinary skill in the art, is set forth in the specification, which makes reference to the appended Figures, in which: 
         FIG.  1    schematically depicts an exemplary aircraft that includes one or more turbomachines; 
         FIGS.  2 A and  2 B  schematically depict exemplary embodiments of a turbomachine that includes a motoring system; 
         FIGS.  3 A and  3 B  schematically depict exemplary sensors and systems for temperature sensor measurements that may be determined from one or more sensors of a turbomachine; 
         FIG.  4    schematically depicts an exemplary thermal bias control module; 
         FIGS.  5 A and  5 B  schematically depict exemplary thermal bias models that may be included in a thermal bias control module; 
         FIGS.  6 A and  6 B  respectively show a chart depicting an example of thermal bias as a function of time after engine shut-off, and a corresponding chart depicting an example residual heat as a function of time after engine shut-off; 
         FIGS.  6 C and  6 D  respectively show a chart depicting another example of thermal bias as a function of time after engine shut-off, and a corresponding chart depicting an example residual heat as a function of time after engine shut-off; 
         FIG.  7    schematically depicts an exemplary control system; and 
         FIGS.  8 A- 8 D  show flow charts depicting an exemplary method of controlling thermal bias. 
     
    
    
     Repeat use of reference characters in the present specification and drawings is intended to represent the same or analogous features or elements of the present disclosure. 
     DETAILED DESCRIPTION 
     Reference now will be made in detail to exemplary embodiments of the presently disclosed subject matter, one or more examples of which are illustrated in the drawings. Each example is provided by way of explanation and should not be interpreted as limiting the present disclosure. In fact, it will be apparent to those skilled in the art that various modifications and variations can be made in the present disclosure without departing from the scope of the present disclosure. For instance, features illustrated or described as part of one embodiment can be used with another embodiment to yield a still further embodiment. Thus, it is intended that the present disclosure covers such modifications and variations as come within the scope of the appended claims and their equivalents. 
     The present disclosure generally pertains to managing residual heat within various regions of a turbomachine after a turbomachine has been shut down and/or prior to starting a turbomachine. The residual heat may be managed based at least in part on a thermal bias between an upward portion of the turbomachine and a downward portion of the turbomachine. The thermal bias may be determined by one or more temperature sensors. In some embodiments, the turbomachine may be rotated so that temperature measurements may be determined for the upward portion and the downward portion of the turbomachine using the same one or more sensors. Additionally, or in the alternative, improved methods of controlling thermal bias may be provided. For example, improved control systems may provide cooling treatment based on thermal bias determined from temperature measurements from one or more temperature sensors configured to determine temperature measurements from any one or more locations around a circumferential perimeter of one or more regions of a turbomachine. 
     The thermal bias may be controlled based at least in part on a comparison of the thermal bias to an upper control limit and/or a lower control limit. Whether the thermal bias is compared to an upper control limit or a lower control limit may depend at least in part on an operating state for a cooling treatment. When the cooling treatment has an active operating state, an initial or current value for thermal bias may be compared to a lower control limit, and the cooling treatment may be deactivated when the initial value for thermal bias is less than the lower control limit. When the cooling treatment has an inactive operating state, an initial/current value or a projection value for thermal bias may be compared to an upper control limit, and the cooling treatment may be activated when the initial/current value or the projection value is greater than the upper control limit. In this way, the cooling treatment may be activated and deactivated one or more times, for example, as part of an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. An engine shut-down method may include shutting down the turbomachine and performing a cooling treatment. An on-ground cooling method may include performing a cooling treatment when the turbomachine is in a non-started state. An engine start-up method may include performing a cooling treatment and starting the turbomachine. 
     By deactivating a cooling treatment when the initial/current thermal bias is less than a lower control limit, more efficient cooling treatments and/or a more efficient reduction in thermal bias may be realized while still providing good protection from bowed rotor conditions. Additionally, or in the alternative, by activating a cooling treatment when either an initial or current value, or a projection value, for thermal bias exceeds an upper control limit, good control of thermal bias may be provided when the thermal bias is sufficiently high or projected to sufficiently high. In some embodiments, a projection value for thermal bias may correspond to a defined time period from the current or initial time. The defined time period may be selected at least in part to allow the cooling treatment to be initiated a sufficient amount of time before the thermal bias exceeds the upper control limit. The thermal bias may cycle between the upper and lower control limits while the cooling treatment is activated and deactivated according to a cooling treatment model. 
     In some embodiments, the turbomachine may be started even when the thermal bias current value is above the lower control limit, such as even when the thermal bias-value is increasing, for example, during a time period prior to a time when a projection value for the thermal bias exceeds the upper control limit. In this way, the presently disclosed thermal bias control may provide a series of time periods when the turbomachine may be started even when the turbomachine has a sufficient level of residual heat that may lead to a thermal bias value that exceeds the upper control limit. As such, the presently disclosed methods of controlling thermal bias may provide increased flexibility in scheduling start-up times for turbomachines, for example, to accommodate schedule changes to a flight itinerary for an aircraft powered by one or more turbomachines that include engine control systems and motoring systems configured to control thermal bias in accordance with the present disclosure. 
     It is understood that terms “upstream” and “downstream” refer to the relative direction with respect to fluid flow in a fluid pathway. For example, “upstream” refers to the direction from which the fluid flows, and “downstream” refers to the direction to which the fluid flows. It is also understood that terms such as “top”, “bottom”, “outward”, “inward”, and the like are words of convenience and are not to be construed as limiting terms. As used herein, the terms “first”, “second”, and “third” may be used interchangeably to distinguish one component from another and are not intended to signify location or importance of the individual components. The terms “a” and “an” do not denote a limitation of quantity, but rather denote the presence of at least one of the referenced item. 
     Approximating language, as used herein throughout the specification and claims, may be applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term or terms, such as “about,” “substantially,” and “approximately,” are not to be limited to the precise value specified. In at least some instances, the approximating language may correspond to the precision of an instrument for measuring the value, or the precision of the methods or machines for constructing or manufacturing the components and/or systems. For example, the approximating language may refer to being within a 10 percent margin. 
     Here and throughout the specification and claims, range limitations are combined and interchanged, such ranges are identified and include all the sub-ranges contained therein unless context or language indicates otherwise. For example, all ranges disclosed herein are inclusive of the endpoints, and the endpoints are independently combinable with each other. 
     Exemplary embodiments of the present disclosure will now be described in further detail. 
       FIG.  1    schematically depicts an exemplary aircraft  100  that may incorporate various embodiments of the present disclosure. The aircraft  100  may include a fuselage  102  and a pair of wings  104  extending laterally outward from the fuselage  102 . It will be appreciated that the embodiment shown is provided by way of example and not to be limiting, and that presently disclosed subject matter may be incorporated into other embodiments of an aircraft without departing from the scope of the present disclosure. As shown, the aircraft  100  may include one or more turbomachines  106  configured to provide thrust for operating the aircraft  100  such as during flight and/or taxiing. As shown, a first turbomachine  106  may be mounted to a first wing  104 , such as in an under-wing configuration, and a second turbomachine  106  may be mounted to a second wing  104 , such as in an under-wing configuration. In some embodiments, a plurality of turbomachines  106  may be mounted to the first and second wings  104 , respectively. Additionally, or in the alternative, one or more turbomachines  106  may be mounted to the aircraft  100  in other suitable locations and/or configurations, such as to the fuselage  102  aft of the wings  104 . 
     The aircraft  100  may include one or more auxiliar power units  108  configured to provide auxiliary power to the aircraft  100 , such as when the aircraft  100  is on the ground and the one or more turbomachines  106  are not running. The one or more auxiliary power units  108  may be configured to provide energy for auxiliary systems of the aircraft  100 . Additionally, or in the alternative, the one or more auxiliary power units  108  may be configured to provide energy to rotate one or more turbomachines  106 , such as in connection with an on-ground cooling method and/or in connection with an engine shut-down or engine start-up method in accordance with the present disclosure. In various embodiments, an auxiliary power unit  108  may be configured to rotate the first and second turbomachines  106 , or a first auxiliary power unit  108  may be configured to rotate the first turbomachine  106  and a second auxiliary power unit  108  may be configured to rotate the second turbomachine  106 . 
     The aircraft  100  may include an engine control system  110  configured to control operations of the aircraft  100  and the various systems thereof, including, for example, operations of the one or more turbomachines  106  and/or operations of the one or more auxiliary power units  108 . The engine control system  110  may include one or more computing devices  112 . The one or more computing devices  112  may include one or more engine control units, electronic engine controllers, and/or full-authority digital engine control (FADEC) device. The one or more computing devices  112  may be located anywhere in the aircraft  100 . By way of example, a first computing device  112 , such as a FADEC device, may be located on or in close proximity to a first turbomachine  106 , a second computing device  112  may be located on or in close proximity to a second turbomachine  106 , and/or a third computing device may be located within the fuselage  102  of the aircraft  100 , such as in the cockpit. The one or more computing devices  112  may be communicatively coupled to the one or more turbomachines  106 , to the one or more auxiliary power units  108 , and/or to one another, via a wired or wireless communications network  114 . The engine control system  110  may also be communicatively coupled with a management system  116  and/or a user interface  118  via a wired or wireless communications network. The management system  116  and the engine control system  110  may be configured to interact with one another in connection with enterprise-level or fleet-level operations pertaining to the aircraft  100  and/or the engine control system  110 . Such enterprise level operations may include transmitting data from the management system  116  to the engine control system  110  and/or transmitting data from the engine control system  110  to the management system  116 . The user interface  118  may include one or more user input/output devices to allow a user to interact with the engine control system  110 . 
     The aircraft  100  may include a plurality of sensors  120  for sensing various operating conditions associated with the aircraft  100 . The plurality of sensors may be communicatively coupled with one or more of the computing devices  112  of the engine control system  110 . The plurality of sensors  120  may include one or more airspeed sensors, temperature sensors, pressure sensors, sensors for recording ambient conditions, and the like. Sensor data from the respective sensors  120  may be provided to the one or more computing devices  112 . 
     Referring to  FIGS.  2 A and  2 B , an exemplary turbomachine  106  will be described. It will be appreciated that the turbomachine  106  shown in  FIG.  2    is provided by way of example and not to be limiting, and that the subject matter of the present disclosure may be implemented with other suitable types of turbomachines, such as steam and other types of gas turbine engines. Further examples of turbomachines may include turbojets, turboprop, turboshaft, aeroderivatives, auxiliary power units, etc. As shown, the turbomachine  106  may include a core engine  200 . The core engine  200  may include one or more shafts  201 , with one or more compressor stages  203  and one or more turbine stages  205  coupled to the one or more shafts  201 . The core engine  200  may also include a combustion chamber coupled to a respective one of the one or more shafts  201 . For example, a first one or more compressor stages  203  and a first one or more turbine stages  205  may be coupled to a first one of the one or more shafts  201 . A second one or more compressor stages  203  and a second one or more turbine stages  205  may be coupled to a second one of the one or more shafts  201 . In some embodiments, a fan section  202  may be positioned upstream of the core engine  200 . 
     In some embodiments, the core engine  200  may include an engine cowl  204  that defines an annular core inlet  206 . The engine cowl  204  may enclose and/or support a booster or low pressure compressor  208 . The low pressure compressor  208  may be configured to pressurize air that enters the core engine  200  through core inlet  206 . The core engine  200  may include a high pressure compressor  210 . The high pressure compressor  210  may include multiple stages arranged axially relative to one another. The high pressure compressor  210  may receive pressurized air from the low pressure compressor  208  and further increases the pressure of the air flowing therethrough. The pressurized air may flow from the high pressure compressor  210  to a combustor  212  where fuel is injected into the pressurized air stream and ignited to raise the temperature and energy level of the pressurized air. High energy combustion gasses in the combustor  212  flow from the combustor  212  to a high pressure turbine  214 . The high pressure turbine  214  and the high pressure compressor  210  may be coupled to a high pressure shaft (“HP shaft”)  216 . The high pressure turbine  214  may drive a rotating portion of the high pressure compressor  210  by way of the HP shaft  216 . The HP shaft  216 , a rotating portion of the high pressure compressor  210  coupled to the HP shaft  216 , and a rotating portion of the high pressure turbine  214  coupled to the HP shaft  216 , may be collectively referred to as a high speed or high pressure spool  218 . 
     The high energy combustion gasses exiting the high pressure turbine  214  flow to a low pressure turbine  220 . The low pressure turbine  220  and the low pressure compressor  208  may be coupled to a low pressure shaft (“LP shaft”)  222 . The low pressure turbine  220  may drive a rotating portion of the low pressure compressor  208  by way of the LP shaft  222 . The LP shaft  222  may be coaxial with the HP shaft  216 . The LP shaft  222  and the HP shaft  216  may freely rotate relative to one another. The LP shaft  222 , a rotating portion of the low pressure compressor  208  coupled to the LP shaft  222 , a rotating portion of the low pressure turbine  220  coupled to the LP shaft  222 , and a rotating portion of the fan section  202 , may be collectively referred to as a low speed or low pressure spool  224 . Combustion gasses exiting the low pressure turbine  220  may flow through an exhaust nozzle  226  to produce propulsive thrust. 
     The fan section  202  may include a rotatable, axial-flow fan rotor  228  surrounded by an annular fan casing  230 . The fan rotor  228  includes a plurality of fan blades  232  extending outward from the fan rotor  228 . The fan casing  230  is supported by the core engine  200  by a plurality of substantially radially-extending, circumferentially-spaced outlet guide vanes  234 . The fan casing  230  encloses the fan rotor  228  and the plurality of fan blades  232  extending outward from the fan rotor  228 . A downstream section of the fan casing  230  extends over an outer portion of the core engine  200  to define a bypass passage  236 . Air that passes through the bypass passage  236  provides propulsive thrust. 
     During operation of the turbomachine  106 , an initial or inlet airflow  238  enters the turbomachine  106  through an inlet  251  defined by the fan casing  230 . The airflow  238  passes through the fan blades  232  and splits into a first air flow  240  that moves through the bypass passage  236  and a second air flow  242  that enters the low pressure compressor  208  through the core inlet  206 . The pressure of the second airflow  242  is progressively increased by the low pressure compressor  208  and then enters the high pressure compressor  210 , as represented by arrow  244 . The discharged pressurized air stream flows downstream to the combustor  212  where fuel is introduced to generate combustion gasses, as represented by arrow  246 . The combustion gasses exit the combustor  212  and flow through the high pressure turbine  214 . The combustion gasses then flow through the low pressure turbine  220  and exit the exhaust nozzle  226  to produce thrust. Concurrently, a portion of the inlet airflow  238  flows through the bypass passage  236  and through an exit nozzle defined between the fan casing  230  and the engine cowl  204  at the downstream section of the fan casing  230 , producing further propulsive thrust. 
     As further shown in  FIGS.  2 A and  2 B , the combustor  212  defines an annular combustion chamber  248  that is generally coaxial with a longitudinal axis. The combustor  212  receives an annular stream of pressurized air from a high pressure compressor discharge outlet  250 . A portion of this compressor discharge air flows into a mixer (not shown). Fuel is injected by a fuel nozzle  252  of a fuel delivery system to mix with the air. This forms a fuel-air mixture that is provided to the combustion chamber  248  for combustion. Ignition of the fuel-air mixture is accomplished by an igniter, and the resulting combustion gasses flow in an axial direction toward and into the high pressure turbine  214 . The high pressure turbine  214  rotates the HP shaft  216 , thereby driving the high pressure compressor  210 . The combustion gasses exit the high pressure turbine  214  and flow into the low pressure turbine  220  via the HP shaft  216 . The low pressure turbine  220  rotates the LP shaft  222 , thereby drives the low pressure compressor  208  and the fan rotor  228  via the LP shaft  222 . 
     Still referring to  FIGS.  2 A and  2 B , exemplary motoring systems  254  for a turbomachine  106  will be described. A motoring system  254  may be utilized to rotate HP shaft  216  and/or the LP shaft  222  of the turbomachine  106  in connection with an on-ground cooling method and/or in connection with an engine shut-down or engine start-up method in accordance with the present disclosure. As shown in  FIG.  2 A , a motoring system  254  may include a motor  256  configured to rotate the HP shaft  216  and/or the LP shaft  222  of the turbomachine  106 . The motor  256  may be drivingly coupled to the HP shaft  216  and/or the LP shaft  222  of the turbomachine  106 , such as by way of a radial drive shaft  258 . For example, the motoring system  254  may include an external gearbox  260  and/or an internal gearbox  262 . The external gearbox  260  may be drivingly coupled to the internal gearbox  262  by way of a radial drive shaft  258 . The external gearbox  260  may be configured to engage with the motor  256  to rotate the radial drive shaft  258 , for example, by way of a series of gears, clutches, or the like. The motor  256  may be configured as an air turbine motor, an electric motor, or the like. In some embodiments, the motor  256  may be configured as a starter motor, such as an air-turbine starter, an electric starter, or the like. The internal gearbox  262  may be configured to rotate the HP shaft  216  and/or the LP shaft  222 . As shown, the internal gearbox  262  may include a series of gears, clutches, or the like configured to drivingly engage the radial drive shaft  258  with the HP shaft  216 . Additionally, or in the alternative, the internal gearbox  262  may include a series of gears, clutches, or the like configured to drivingly engage the radial drive shaft  258  with the LP shaft  222 . The external gearbox  260  may sometimes be referred to as an accessory gearbox, as the external gearbox  260  may include a series of power takeoffs for operating various accessories to the turbomachine  106 , such as a fuel pump, an oil pump, an electrical generator, a tachometer, and so forth. 
     When the motor  256  rotates the radial drive shaft  258  by way of the external gearbox  260 , the radial drive shaft  258  may, in turn, rotate the HP shaft  216  by way of the internal gearbox  262 . Rotation of the HP shaft  216  may cause the high pressure spool  218  to rotate, thereby causing air to circulate within and/or flow through the respective portions of the high pressure spool  218 , from the high pressure compressor  210  to the combustion chamber  248 , and/or from the combustion chamber to the high pressure turbine  214 . Air from the high pressure turbine  214  may flow through the low pressure turbine, causing the LP shaft  222  to rotate. Rotation of the LP shaft  222  may cause the low pressure spool  224  to rotate, including, for example, the low pressure compressor  208  and or the fan rotor  228 . 
     In some embodiments, such as when the motor  256  is configured as an air turbine motor, the motor  256  may receive motoring air from a motoring air supply line  264 . The motoring air supply line  264  may be configured to supply motoring air to rotate the motor  256 . The motoring air supply line  264  may be in fluid communication with an auxiliary power unit  108 . Additionally, or in the alternative, the motoring air supply line  264  may be in fluid communication with a ground start unit  266 , with another turbomachine  106  on the aircraft  100 , and/or any other suitable source of motoring air. A motoring air supply valve  268  may be opened, closed, and/or modulated to control a flow of motoring air to the motor  256 , thereby controlling the power output of the motor  256 . In some embodiments, the rate of rotation of the high pressure spool  218  and/or the rate of rotation of the low pressure spool  224  may depend at least in part on the power output of the motor  256 . 
     As shown in  FIG.  2 B , in some embodiments, the motoring system  254  may include a motoring air supply line  264  configured to supply motoring air to one or more regions of the turbomachine  106 , such as one or more regions of the core engine  200  of the turbomachine  106 . The motoring air supplied to the one or more regions of the core engine  200  may circulate within and/or flow through the respective regions of the core engine  200 . Motoring air supplied to the one or more regions of the core engine  200  may cause the high pressure spool  218  and/or the low pressure spool  224  to rotate. As shown in  FIG.  2 B , the motoring air supply line  264  may be configured to supply motoring air to the high pressure compressor  210  of the turbomachine  106 . Additionally, or in the alternative, the motoring air supply line  264  may be configured to supply motoring air to the low pressure compressor  208 , the high pressure turbine  214 , and/or the low pressure turbine  220 . By way of example, when the motoring air causes the high pressure spool  218  to rotate, such rotation of the high pressure spool  218  may cause additional air to circulate within and/or flow through the respective portions of the high pressure spool  218 , from the high pressure compressor  210  to the combustion chamber  248 , and/or from the combustion chamber to the high pressure turbine  214 . Air from the high pressure turbine  214  may flow through the low pressure turbine, causing the LP shaft  222  to rotate. Rotation of the LP shaft  222  may cause the low pressure spool  224  to rotate, including, for example, the low pressure compressor  208  and or the fan rotor  228 . The motoring air may be supplied to the core engine  200  by any suitable source, such as an auxiliary power unit  108 , a ground start unit  266 , and/or any other suitable air source. The motoring air supply valve  268  may be opened, closed, and/or modulated to control a flow of motoring air to the core engine  200 , thereby controlling the rate of rotation of the high pressure spool  218  and/or the rate of rotation of the low pressure spool  224 . 
     Additionally, or in the alternative, in some embodiments, a motoring air supply line  264  may be configured to supply motoring air to the fan section  202  of the turbomachine. Motoring air supplied to the fan section  202  of the turbomachine  106  may cause the fan rotor  228  to rotate, thereby rotating the low pressure spool  224 . Rotation of the low pressure spool  224  may cause air to flow through the high pressure spool  218 , thereby rotating the high pressure spool  218 . 
     Still referring to  FIGS.  2 A and  2 B , the turbomachine  106  may include one or more sensors  120 , such as one or more high pressure compressor-sensors  270 , one or more high pressure turbine-sensors  272 , and/or one or more low pressure turbine-sensors  274 . The one or more sensors may respectively include a temperature sensor and/or a pressure sensor. For example, the one or more high pressure compressor-sensors  270  may include one or more temperature sensors configured to obtain temperature measurements from the high pressure compressor  210  and/or from the core engine  200  adjacent to the high pressure compressor  210 . The one or more high pressure turbine-sensors  272  may include one or more temperature sensors configured to obtain temperature measurements from the high pressure turbine  214  and/or from the core engine  200  adjacent to the high pressure turbine  214 . The one or more low pressure turbine-sensors  274  may include one or more temperature sensors  120  configured to obtain temperature measurements from the low pressure turbine  220  and/or from the core engine  200  adjacent to the low pressure turbine  220 . 
     The one or more sensors  120  may be respectively located at any desired axial position within the turbomachine  106 , including at a position corresponding to a respective stage or stages of the high pressure compressor  210 , the high pressure turbine  214 , and/or the low pressure turbine  220 . The one or more sensors  120  may be respectively configured to determine sensor measurements, such as temperature measurements and/or pressure measurements, at any desired circumferential position of a shaft  201  of the turbomachine  106 , such as any desired circumferential position of the HP shaft  216  and/or the high pressure spool  218 , and/or any desired position of the LP shaft  222  and/or the low pressure spool  224 . A circumferential position of a shaft  201  of the turbomachine  106  may be described with reference to a 360-degree circumferential axis and/or with reference to time positions on a clock. A vertical axis may intersect the circumferential axis at 0-radians, or (2π)-radians, or twelve o&#39;clock at the top of the circumferential axis, and at (π)-radians, or six o&#39;clock at the bottom of the circumferential axis. For example, one or more of the sensors  120  may be configured to determine sensor measurements corresponding to an upward position of the turbomachine  106 , such as an upward position of the high pressure spool  218  and/or an upward position of the low pressure spool  224 . The upward position of the turbomachine  106  may include 0-radians on the circumferential axis of the turbomachine  106 . For example, the upward position may include a location within a range of from about (π/3)-radians to about (5π/3)-radians, such as from about (π/6)-radians to about (11π/6)-radians, such as from about (2π)-radians+/−(π/3)-radians, such as from about (2π)-radians+/−(π/6)-radians, or such as from about (2π)-radians+/−(π/18)-radians. Additionally, or in the alternative, one or more of the sensors  120  may be configured to determine sensor measurements corresponding to a downward position of the turbomachine  106 , such as an upward position of the high pressure spool  218  and/or an upward position of the low pressure spool  224 . The downward position of the turbomachine  106  may include (π)-radians on the circumferential axis of the turbomachine  106 . For example, the downward position may include a location within a range of from about (2π/3)-radians to about (4π/3)-radians, such as from about (5π/6)-radians to about (7π/6)-radians, such as from about (π)-radians+/−(π/3)-radians, such as from about (π)-radians+/−(π/6)-radians, or such as from about (π)-radians+/−(π/18)-radians. Additionally, or in the alternative, one or more of the sensors  120  may be configured to determine sensor measurements corresponding to a side position of the turbomachine  106 . The side position of the turbomachine  106  may include (3π/2)-radians on the circumferential axis of the turbomachine  106 . For example, the side position may include a location within a range of from about (π/6)-radians to about (5π/6)-radians, such as from about (π/3)-radians to about (2π/3)-radians, or such as from about (π/2)-radians+/−(π/18)-radians. Additionally, or in the alternative, the side position may include a location within a range of from about (7π/6)-radians to about (11π/6)-radians, such as from about (4π/3)-radians to about (5π/3)-radians, or such as from about (3π/2)-radians+/−(π/18)-radians. A turbomachine  106  may include one or more sensors  120  located about the circumferential axis of the turbomachine  106 , for example, with a circumferential separation of from about (π/36)-radians to about (π)-radians, such as from about (π/36)-radians to about (π/6)-radians, such as from about (π/6)-radians to about (π/3)-radians, such as from about (π/3)-radians to about (π/2)-radians, or such as from about (π/2)-radians to about (π)-radians. 
     Referring now to  FIGS.  3 A and  3 B , exemplary sensor measurements that may be utilized in connection with thermal bias control will be described. The sensor measurements may be determined from the one or more sensors  120 . The one or more sensors  120  may be communicatively coupled to a computing device  112 , such as may be included in an electronic engine controller, a full-authority digital engine control (FADEC) device, or the like. The computing device  112  may be configured to determine sensor values, such as a temperature or pressure, from the sensor measurements obtained from the one or more sensors  120 . The computing device  112  may be configured to cause the motoring system  254  to perform an on-ground cooling method, an engine shut-down method, and/or an engine start-up method, based at least in part on an input from the one or more sensors  120 . As shown in  FIG.  3 A , a turbomachine  106  may include one or more sensors  120  configured to obtain temperature measurements from one or more locations of the turbomachine  106 . For example, one or more high pressure compressor-sensors  270  may be configured as a temperature sensor  120 , such as a thermocouple, a thermopile, an RTD (resistance temperature detector), a thermistor, or the like. 
     In some embodiments, the one or more sensors  120 , such as one or more high pressure compressor-sensor  368 , may include an upward temperature sensor  300  and/or a downward temperature sensor  302 . The upward temperature sensor  300  may be located at an upward portion of the core engine  200  and/or otherwise configured to obtain temperature measurements at an upward portion of the turbomachine  106 , such as at an upward portion of the core engine  200 . The downward temperature sensor  302  may be configured to obtain temperature measurements at a downward portion of the turbomachine  106 , such as at a downward portion of the core engine  200 . One or more upward temperature sensors  300  and/or one or more downward temperature sensors  302  may be distributed axially along the core engine  200 . As shown, the upward temperature sensor  300  and/or the downward temperature sensor  302  may be configured to obtain temperature measurements at a location of the core engine  200  corresponding to the high pressure compressor  210 . For example, the upward temperature sensor  300  and/or the downward temperature sensor  302  may include a probe extending into the core engine  200  adjacent to the high pressure compressor  210 . Additionally, or in the alternative, the upward temperature sensor  300  and/or the downward temperature sensor  302  include a probe extending into the high pressure compressor  210 , such as through a stator vane of the high pressure compressor  210 . Additionally, or in the alternative, the turbomachine  106  may include one or more upward temperature sensors  300  and/or the downward temperature sensor  302  with a probe extending into at least one of the core engine  200  adjacent to the low pressure compressor  208  or into the low pressure compressor  208  (e.g., through a stator vane of the low pressure compressor  208 ), adjacent to the combustion chamber  248  or into the combustion chamber  248 , adjacent to the high pressure turbine  214  or into the high pressure turbine  214  (e.g., through a stator vane of the high pressure turbine  214 ), or adjacent to the low pressure turbine  220  or into the low pressure turbine  220  (e.g., through a stator vane of the low pressure turbine  220 ). 
     As shown in  FIG.  3 B , a computing device  112  may be configured to determine one or more upward temperature-values  304  and/or one or more downward temperature-values  306 . The one or more upward temperature-values may correspond to a first one or more temperature measurements  308  of an upward portion of the turbomachine  106 . The upward portion of the turbomachine  106  may include any suitable location about the circumferential axis of the turbomachine  106 . The downward portion of the turbomachine  106  may include any suitable location about the circumferential axis of the turbomachine that is below the upward portion of the turbomachine  106 . Additionally, or in the alternative, the upward portion of the turbomachine  106  may include any suitable location about the circumferential axis of the turbomachine  106  that is above the downward portion of the turbomachine  106 . For example, the upward portion of the turbomachine  106  may include a location above a horizontal midline of the turbomachine  106 . In some embodiments, the upward portion of the turbomachine  106  may include an intersection of the vertical axis and the top of the circumferential axis of the turbomachine  106 . Additionally, or in the alternative, the downward portion of the turbomachine  106  may include a location below a horizontal midline of the turbomachine  106 . In some embodiments, the downward portion of the turbomachine  106  may include an intersection of the vertical axis and the bottom of the circumferential axis of the turbomachine  106 . 
     In some embodiments, the upward temperature-values  304  may be determined from temperature measurements  308  obtained from an upward temperature sensor  300 , and/or the downward temperature-values  306  may be determined from temperature measurements  308  obtained from a downward temperature sensor  302 . Additionally, or in the alternative, in some embodiments, the upward temperature-values  304  and the downward temperature-values  306  may be determined from temperature measurements  308  obtained from the same sensor  120 . For example, temperature measurements  308  obtained from an upward temperature sensor  300  may be used to determine both the upward temperature-values  304  and the downward temperature-values  306  by rotating the high pressure spool  218  and/or the low pressure spool  224  using the motoring system  254  and obtaining temperature measurements  308  at different points of rotation. As another example, temperature measurements  308  obtained from a downward temperature sensor  302  may be used to determine the both upward temperature-values  304  and the downward temperature-values  306  by rotating the high pressure spool  218  and/or the low pressure spool  224  using the motoring system  254  at different points of rotation. For example, an upward temperature-value  304  may be determined from one or more temperature measurements  308  obtained with a shaft  201  of the turbomachine  106  oriented at (0)-radians of rotation, and/or a downward temperature-value may be determined from one or more temperature measurements  308  obtained with the shaft  201  of the turbomachine  106  oriented at (π)-radians of rotation. The orientation of a shaft  201  of the turbomachine  106  with respect to the circumferential axis may include an orientation of the HP shaft  216  and/or the high pressure spool  218 , and/or an orientation of the LP shaft  222  and/or the low pressure spool  224 . An upward temperature-value  304  may be determined from an upward temperature sensor  300  at the initiation of a spool rotation sequence that includes rotating the high pressure spool  218  and/or the low pressure spool  224 . A downward temperature-value  306  may be determined from an upward temperature sensor  300  during a spool rotation sequence that includes rotating the high pressure spool  218  and/or the low pressure spool  224  by about (π)-radians, for example, such that a downward portion of the high pressure spool  218  and/or a downward portion of the low pressure spool  224  becomes adjacent to the upward temperature sensor  300 . Additionally, or in the alternative, a downward temperature-value  306  may be determined from a downward temperature sensor  302  at the initiation of a spool rotation sequence that includes rotating the high pressure spool  218  and/or the low pressure spool  224 . An upward temperature-value  304  may be determined from a downward temperature sensor  302  during a spool rotation sequence that includes rotating the high pressure spool  218  and/or the low pressure spool  224  by about (π)-radians, for example, such that an upward portion of the high pressure spool  218  and/or an upward portion of the low pressure spool  224  becomes adjacent to the downward temperature sensor  302 . 
     In still further embodiments, an upward temperature-value  304  and/or a downward temperature-value  306  may be determined from a sensors  120  located at any desired position about a circumference of a turbomachine  106 , such as from a side position or from any desired circumferential position of the core engine  200 , in addition or in the alternative to an upward position or a downward position. In some embodiments, temperature measurements  308  from a sensor  120  may be utilized to determine an upward temperature-value  304  and/or a downward temperature-value  306  regardless of the circumferential location of a sensor  120 . For example, an upward temperature-value  304  and/or a downward temperature-value  306  may be determined from a sensors  120  located at a side position, by coordinating the radians of rotation to the circumferential location of the sensor  120 . By way of illustration, for a sensor  120  located at about (π/2)-radians, an upward temperature-value  304  may be determined from the sensor  120  using temperature measurements  308  obtained upon having rotated the high pressure spool  218  and/or the low pressure spool  224  such that an upward portion thereof becomes adjacent to the side position corresponding to the sensor  120 . In some embodiments, the temperature measurements  308  from a sensor  120  at the side position of the core engine  200  utilized to determine the upward temperature-value  304  may correspond to the high pressure spool  218  and/or the low pressure spool  224  having been rotated about (π/2)-radians, thereby aligning the upward portion of the high pressure spool  218  and/or the low pressure spool  224  with the sensor  120 . Additionally, or in the alternative, the temperature measurements  308  from the sensor  120  at the side position utilized to determine the downward temperature-value  306  may correspond to the high pressure spool  218  and/or the low pressure spool  224  having been rotated about (3π/2)-radians, thereby aligning the downward portion of the high pressure spool  218  and/or the low pressure spool  224  with the sensor  120 . 
     Still referring to  FIG.  3 B , in some embodiments, a plurality of temperature measurements  308  may be determined about a circumference of one or more locations of the core engine  200 , and/or about a circumference of one of more locations of the high pressure spool  218  and/or the low pressure spool  224 . In some embodiments, a circumferential temperature profile  310  for one or more locations of the high pressure spool  218  and/or the low pressure spool  224  may be determined from the plurality of temperature measurements  308 . The temperature profile may be determined from a regression or best fit of the temperature measurements  308 . The temperature measurements  308  may be obtained periodically or continuously over a period of time. Additionally, or in the alternative, the temperature profile may be determined periodically or continuously or over a period of time. 
     The upward temperature-value  304  and/or the downward temperature-value  306  may be determined from a plurality of temperature measurements  308 , such as from a plurality of temperature measurements  308  determined about a circumference of one or more locations of the core engine  200 , and/or about a circumference of one of more locations of the high pressure spool  218  and/or the low pressure spool  224 . In some embodiments, the upward temperature-value  304  and/or the downward temperature-value  306  may be determined by synchronizing the rotational position of the high pressure spool  218  and/or the low pressure spool  224  with the temperature measurements  308 , such as by a tachometer  276  coupled to the turbomachine  106  and/or the motoring system  254 . Additionally, or in the alternative, temperature measurements  308  that represent relatively higher values may be attributed to an upward portion of respective region of the turbomachine  106  and/or temperature measurements  308  that represent relatively lower values may be attributed to a downward portion of the respective region of the turbomachine  106 . By obtaining a plurality of temperature measurements  308  in coordination with rotating the high pressure spool  218  and/or the low pressure spool  224 , an upward portion of the high pressure spool  218  and/or the low pressure spool  224  may be determined, for example, based on a difference in temperature at various circumferential locations thereof. In some embodiments, the temperature measurements  308  and/or the difference in temperature at various circumferential locations of the high pressure spool  218  and/or the low pressure spool  224  may be utilized to determine the upward temperature-value  304  and/or the downward temperature-value  306 , for example, without reference to a circumferential position prior to commencing rotation. For example, a circumferential position that has a relatively higher temperature may correspond to an upward portion of the high pressure spool  218  and/or the low pressure spool  224 . Additionally, or in the alternative, a circumferential position that has a relatively lower temperature may correspond to a downward portion of the high pressure spool  218  and/or the low pressure spool  224 . In this way, the upward portion and the downward portion of the high pressure spool  218  and the low pressure spool  224  can be determined even when the high pressure spool  218  and the low pressure spool  224  rotate independently from one another and/or at different rates of rotation. 
     The upward temperature-value  304  may be determined from or more temperature measurements  308  corresponding to relatively higher values. For example, a plurality of temperature measurements  308  that represent a relatively higher temperature across an arc of the circumferential temperature profile  310  may be utilized to determine the upward temperature-value  304 . The temperature measurements  308  that represent a relatively higher temperature may be determined, for example, with reference to a circumferential temperature profile  310  and/or an average circumferential temperature  312 . At least some of the plurality of temperature measurements  308  representing a relatively higher temperature may define an upward temperature arc  314 . The upward temperature arc  314  may include a group or cluster of temperature measurements  308  that have a relatively higher temperature, for example, relative to the average circumferential temperature  312 . The upward temperature-value  304  may be determined based at least in part on one or more temperature measurements  308  from the upward temperature arc  314 . For example, the upward temperature-value  304  may correspond to a maximum temperature, an average temperature value, or the like, from the upward temperature arc  314 . 
     The downward temperature-value  306  may be determined from or more temperature measurements  308  corresponding to relatively lower values. For example, a plurality of temperature measurements  308  that represent a relatively lower temperature across an arc of the circumferential temperature profile  310  may be utilized to determine the downward temperature-value  306 . The temperature measurements  308  that represent a relatively lower temperature may be determined, for example, with reference to the circumferential temperature profile  310  and/or the average circumferential temperature  312 . At least some of the plurality of temperature measurements  308  representing a relatively lower temperature may define a downward temperature arc  316 . The downward temperature arc  316  may include a group or cluster of temperature measurements  308  that represent a relatively lower temperature, for example, relative to the average circumferential temperature  312 . The downward temperature-value  306  may be determined based at least in part on one or more temperature measurements  308  from the downward temperature arc  316 . For example, the downward temperature-value  306  may correspond to a maximum temperature, an average temperature value, or the like, from the downward temperature arc  316 . 
     In some embodiments, the arc length of the upward temperature arc  314  and/or the downward temperature arc  316  may be determined based at least in part on a degree of variability of the temperature measurements  308  corresponding to the circumferential temperature profile  310 . For example, the arc length of the upward temperature arc  314 , and/or the arc length of the downward temperature arc  316 , may include a range of temperature measurements  308  that correspond a variance (T σ (i))  318  in the temperature measurements  308 , such as a statistical variance, a specified variance, a standard deviation, a confidence interval, a control limit, or the like. The arc length of the upward temperature arc  314 , and/or the arc length of the downward temperature arc  316 , may extend to include a group of temperature measurements  308  that fall within a variance  318 . The upward temperature arc  314  may include a group of relatively higher temperature measurements  308 , for example, including and/or clustered around a maximum temperature value. The upward temperature arc  314  may include, for example, all temperature measurements  308  that fit within the variance  318  relative to a maximum temperature value, and/or a group of temperature measurements  308  that fit within a confidence interval relative to a maximum temperature value. The downward temperature arc  316  may include a group of relatively lower temperature measurements  308 , for example, including and/or clustered around a minimum temperature. The downward temperature arc  316  may include, for example, all temperature measurements  308  that fit within the variance  318  relative to a minimum temperature value, and/or a group of temperature measurements  308  that fit within a confidence interval relative to a minimum temperature value. Such maximum temperature value may include an absolute maximum, an average maximum, or the like. Additionally, or in the alternative, such minimum temperature value may include an absolute minimum, an average minimum, or the like. Such maximum or minimum temperature value may be determined from a filtered data set. For example, a data set may be filtered to remove outliers and the like. The upward temperature arc  314  and/or the downward temperature arc  316  may extend a sufficient length to include temperature measurements  308  that fall within a variance  318 , such as an arc length sufficient to include temperature measurements  308  that fit within a confidence interval. For example, the upward temperature arc  314  and/or the downward temperature arc  316  may have an arc length that includes temperature measurements  308  within the variance  318  to a 90%, 95%, or 99% confidence interval. 
     A thermal bias (ΔT)  320  may be determined based at least in part on a difference between one or more temperature measurements  308 . As shown in  FIG.  3 B , a thermal bias  320  may be determined based at least in part on a difference between an upward temperature-value  304  and a downward temperature-value  306 . The thermal bias  320  may correspond to a difference in temperature as between an upward portion of the turbomachine  106  and a downward portion of the turbomachine  106 . Additionally, or in the alternative, a thermal bias  320  may correspond to a difference in temperature as between at least two portions of the turbomachine  106 , such as between portions of a turbomachine  106  located at about opposite circumferential positions, such as at circumferential positions separated by about (π)-radians, or such as by about (π)-radians+/−(π/18)-radians. A thermal bias  320  may correspond to a difference in temperature as between a temperature value within an upward temperature arc  314 , such as an upward temperature-value  304 , and a temperature value within a downward temperature arc  316 , such as a downward temperature-value  306 . For example, in some embodiments, a thermal bias  320  may be determined as between a maximum temperature value within an upward temperature arc  314  and a minimum temperature value within a downward temperature arc  316 . In some embodiments, the maximum temperature value within an upward temperature arc  314  may be determined to be the upward temperature-value  304  and/or a minimum temperature value within a downward temperature arc  316  may be determined to be the downward temperature-value  306 . 
     Still referring to  FIG.  3 B , a thermal data set  322  may be determined for one or more time periods (i). The thermal data set  322  may be utilized to provide a cooling treatment to reduce and/or control thermal bias, such as in connection with an on-ground cooling method and/or in connection with an engine shut-down or engine start-up method, for example, to determine when to operate motoring system  254  for purposes of controlling thermal bias  320 , such as within a range defined by upper and lower control limits. By way of example, the thermal data set  322  may include, for respective time periods, a plurality of temperature measurements  308 , one or more upward temperature-values  304  and/or one or more downward temperature-values  306 , a circumferential temperature profile  310 , average circumferential temperature  312 , an upward temperature arc  314 , a downward temperature arc  316 , a variance  318 , a thermal bias  320 , and/or one or more statistical values related thereto, such as a statistical variance, a standard deviation, a confidence interval, a control limit, or the like. 
     Referring again to  FIG.  3 A , in an exemplary embodiment, a cooling treatment may include rotating a shaft  201  of the turbomachine  106 , such as the HP shaft  216  and/or the LP shaft  222 , with the motoring system  254  for a period of time to sufficiently reduce the thermal bias  320 . Rotation of the shaft  201 , such as the HP shaft  216 , for example, by the radial drive shaft  258  of the motoring system  254 , may cause the high pressure spool  218  to rotate, thereby causing air to circulate within and/or flow through at least a portion of the high pressure spool  218 . Air in the respective portions of the high pressure spool  218  may flow circumferentially and/or axially, depending, for example, on the rotational speed of the shaft  201 , such as the HP shaft  216 , and/or the pressure differential across respective portions of the high pressure spool  218  generated by rotation of the shaft  201  and/or air flow through the high pressure spool  218 . At a relatively low rotational speed, air may flow circumferentially through one or more of the respective portions of the high pressure spool  218 , for example, with minimal axial flow. Circumferential airflow through the high pressure spool  218  is represented by arrows  324 . With sufficient rotational speed, air may flow axially through one or more of the respective portions of the high pressure spool  218 . Such axial airflow through the high pressure spool  218  is represented by arrows  326 . Additionally, or in the alternative, at a relatively low rotational speed, air may flow circumferentially through one or more of the respective portions of the low pressure spool  224 , for example, with minimal axial flow. Circumferential airflow through the low pressure spool  224  is represented by arrows  328 . With sufficient rotational speed, air may flow axially through one or more of the respective portions of the low pressure spool  224 . Such axial airflow through the low pressure spool  224  is represented by arrows  330 . 
     For embodiments in which the LP shaft  222  is not coupled to a radial drive shaft  258  of the motoring system  254 , rotation of the high pressure spool  218 , such as by way of a radial drive shaft  258  coupled to the HP shaft  216 , and/or sufficient pressure differential across one or more portions of the high pressure spool  218 , may generate sufficient airflow through at least a portion of the low pressure spool  224  to cause the low pressure spool  224  to rotate. Additionally, or in the alternative, the motoring system  254  may include a radial drive shaft  258  coupled to the LP shaft  222  and configured to rotate the low pressure spool  224 . For embodiments in which the HP shaft  216  is not coupled to a radial drive shaft  258  of the motoring system  254 , rotation of the low pressure spool  224 , such as by waw of a radial drive shaft  258  coupled to the LP shaft  222 , and/or sufficient pressure differential across one or more portions of the low pressure spool  224 , may generate sufficient airflow through at least a portion of the high pressure spool  218  to cause the high pressure spool  218  to rotate. The HP spool  216  and/or the LP spool  224  may be rotated by the motoring system  254 , for example, to control a thermal bias  320  ( FIG.  3 B ) in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. The rotation of the HP spool  216  and/or the LP spool  224  may be determined based at least in part on a thermal bias  320  and/or one or more other parameters of a thermal data set  322  ( FIG.  3 B ). 
     Referring now to  FIG.  4   , an exemplary thermal bias control module  400  will be described. Thermal bias control module  400  may include one or more thermal bias models  402  configured to reduce and/or control thermal bias. The thermal bias control module  400  may be configured to reduce and/or control thermal bias when the turbomachine  106  is in a non-started state, such as only when the turbomachine  106  is in a non-started state. For example, the thermal bias control module  400  may be configured to control thermal bias  320  in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. The thermal bias control module  400  may be implemented on one or more computing devices  112 , such as one or more engine control units, electronic engine controllers, full-authority digital engine control (FADEC) devices, or the like. The thermal bias control module  400  may control thermal bias  320  at least in part by providing control commands to various controllable components of a turbomachine  106  and/or to various controllable components related to the turbomachine  106 , such as controllable components of a motoring system  254 , an auxiliary power unit  108 , a ground start unit  266 , or the like. 
     As shown in  FIG.  4   , the thermal bias control module  400  may receive one or more model inputs model inputs  404 . The model inputs  404  may be utilized by one or more thermal bias models  402 , for example, to provide one or more model outputs  406 . The model inputs  404  may include sensor data  408 , such as temperature measurements  308  from one or more sensors  120  configured to obtain temperature measurements  308  from one or more locations of the turbomachine  106 . The sensor data  408  may additionally include data from various other sources associated with the turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , a ground start unit  266 , and so forth. The model inputs  404  may additionally include one or more setpoints  410 , such as setpoints  410  for upper and/or lower control limits corresponding to operations of the one or more thermal bias models and/or setpoints  410  corresponding to the one or more model outputs  406 . The model inputs  404  may additionally include a counter time  412 , such as a clock time, to which various actions of the thermal bias models  402  may be synchronized. Additionally, or in the alternative, one or more model outputs  406  may be synchronized to the counter time  412 . In some embodiments, the model inputs  404  may include fleet data  414 , such as data from a management system  116 . The fleet data  414  may include past operating data pertaining to an aircraft  100  and/or turbomachine  106  utilizing the thermal bias control module  400 . Additionally, or in the alternative, the fleet data  414  may include current or past operating data from other aircraft  100  and/or other turbomachines  106 , such as current or past operating data pertaining to thermal bias control module  400  implemented with such other aircraft  100  and/or such other turbomachines  106 . 
     The thermal bias control module  400  may include one or more model outputs  406 . The model outputs may include control commands to one or more controllable components of a turbomachine  106  and/or to one or more controllable components related to the turbomachine  106 , such as one or more controllable components of a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . In some embodiments, the model outputs  406  may include cooling control commands  416 . A cooling control command  416  may be configured to control one or more controllable components in connection with rotating one or more portions of the turbomachine  106 , such as a HP spool  216  and/or an LP spool  224 , for example, in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. Additionally, or in the alternative, the model outputs  406  may include one or more engine protection control commands  418 . An engine protection control command  418  may be configured to protect a turbomachine  106  from damage or unnecessary wear, for example, by enabling or disabling operation of one or more controllable components. For example, an engine protection control command  418  may be configured to activate or deactivate operation of one or more controllable components of the turbomachine  106  and/or one or more controllable components related to the turbomachine  106 , such as one or more controllable components of a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . For example, an engine protection control command  418  may disable an engine start-up method or a portion thereof in the event of a thermal bias  320  that may damage or cause unnecessary wear to the turbomachine  106  if the turbomachine  106  were operated prior to sufficiently reducing thermal bias. Additionally, or in the alternative, an engine protection control command  418  may disable an engine start-up method or a portion thereof during an on-ground cooling method. As another example, an engine protection control command  418  may disable an on-ground cooling method or a portion thereof prior to or during an engine start-up method for starting a turbomachine  106 . 
     The thermal bias control module  400  may include one or more thermal bias models  402 . In some embodiments, a thermal bias model  402  may include an initial value model  420 . An initial value model  420  may be configured to determine an initial value for a thermal bias  320 . An initial value for a thermal bias  320  may be described by the relationship: ΔT(i)=T U (i)−T D (i), where for an initial time, (i), ΔT(i) is the thermal bias  320 , T U (i) is an upward temperature-value  304 , and T D (i) is a downward temperature-value  306 . The initial time, (i), may be any selected initial time, such as a current time, an engine shut-off time, or any other selected time. Additionally, or in the alternative, an initial value for a thermal bias  320  may be described by the relationship: ΔT(i)=f(m_ initial )[T U (i)−T D (i)], where f(m_ initial ) is a machine-learned function pertaining to the thermal bias  320  for the initial time. The machine-learned function may include a static factor, such as a multiplier. Additionally, or in the alternative, the machine-learned function may include one or more variables and/or one or more single-order or multiple-order functions. 
     In addition, or in the alternative to an initial value model  420 , a thermal bias model  402  may include a rate of change model  422 . A rate of change model  422  may be configured to determine a rate of change for a thermal bias  320 . A rate of change for a thermal bias  320  may be described by the relationship: dT(i)/dt=[ΔT(i)−ΔT(i−1)]/δt, where dT(i)/dt is the rate of change for the thermal bias  320 , and δt is a time period. Additionally, or in the alternative, a rate of change for a thermal bias  320  may be described by the relationship: ΔT(i)=f(m_ rate )[ΔT(i)−ΔT(i−1)]/δt, where f(m_ rate ) is a machine-learned function pertaining to the thermal bias  320  and/or the rate of change for the thermal bias  320 . The machine-learned function may include a static factor, such as a multiplier. Additionally, or in the alternative, the machine-learned function may include one or more variables and/or one or more single-order or multiple-order functions. 
     In addition, or in the alternative to an initial value model  420  and/or a rate of change model  422 , a thermal bias model  402  may include a projection model  424 . A projection model  424  may be configured to determine a projection value for a thermal bias  320  at a future time. A projection value for a thermal bias  320  may be described by the relationship: ΔT(i+t SET )=ΔT(i)+(t SET ·dT(i)/dt), where ΔT(i+t SET ) is the projection value for the thermal bias  320 , and t SET  is the time period from the initial time, (i) for which the projection value for a thermal bias  320  is determined. In addition, or in the alternative to the foregoing, a thermal bias model  402  may utilize numerous other operating data, such as inputs from other sensors associated with a turbomachine, operating conditions and/or operating duration prior to shut-down, and the like. Additionally, or in the alternative, a projection value for a thermal bias  320  may be described by the relationship: ΔT(i)=f(m_ projection )[ΔT(i)+(t SET ·dT(i)/dt)], where f(m_ projection ) is a machine-learned function pertaining to the thermal bias  320  and/or the projection value for a thermal bias  320 . The machine-learned function may include a static factor, such as a multiplier. Additionally, or in the alternative, the machine-learned function may include one or more variables and/or one or more single-order or multiple-order functions. 
     A thermal bias model  402  may additionally, or alternatively include one or more cooling treatment models  426 . A cooling treatment model  426  may be configured to determine one or more control parameters. Such control parameters may be utilized by the cooling treatment model  426 , for example, to provide model outputs  406  such as control commands. Exemplary control parameters may include setpoints, gain values, filters, feedback parameters, cascade hierarchies, transfer functions, differential equations, and so forth. Such control parameters may be based at least in part on operating data  408  and/or fleet data  414 . Additionally, or in the alternative, a cooling treatment model  426  may be configured to determine a control regime, such as a control loop. Exemplary control regimes may include on-off control, linear control, proportional control, proportional-integral-derivative (PID) control, single-input-single-output (SISO) control, multi-input-multi-output (MIMO) control, multi-input-single-output (MISO) control, H2-optimal control, H-infinity control, Mu-synthesis control, distributed parameter control, hierarchical control, model predictive control, and so forth. Such a control regime may utilize the one or more control parameters determined by the cooling treatment model  426 . Additionally, or in the alternative, a cooling treatment model  426  may be configured to determine one or more control commands, such as cooling control commands  416  and/or engine protection control commands  418 . The control commands determined by the cooling treatment model  426  may be determined based at least in part on one or more control parameters and/or based at least in part on a control regime determined by the cooling treatment model  426 . Additionally, or in the alternative, the cooling treatment model  426  may be configured to determine one or more control parameters, control regimes, and/or control commands, based at least in part on data from an additional thermal bias model  402 , such as data from an initial value model  420 , a rate of change model  422 , and/or a projection model  424 . The control commands determined by the cooling treatment model  426  may be utilized to control one or more controllable components of a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . By way of example, the control commands may be utilized in an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. 
     In some embodiments, the model outputs  406  may include cooling control commands  416 . A cooling command  416  may be configured to control one or more controllable components in connection with a cooling treatment, such as rotating one or more portions of the turbomachine  106 , such as a HP spool  216  and/or an LP spool  224 . Such cooling treatments may be performed, for example, in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. Additionally, or in the alternative, the model outputs  406  may include one or more engine protection control commands  418 . 
     As shown in  FIG.  4   , in some embodiments, thermal bias control module  400  may include a model trainer  428 . The model trainer  428  may use training data, such as a representative data set, to train, adjust, and/or develop a thermal bias model  402 . The training data may include model inputs  404 , such as operating data  408  accumulated concurrently with operation of existing thermal bias models  402  and/or from previous operations of a thermal bias model  402 . Additionally, or in the alternative, the training data may include model inputs  404 , such as fleet data  414 , from previous operations of other aircraft  100  or other turbomachines  106 . Any one or more various training or learning techniques may be utilized by the model trainer  428 , such as backwards propagation of errors, including, for example, performing truncated backpropagation through time. In some embodiments, supervised training techniques may be used on a set of labeled training data. The model trainer  428  may perform a number of generalization techniques (e.g., weight decays, dropouts, etc.) to improve the generalization capability of the thermal bias models  402  being trained. 
     An exemplary model trainer  428  may include a machine-learned model  430 . The model trainer  428  may utilize one or more model inputs  404  as inputs to a machine-learned model  430 . The model trainer  428  may output one or more model adjustments  432 . The model adjustments  432  may include updates or adjustments to one or more thermal bias models  402 , such as to settings, values, and/or schedules included in or utilized by such a thermal bias model  402 . Additionally, or in the alternative, the model adjustments  432  may include new thermal bias models  402 , and/or new settings, values, and/or schedules to be included in or utilized by one or more thermal bias model  402 . The machine-learned model  430  may provide a machine-learned function, f(m). Such a machine-learned function may be utilized by or included in a thermal bias model  402 . For example, the machine-learned model  430  may include a machine-learned function pertaining to the thermal bias  320 , a rate of change for the thermal bias  320 , and/or a projection value for a thermal bias  320 . In some embodiments, the machine-learned model  430  may include and/or may be based at least in part on one or more thermal data sets  322 , such as a plurality of thermal data sets  322  accumulated over the course of one or more on-ground cooling method and/or in connection with an engine shut-down or engine start-up method. The thermal data sets  322  utilized by the machine-learned model  430  may be determined by a computing device  112  associated with an aircraft  100  or turbomachine  106  that receives control commands from the thermal bias control module  400 , such as by the computing device that executes the thermal bias control module  400 . Additionally, or in the alternative, the thermal data sets  322  utilized by the machine-learned model  430  may be provided by a management system  116 . In some embodiments, the thermal data sets  322  utilized by the machine-learned model  430  may include thermal data sets  322  determined from previous operations of the aircraft  100  or turbomachine  106  that receives control commands from the thermal bias control module  400 , and/or from previous operations of other aircraft  100  or other turbomachines  106 . The machine-learned model  430  may be configured to determine one or more model outputs  406  to be provided by a thermal bias model  402 , and/or one or more criterion, algorithm, or formula based upon which one or more model outputs  406  may be provided. The machine-learned model  430  may provide an improved criterion, algorithm, or formula for determining a thermal bias  320 . For example, the machine-learned model  430  may provide an improved initial value model  420 , an improved rate of change model  422 , and/or an improved projection value model  424 . Additionally, or in the alternative, the machine-learned model  430  may provide an improved on-ground cooling method, engine shut-down method, or engine start-up method, for example, to provide improved control of such thermal bias  320 . 
     A machine-learned model  430  may use any suitable machine learning technique, operating regime, or algorithm. A machine-learned model  430  may be configured to use pattern recognition, computational learning, artificial intelligence, or the like to derive algorithms that allow the machine-learned model  430  to determine model adjustments  432  to one or more thermal bias models  402 , new thermal bias models  402 , and/or new settings, values, and/or schedules to be included in or utilized by a thermal bias model  402 . A machine-learned model  430  may include an unsupervised or a supervised learning regime, including a semi-supervised learning regime, an active learning regime, a reinforcement learning regime, and/or a representation learning regime. A machine-learned model  430  may utilize neural networks, decision trees, association rules, inductive logic algorithms, cluster analysis algorithms, and the like. In some embodiments, the model inputs  404  utilized by a thermal bias model  402  may include data associated with or generated by a machine-learned model  430 . 
     In some embodiments, as illustrated in  FIG.  4   , the machine-learned model  430  may include a neural network. Additionally, or in the alternative, an exemplary machine-learned model  430  may include any other suitable model, including a linear discriminant analysis model, a partial least squares discriminant analysis model, a support vector machine model, a random tree model, a logistic regression model, a naive Bayes model, a K-nearest neighbor model, a quadratic discriminant analysis model, an anomaly detection model, a boosted and bagged decision tree model, an artificial neural network model, a C4.5 model, a k-means model, and combinations thereof. Further types of machine or statistical learning models are also contemplated. It will also be appreciated that the machine-learned model  430  can use certain mathematical methods alone or in combination with one or more other machine or statistical learning models. 
     In addition to outputting a model adjustments  432 , in some embodiments, a machine-learned model  430  may output a confidence score  434 . The confidence score  434  may provide an indication as to a level of confidence attributable to one or more outputs of the machine-learned model  430 . Additionally, or in the alternative, a thermal bias model  402  may output a confidence score  434  that provides an indication as to a level of confidence attribute one or more model outputs  406  of the thermal bias model  402 . The confidence score  434  can be used, for example, to set a margin of error to be used by the thermal bias control module  400  in determining a model adjustments  432 . For example, in the event of a low confidence score  434  the thermal bias control module  400  may account for a more conservative or wide margin for error when determining a model adjustments  432 , whereas in the event of a high confidence score  434  the thermal bias control module  400  may allow for a more aggressive or narrow margin for error when determining a model adjustments  432 . In some embodiments, the model inputs  404  may include a confidence score  434 . 
     Referring now to  FIGS.  5 A and  5 B , exemplary thermal bias models  402 , such as exemplary cooling treatment models  426 , are further described. A thermal bias model  402  may be configured to provide cooling treatments to reduce and/or control thermal bias. As shown in  FIG.  5 A , a thermal bias model  402  may be configured to determine one or more thermal bias-values  500 . The thermal bias-values  500  may be utilized by the cooling treatment model  426  to provide model outputs  406 , such as cooling control commands  416  and/or engine protection control commands  418 . For example, a thermal bias model  402  may be configured to determine a thermal bias-value  500  that includes a thermal bias-initial value ΔT(i)  502 . The thermal bias-initial value  502  may indicate an initial value for thermal bias  320 , at an initial time (i). The thermal bias-initial value  502  may be utilized by the cooling treatment model  426  as a thermal bias-value  500 . By way of example, a thermal bias-initial value  502  may be determined by an initial value model  420 . The thermal bias-initial value  502  may be determined based at least in part on a difference between an upward temperature-value  304  and a downward temperature-value  306 . As another example, a thermal bias model  402  may be configured to determine a thermal bias-value  500  that includes a thermal bias-projection value  504 . The thermal bias-projection value  504  may indicate a projected value for thermal bias  320 , at a future time (i+t FV ). The thermal bias-projection value  504  may be utilized by the cooling treatment model  426  as a thermal bias-value  500 . By way of example, a thermal bias-projection value  504  may be determined by a projection model  424 . The thermal bias-projection value  504  may be determined based at least in part on a difference between an upward temperature-value  304  and a downward temperature-value  306 . 
     In addition, or in the alternative to thermal bias-values  500 , a thermal bias model  402  may include a thermal bias model  402  configured to determine one or more control limits  506 . The one or more control limits may be utilized by the cooling treatment model  426 . The one or more control limits  506  may be compared to thermal bias-values  500 , for example, to provide model outputs  406 , such as cooling control commands  416  and/or engine protection control commands  418 . For example, a thermal bias model  402  may be configured to determine a control limit  506  that includes an upper control limit ΔT UCL    508  for a thermal bias-value  500 , such as a thermal bias-initial value  502  and/or a thermal bias-projection value  504 . In an exemplary embodiment, a thermal bias-initial value  502  and/or a thermal bias-projection value  504  may be compared to an upper control limit  508 . When a thermal bias-value  500  exceeds the upper control limit  508 , a cooling treatment may be activated, for example, in an on-ground cooling method, an engine shut-down method, or an engine start-up method. As another example, a thermal bias model  402  may be configured to determine a control limit  506  that includes a lower control limit ΔT LCL    510  for a thermal bias-value  500 . In an exemplary embodiment, a thermal bias-initial value  502  may be compared to a lower control limit  510 . When a thermal bias-value  500  such as a thermal bias-initial value  502  is below the lower control limit  510 , the cooling treatment may be postponed or deactivated, for example, unless or until the thermal bias-value  500  increases by a sufficient amount to activate the cooling treatment according to one or more conditions of the cooling treatment model  426 . Additionally, or in the alternative, a thermal bias-projection value  504  may be compared to a lower control limit  510 , and the cooling treatment may be postponed or deactivated when the thermal bias-projection value  504  is below the lower control limit  510 , for example, unless or until the thermal bias-projection value  504  increases by a sufficient amount to activate cooling treatment according to one or more conditions of the cooling treatment model  426 . The thermal bias-values  500  and/or the control limits  506  may be determined based at least in part on any one or more inputs  404 , including, for example, temperature measurements  308  such as an upward temperature-value T U (i)  304  and/or a downward temperature-value T D (i)  306 , setpoints  410 , etc. Additionally, or in the alternative, the thermal bias-values  500  and/or the control limits  506  may be determined from inputs  404  such as operating data  408 , setpoints  410 , and/or fleet data  414 . 
     In exemplary embodiments, thermal bias  320  may be controlled by performing a cooling treatment. The cooling treatment may include circulating air through at least a portion of the turbomachine  106 . Additionally, or in the alternative, the cooling treatment may include rotating a shaft of the turbomachine  106  with a motoring system  254 , such as the HP shaft  216  and/or the LP shaft  222 . Rotating such a shaft of the turbomachine  106  may reduce thermal bias at least in part by exposing respective circumferential portions of the high pressure spool and/or the low pressure spool to relatively higher-temperature air located in the upward portion of the turbomachine  106 . Additionally, or in the alternative, rotating a shaft of the turbomachine  106 , such as the HP shaft  216  and/or the LP shaft  222 , may cause air to circulate through at least a portion of the turbomachine  106 . In some embodiments, a thermal bias model  402  may be configured to control thermal bias  320  in a turbomachine by modulating a cooling treatment based at least in part on a thermal bias-value  500 . For example, modulating the cooling treatment may include modulating one or more controllable components of a motoring system  245 , for example, to control a rate of air flowing through a motoring air supply line  264  and/or to control a rate of rotation of a shaft of the turbomachine  106 , such as a rate of rotation of the HP shaft  216  and/or the LP shaft  222 . Additionally, or in the alternative, a cooling treatment may be activated and/or deactivated based at least in part on a thermal bias-value  500 . 
     As shown in  FIG.  5 B , an exemplary thermal bias model  402 , such as a cooling treatment model  426 , may be configured to provide model outputs  406 . As indicated at block  512 , the cooling treatment model  426  may provide model outputs  406  configured to perform a cooling treatment when the turbomachine  106  is in a non-started state. The non-started state may include an engine-operating parameter having been disabled for the turbomachine  106 . As shown, an exemplary cooling treatment model  426  may determine model outputs  406  such as cooling control commands  416  and/or engine protection control commands  418 . The model outputs  406  may be provided based at least in part on one or more thermal bias-values  500  and/or control limits  506 , such as determined as described with reference to  FIG.  5 A . The cooling treatment model  426  may compare one or more thermal bias-values  500  to one or more control limits  506 . The cooling treatment model  426  may determine the one or more model outputs  406  based at least in part on the comparison of the one or more thermal bias-values  500  to one or more control limits  506 . 
     The non-started state of the turbomachine  106  may be determined based at least in part on a status indication from a computing device  112  associated with the cooling treatment model  426 , for example, by way of an engine-operating parameter associated with the cooling treatment model  426 . Additionally, or in the alternative, a non-started state of the turbomachine  106  may include a status indication from any one or more controllable components associated with the turbomachine  106 . Such status indication may be provided directly or indirectly from such a controllable component and/or the computing device  112 . A non-started state may include one or more status indications configured to ensure that the turbomachine  106  is not operating prior to performing a cooling treatment. For example, a non-started state of the turbomachine  106  may be determined based at least in part on an indication corresponding to combustion in the combustion chamber  248 , fuel flow through a fuel nozzle  252 . Additionally, or in the alternative, a non-started state may be determined based at least in part on an indication that a rotation of the turbomachine  106  is below a specified rate of rotation. The indication may include a tachometer value corresponding to a tachometer  276  configured to determine a rate of rotation of one or more rotatable components of the turbomachine  106 . For example, the specific rate of rotation for a non-started state may include a rate of rotation of the HP shaft  216  and/or the LP shaft  222 , an external gearbox  260 , an internal gearbox  262 , and/or a radial drive shaft  258 . The tachometer value may indicate a rate of rotation commensurate with the turbomachine  106  having been shut down and a rate of rotation of the turbomachine  106  having come to a stop or decreased below the specified rate. 
     An exemplary cooling treatment model  426  may determine an operating state for a cooling treatment performed to reduce thermal bias in the turbomachine  106 , as indicated at block  514  of  FIG.  5 B . The cooling treatment may include rotating a shaft  201  of a turbomachine  106  with a motoring system  254 , thereby causing air to circulate through at least a portion of the turbomachine  106 . Possible operating states for a cooling treatment may include active and inactive. In some embodiments, the cooling treatment model  426  may determine whether the cooling treatment has an active state. Additionally, or in the alternative, the cooling treatment model  426  may determine whether the cooling treatment has an inactive state. The cooling treatment may be determined to have an active state (or an inactive state) based at least in part on a status indication from a computing device  112  associated with the cooling treatment model  426  and/or based at least in part on an indication that one or more controllable components associated with the cooling treatment have an active state (or an inactive state). 
     A status indication from a computing device  112  associated with the cooling treatment model  426  may indicate an active state of the cooling treatment, for example, by way of a control parameter associated with the cooling treatment model  426 . Additionally, or in the alternative, an active state may include a status indication from any one or more controllable components associated with the motoring system  254 . Such status indication may be provided directly or indirectly from such a controllable component and/or the computing device  112 . For example, an active state for the cooling treatment may be determined based at least in part on an indication corresponding to a motor  256 , an external gearbox  260 , an internal gearbox  262 , and/or a motoring air supply valve  268 . The indication may include a tachometer value corresponding to a tachometer  276  configured to determine a rate of rotation of a motor  256 , an external gearbox  260 , an internal gearbox  262 , or another component of a motoring system  254 . The tachometer value may indicate a rate of rotation commensurate with the cooling treatment, for example, in contrast to a rate of rotation commensurate with startup. By way of example, during a cooling treatment, the tachometer value may be from about 0.01% to about 50% of a rate of rotation used for starting the turbomachine  106 , such as from about 0.01% to about 10% of the rate of rotation used for starting the turbomachine  106 . Additionally, or in the alternative, an active state for a cooling treatment may be determined based at least in part on a valve position value corresponding to a motoring air supply valve  268 . By way of example, during the cooling treatment, the valve position value for the motoring air supply valve  268  may be from about 1% to about 50% of a valve position for the air supply valve  268  used for starting the turbomachine  106 , such as from about 1% to about 10% of the valve position used for starting the turbomachine  106 . 
     As shown in  FIG.  5 B , the one or more thermal bias-values  500  and the one or more control limits  506  that are compared to one another by the cooling treatment model  426  may depend at least in part on the determination as to whether the cooling treatment is active, at block  514 , such as a determination of an active state for the cooling treatment. When the cooling treatment model  426  determines an active state for the cooling treatment, at block  516 , the cooling treatment model  426  may compare a thermal bias-initial value  502  to a lower control limit ΔT LCL    510 . When the cooling treatment model  426  determines an active state for the cooling treatment, the cooling may continue for a period of time and/or the cooling treatment may be deactivated. When the cooling treatment model  426  determines an inactive state for the cooling treatment, at block  518 , the cooling treatment model  426  may compare a thermal bias-initial value  502  to an upper control limit ΔT UCL    508  for thermal bias. Additionally, or in the alternative, when the cooling treatment model  426  determines an inactive state for the cooling treatment, at block  518 , the cooling treatment model  426  may compare a thermal bias-projection value  504  to an upper control limit  508  for thermal bias. When the cooling treatment model  426  determines an inactive state for the cooling treatment, the cooling treatment may be activated and/or the cooling treatment may remain in an inactive state for a period of time. 
     With an active state for the cooling treatment, at block  516 , the cooling treatment model  426  may compare a thermal bias-initial value  502  to a lower control limit  510 . When the thermal bias-initial value  502  is less than the lower control limit  510 , the cooling treatment model  426  may provide a model output  406  that includes one or more control commands configured to deactivate the cooling treatment, at block  520 . The one or more control commands configured to deactivate the cooling treatment, at block  520 , may include one or more control commands configured to change an operating state of one or more controllable components of the motoring system  254  and/or the turbomachine  106 , such as a motor  256 , an external gearbox  260 , an internal gearbox  262 , and/or a motoring air supply valve  268 , thereby deactivating the cooling treatment. Additionally, or in the alternative, the one or more control commands configured to deactivate the cooling treatment, at block  520 , may include a control command configured to set an inactive state for the cooling treatment. 
     When the thermal bias-initial value  502  is less than the lower control limit  510 , the cooling treatment model  426  may provide a model output  406  that includes one or more control commands configured to enable engine start, at block  522 . An engine start status may have an operating state of enabled or disabled. The one or more control commands configured to enable engine start may be conditioned or contingent upon the cooling treatment model  426  having deactivated the cooling treatment, at block  520 , and/or upon the cooling treatment having an inactive operating state. In some embodiments, the cooling treatment model  426  may be configured to set an engine start status to an enabled operating state only when the cooling treatment has an inactive operating state and/or only when the cooling treatment model  426  has deactivated the cooling treatment, at block  520 . The one or more control commands configured to deactivate the cooling treatment, at block  520 , and the one or more control commands configured to enable engine start, at block  522 , may be provided in series or in parallel. In exemplary embodiments, the cooling treatment model  426  may be configured to prevent or condition the engine start status from being set to an enabled operating state unless the cooling treatment has an inactive operating state. The enabled operating state for the engine start status may become effective when the cooling treatment has an inactive operating state. In some embodiments, following the one or more model outputs  406  to deactivate the cooling treatment, at block  520 , and/or to enable engine start, at block  522 , the cooling treatment model  426  may return to block  514 , to determine whether the cooling treatment has an active state and/or to determine whether the cooling treatment has an inactive state. When, at block  522 , the engine start status has an enabled operating state, the turbomachine  106  may be started. When commencing startup of the turbomachine  106 , and/or after the turbomachine  106  has been started, an operating state of the turbomachine  106  may indicated an operating status that includes a started state, at block  526 . The started state may include an engine-operating parameter having been enabled for the turbomachine  106 . 
     Referring again to block  518  of the cooling treatment model  426 , when the cooling treatment has an inactive state, the cooling treatment model  426  may compare a thermal bias-initial value  502  to an upper control limit  508 , and/or the cooling treatment model  426  may compare a thermal bias-projection value  504  to an upper control limit  508 . When the thermal bias-initial value  502  is greater than the upper control limit  508 , the cooling treatment model  426  may provide a model output  406  that includes one or more control commands configured to disable engine start, at block  522 . Additionally, or in the alternative, when the thermal bias-projection value  504  is greater than the upper control limit  508 , the cooling treatment model  426  may provide a model output  406  that includes one or more control commands configured to disable engine start, at block  522 . The cooling treatment model  426  may provide the model output  406  that includes one or more control commands configured to disable engine start, at block  524 , when either the thermal bias-initial value  502  or the thermal bias-projection value  504  is greater than the upper control limit  508 . In this way, the turbomachine  106  may be prevented from starting when the cooling treatment model  426  determines that thermal bias  320  is too great, such as at an initial time (i), or that the thermal bias  320  is projected to be too great, such as at a future time (i+t FV ). The turbomachine  106  may be prevented from starting by the model output  406  configured to disable engine start, at block  524 . For example, the model output  406 , at block  524 , may include a control command configured to set the engine start status to a disabled operating state when either the thermal bias-initial value  502  or the thermal bias-projection value  504  is greater than the upper control limit  508 . 
     When either the thermal bias-initial value  502  or the thermal bias-projection value  504  is greater than the upper control limit  508 , the cooling treatment model  426  may additionally or alternatively provide a model output  406  that includes one or more control commands configured to activate the cooling treatment, at block  528 . The one or more control commands configured to activate the cooling treatment, at block  528 , may include one or more control commands configured to change an operating state of one or more controllable components of the motoring system  254  and/or the turbomachine  106 , such as a motor  256 , an external gearbox  260 , an internal gearbox  262 , and/or a motoring air supply valve  268 , thereby activating the cooling treatment. Additionally, or in the alternative, the one or more control commands configured to activate the cooling treatment, at block  528 , may include a control command configured to set an active state for the cooling treatment. In some embodiments, activating a cooling treatment may include modulating the cooling treatment based at least in part on a thermal bias-value  500 . 
     The one or more control commands configured to activate the cooling treatment may be conditioned or contingent upon the cooling treatment model  426  having disabled engine start, at block  524 , and/or upon engine start having a disabled operating state. In this way, the cooling treatment model  426  may be configured to change the operating state of the cooling treatment to an enabled operating state, and/or to activate the cooling treatment, only when the engine start status has a disabled operating state. The one or more control commands configured to disable the engine start status, at block  524 , and the one or more control commands configured to activate the cooling treatment, at block  528 , may be provided in series or in parallel. In exemplary embodiments, the cooling treatment model  426  may be configured to prevent or condition the cooling treatment from being activated or changed to an active operating state unless engine start status is set to a disabled operating state. The active operating state for the cooling treatment may become effective when the engine start status has disabled operating state. In some embodiments, following the one or more model outputs  406  to disable engine start, at block  524 , and/or to activate the cooling treatment, at block  528 , the cooling treatment model  426  may return to block  514 , to determine whether the cooling treatment has an active state and/or to determine whether the cooling treatment has an inactive state. 
     When, at block  516 , the cooling treatment model  426  determines that the thermal bias-initial value  502  is greater than or equal to (or not less than) the lower control limit  510 , the cooling treatment model may proceed to block  518 , to determine whether either of the thermal bias-initial value  502  or the thermal bias-projection value  504  is greater than the upper control limit  508 . At block  518 , when both the thermal bias-initial value  502  and the thermal bias-projection value  504  is less than or equal to (or not greater than) than the upper control limit  508 , the cooling treatment model  426  may provide a model output  406 , at block  520 . The model output  406  may include one or more control commands configured to deactivate the cooling treatment. The cooling treatment model  426  may be configured to continue the cooling treatment until the thermal bias-initial value  502  and the thermal bias-projection value  504  are less than or equal to (or not greater than) than the upper control limit  508 . In some embodiments, the engine start status may be set to an enabled operating state, and an engine start-up method for the turbomachine  106  may be initiated and/or the turbomachine  106  may be started within the time period from an initial time (i) to a future time (i+t FV ), such as when the thermal bias-value  500  is less than the upper control limit  508 , such as when the thermal bias-value  500  is between the lower control limit  510  and the upper control limit  508 . The future time (i+t FV ) may be selected to provide sufficient level of confidence that the thermal bias  320  will not be above the upper control limit  508  if the turbomachine  106  is started within the time period from the initial time (i) to the future time (i+t FV ). For example, the level of confidence may be determined based at least in part on a variance  318  in the temperature measurements  308 , such as a variance  318  in the temperature measurements  308  used to determine the thermal bias-value  500  and/or the upper control limit  508 . Additionally, or in the alternative, such level of confidence may be determined based at least in part on a statistical variance, a specified variance, a standard deviation, a confidence interval, or the like. 
     In some embodiments, the turbomachine  106  may be prevented from being started when the engine start status is set to a disabled operating state. The computing device  112  and/or the cooling treatment model  426  may include control logic that prevents the turbomachine  106  from starting when the engine start status is set to a disabled operating state. Additionally, or in the alternative, the computing device  112  and/or the cooling treatment model  426  may include control logic that allows the turbomachine  106  to be started only when the engine start status is set to an enabled operating state. The disabled operating state may cause the cooling treatment model  426  to provide one or more outputs  406  that physically prevent the turbomachine  106  from being started when the engine start status is set to a disabled operating state, and/or that allow the turbomachine  106  to be started when the engine start status is set to an enabled operating state. For example, the turbomachine  106  may be prevented from being started by a control command that causes the motoring system  254  to exhibit an operating state and/or a physical configuration that prevents the turbomachine  106  from being started. Additionally, or in the alternative, the turbomachine  106  may be physically capable of being started when a control command causes the motoring system  254  to exhibit an operating state and/or a physical configuration that allows the turbomachine  106  to be started. Such a physical configuration may include a position of one or more gears, clutches, or the like of an external gearbox  260  and/or an internal gearbox  262 . For example, with a disabled operating state, the motoring system  254  may exhibit a gear ratio that is suitable for cooling treatment, but unsuitable for starting the turbomachine  106 . Additionally, or in the alternative, such a physical configuration may include a position of a motoring air supply valve  268 . For example, with a disabled operating state, the motoring air supply valve  268  may exhibit a valve position that is suitable for cooling treatment, but unsuitable for starting the turbomachine  106 . In this way, the turbomachine  106  may be prevented from rotating at high rate of speed such during an engine start-up in the event of a thermal bias-value  500  that exceeds an upper control limit  508  and/or when the cooling treatment is active. 
     Referring now to  FIGS.  6 A and  6 B, and  6 C and  6 D , exemplary methods of controlling thermal bias are further described.  FIG.  6 A  shows a chart depicting an example thermal bias-value  500  as a function of time after engine shut-off of a turbomachine  106 .  FIG.  6 B  shows a chart corresponding to  FIG.  6 A , depicting residual heat in one or more areas of a turbomachine  106  as a function of time after engine shut-off of the turbomachine  106 .  FIG.  6 C  shows a chart depicting another example thermal bias-value  500  as a function of time after engine shut-off of a turbomachine  106 .  FIG.  6 D  shows a chart corresponding to  FIG.  6 C , depicting residual heat in one or more areas of a turbomachine  106  as a function of time after engine shut-off of the turbomachine  106 . As illustrated by a solid line  600  in  FIGS.  6 A and  6 B , and  FIGS.  6 C and  6 D , thermal bias may be controlled using a thermal bias model  402 , such as a cooling treatment model  426 . An exemplary uncontrolled thermal bias  320  is illustrated by a dashed line  602  in  FIGS.  6 A and  6 B , and  FIGS.  6 C and  6 D . By way of comparison, another exemplary method of controlling thermal bias is illustrated by a dash-and-dotted line  604  in  FIGS.  6 A and  6 B , and  FIGS.  6 C and  6 D . 
     The method of controlling thermal bias depicted in  FIGS.  6 A and  6 B, and  6 C and  6 D , may include one or more cooling periods  606  during which a cooling treatment may be provided. As shown in  FIG.  6 A , a cooling period  606  may include performing a cooling treatment that provides a relatively rapid decrease in thermal bias-value  500 . In some embodiments, thermal bias may be controlled at least in part using a plurality of cooling periods  606 , as shown, for example, in  FIG.  6 A . As shown in  FIG.  6 A , the thermal bias-value  500  may be reduced by some amount, for example, from about an upper control limit ΔT UCL    508  to a lower control limit ΔT LCL    510 , over a period of time corresponding to the cooling period  606 . After a cooling period, the thermal bias-value  500  may increase, for example as a result of residual heat that still exists within the turbomachine  106 . As the thermal bias-value  500  approaches the upper control limit  508 , a next cooling period may begin, again reducing the thermal bias-value  500 , for example, from about an upper control limit  508  to a lower control limit  510 . In some embodiments, sequential cooling periods  606  may be provided until a sufficient amount of residual heat is removed from the turbomachine  106  and/or until commencing an engine startup. Additionally, or in the alternative, as shown in  FIG.  6 C , a single cooling period  606  may be provided. In some embodiments, the cooling period  606  may include providing a controlled amount of cooling, for example, to control thermal bias-value  500  to a setpoint and/or a range. For example, the thermal bias-value  500  may be controlled to a setpoint such as an upper control limit  508 , and/or to a range, such as a range defined by an upper control limit  508  and a lower control limit  510 . 
     By way of example, as shown in  FIGS.  6 A and  6 B, and  6 C and  6 D , a turbomachine  106  may be shut-off at a time t( 0 ). At the time of engine shut-off, the turbomachine  106  may exhibit a low thermal bias-value  500  ( FIGS.  6 A and  6 C ) that increases due to a high amount of residual heat in the turbomachine ( FIGS.  6 B and  6 D ). When uncontrolled, the thermal bias-value  500  may increase beyond the upper control limit ΔT UCL    508 , as shown in  FIGS.  6 A and  6 C . The thermal bias  320  may be attributable at least in part to asymmetric cooling within the turbomachine after engine shut-off. For example, asymmetric cooling may be attributable to hot gasses rising to upward portions of the turbomachine  106 , which may transfer heat to such upward portions and/or may allow downward portions to cool more than upward portions. Over time, and uncontrolled thermal bias  320  may gradually decrease, for example, as residual heat within the turbomachine declines as shown in  FIGS.  6 B and  6 D . When uncontrolled, at a time t(n), the thermal bias  320  may exhibit a thermal bias-value may remain above the upper control limit  508 . As shown by the dash-and-dotted line  604 , thermal bias  320  may be controlled, for example, using a motoring system  254  to rotate a shaft  201  of the turbomachine  106 , such as the HP shaft  216  and/or the LP shaft  222  ( FIG.  2   ). Rotation of the HP shaft  216  may cause the high pressure spool  218  to rotate. Rotation of the LP shaft  222  may cause the low pressure spool  224  to rotate ( FIG.  2   ). The dash-and-dotted line  604  may represent a cooling treatment that is activated and continued until a time when enough residual heat has been removed from the turbomachine  106 , as shown in  FIGS.  6 B and  6 D  at a time t(x). The solid line  600  represents a method of controlling thermal bias performed, for example, according to a thermal bias model  402 , such as a cooling treatment model  426 , as described with reference to  FIGS.  4 ,  5 A, and  5 B . 
     Referring to  FIGS.  6 A and  6 B , as illustrated by the solid line  600  in  FIGS.  6 A and  6 B , a first cooling period  606  may begin at a time t( 1 ). The time t( 1 ) may correspond to a determination, at block  518  of  FIG.  5 B , that a thermal bias-initial value  502  or a thermal bias-projection value  504  is greater than the upper control limit ΔT UCL    508 . For example, as shown in  FIG.  6 A , the first cooling period  606  may begin prior to the thermal bias-value  500  reaching the upper control limit  508 , indicating that the cooling treatment model  426  determined the thermal bias-projection value  504  is greater than the upper control limit  508 . The cooling period may continue for a first cooling period  606  indicated by a first shaded region in  FIGS.  6 A and  6 B , for example, until such time as the cooling treatment model  426  determines that the thermal bias-initial value  502  is less than the lower control limit ATLCL  510  ( FIG.  5 B , block  516 ). As shown in  FIG.  6 A , the cooling period  606  may end when the thermal vias  320  is slightly below the lower control limit  510 . 
     As shown in  FIG.  6 B , the first cooling period  606  may reduce the amount of residual heat in one or more regions of the turbomachine  106 . Remaining residual heat within the turbomachine  106  may cause thermal bias to increase, as shown in  FIG.  6 A . At a second time t( 2 ), a second cooling period  608  may begin, as indicated by a second shaded region in  FIGS.  6 A and  6 B , for example, when the cooling treatment model  426  determines the thermal bias-projection value  504  is greater than the upper control limit  508  ( FIG.  5 B ). The second cooling period  608  may continue until such time as the cooling treatment model  426  determines that the thermal bias-initial value  502  is less than the lower control limit  510  ( FIG.  5 B , block  516 ). The second cooling period  608  may further reduce the amount of residual heat in one or more regions of the turbomachine  106 , as shown in  FIG.  6 B . Remaining residual heat within the turbomachine  106  may cause thermal bias to increase, as shown in  FIG.  6 A . At a third time t( 3 ), a third cooling period  610  may begin, as indicated by a third shaded region in  FIGS.  6 A and  6 B , for example, when the cooling treatment model  426  determines the thermal bias-projection value  504  is greater than the upper control limit  508  ( FIG.  5 B , block  518 ). The third cooling period  610  may continue until such time as the cooling treatment model  426  determines that the thermal bias-initial value  502  is less than the lower control limit  510  ( FIG.  5 B , block  516 ). The third cooling period  610  may further reduce the amount of residual heat in one or more regions of the turbomachine  106 , as shown in  FIG.  6 B . Remaining residual heat within the turbomachine  106  may cause thermal bias to increase, as shown in  FIG.  6 A . 
     Referring to  FIGS.  6 C and  6 D , as illustrated by the solid line  600  in  FIGS.  6 A , a cooling period  606  may begin at a time t( 1 ). The time t( 1 ) may correspond to a determination, at block  518  of  FIG.  5 B , that a thermal bias-initial value  502  or a thermal bias-projection value  504  is greater than the upper control limit ΔT UCL    508 . For example, as shown in  FIG.  6 C , the cooling period  606  may begin prior to the thermal bias-value  500  reaching the upper control limit  508 , indicating that the cooling treatment model  426  determined the thermal bias-projection value  504  is greater than the upper control limit  508 . During the cooling period  606 , the thermal bias-value  500  may be controlled based at least in part on a setpoint, such as the upper control limit  508 . The thermal bias model  402  may provide a controlled amount of cooling, for example, during the cooling period  606 . The thermal bias model  402  may provide model outputs  406 , such as cooling control commands  416 . The cooling control commands  416  may control the thermal bias-value  500  to a setpoint such as an upper control limit  508 , and/or to a range, such as a range defined by an upper control limit  508  and a lower control limit  510 . For example, as shown in  FIG.  6 C , the thermal bias may exhibit thermal bias-values  500  at and/or below the upper control limit  508 , and/or between the upper control limit  508  and the lower control limit  510 . 
     In some embodiments, the cooling period  606  may continue until a sufficient amount of residual heat has been removed from the turbomachine  106 . For example, as shown in  FIG.  6 D , after the cooling period  606  had ended, the thermal bias may exhibit thermal bias-values  500  that exceed the upper control limit  508  for a period of time, but then decrease below the upper control limit  508 , for example, as residual heat dissipates from the turbomachine  106 . In some embodiments, the cooling period  606  may end when sufficient residual heat has been removed from the turbomachine  106  such that the thermal bias is projected to exhibit a thermal bias-value that is below the upper control limit  508  at a scheduled engine start time, for example, even with an initial increase in thermal bias upon ending the cooling period  606 . 
     It will be appreciated that any number of cooling periods  606  may be provided, for example, until such time as the thermal bias no longer increases to the extent that the thermal bias-initial value  502  and/or the thermal bias-projection value  504  is greater than the upper control limit  508  ( FIG.  5 B , block  518 ). By way of example, as shown in  FIGS.  6 A , the thermal bias-value  500  may increase but remain below the upper control limit  508  after a number of cooling periods  606 , such as after the third cooling period  610  as shown in  FIGS.  6 A and  6 B . Additionally, or in the alternative, as shown, in  FIG.  6 B , the residual heat in one or more regions of the turbomachine  106  may decrease below a residual heat-threshold QLIMIT  612 , such that the remaining residual heat may be insufficient to increase the thermal bias-value  500  above the upper control limit  508 . It will be appreciated that the three cooling periods  606 ,  608 ,  610  described with reference to  FIGS.  6 A and  6 B  are provided by way of example only, and that any number of cooling periods  606  may be provided in accordance with the present disclosure. The specific number of cooling periods  606  may depend on a variety of parameters associated with the turbomachine  106 , such as material properties and physical configuration of the turbomachine  106 . Additionally, or in the alternative, the specific number of cooling periods  606  may depend on a variety of parameters associated with the thermal bias control module  400 , such as the amount of residual heat in one or more regions of the turbomachine  106 , the rate of residual heat removal during the respective cooling periods  606 , and the duration of the respective cooling periods  606 . 
     In some embodiments, as illustrated in  FIGS.  6 A and  6 B , thermal bias  320  may be controlled more efficiently and/or more rapidly when using a plurality of cooling periods  606 , such as according to a thermal bias model  402 , such as a cooling treatment model  426 . The more efficient and/or more rapid cooling treatment may be realized relative to a cooling treatment that is activated and continued until a time when the residual heat in the turbomachine  106  is reduced below a residual heat-threshold Q LIMIT    612  such that the remaining residual heat may be insufficient to increase the thermal bias-value  500  above the upper control limit  508 , as indicated in  FIGS.  6 A and  6 B  by a dash-and-dotted line  604 . For example, in some embodiments, by allowing the thermal bias  320  to increase to some extent (e.g., to the upper control limit  508 ) prior to initiating a cooling period  606 , the passage of time prior to initiating a cooling period  606  may allow residual heat to be conducted from inward regions of the turbomachine  106  in a direction towards toward outward regions of the turbomachine  106 . Such passage of time prior to initiating the cooling period  606  may thereby provide a relatively higher temperature gradient between regions of the turbomachine  106  that have a heat transfer relationship with airflow that passes through the turbomachine  106  during the cooling treatment. Additionally, or in the alternative, by deactivating the cooling treatment when the thermal bias  320  has decreased to some extent (e.g., to the lower control limit  510 ), time and energy consumed by the cooling treatment may be postponed, for example, during a period following a previous cooling period  606  and/or preceding a next cooling period  606  when the temperature gradient is relatively low between regions of the turbomachine  106  that have a heat transfer relationship with airflow that passes through the turbomachine  106  during the cooling treatment. In some embodiments, the upper control limit  508  and/or the lower control limit  510  may be selected at least in part to provide improved cooling, including, for example, improved heat dissipation during cooling periods  606  and/or improved energy efficiency for the cooling treatment. 
     In some embodiments, a method of controlling thermal bias may include and/or provide for a plurality of engine start windows  614  between respective cooling periods. An engine start window  614  may include a period of time within which the turbomachine  106  may be started, such as when the thermal bias-value  500  is below a setpoint, such as an upper control limit  508 . During an engine start window  614 , cooling may be deactivated ( FIG.  5 B , block  520 ), for example, to assure that the turbomachine is not started during a cooling period  606 . Additionally, or in the alternative, during an engine start window  614 , an engine start status for the turbomachine  106  may have an operating state of enabled ( FIG.  5 B , block  522 ). 
     In some embodiments, temperature measurements may be determined periodically with incremental rotations of a shaft  201  of a turbomachine  106 , such as at intervals of about (π)-radians, such as at intervals of (2π)-radians, such as at intervals of from about (π/6)-radians to about (2π)-radians, such as at intervals of from about (π/2)-radians to about (3π/2)-radians, such as at intervals of from about (5π/6)-radians to about (7π/6)-radians, such at intervals of about (2π)-radians+/−(7π/4)-radians, or such as at intervals of about (π)-radians+/−(π/4)-radians, as well as at any multiple of such an interval. 
     In some embodiments, an exemplary the upper control limit  508  ( FIGS.  5 A and  5 B ) may be from about 10 degrees Celsius (° C.) to about 30° C., such as from about to about 10° C. to about 20° C., such as from about to about 15° C. to about 25° C., or such as from about to about 20° C. to about 30° C. The upper control limit  508  may be at least about 10° C., such as at least about 15° C., such as at least about 20° C., or such as at least about 25° C. 
     In some embodiments, an exemplary lower control limit  510  ( FIGS.  5 A and  5 B ) may be from about 1° C. to about 10° C., such as from about to about 2° C. to about 8° C., or such as from about to about 4° C. to about 6° C., such as from about to about 6° C. to about 8° C. The upper control limit  508  may be at least about 2° C., such as at least about 4° C., such as at least about 6° C., or such as at least about 8° C. 
     In some embodiments, the future time (i+t FV ) ( FIGS.  5 A and  5 B ) for a thermal bias-projection value  504  may be from about 1 minute to about 10 minutes, such as from about 2 minutes to about 8 minutes, such as from about 3 minutes to about 5 minutes. The future time (i+t) may be at least about 2 minutes, such as at least about 5 minutes, or such as at least about 8 minutes. 
     In some embodiments, a cooling period  606  ( FIGS.  6 A- 6 D ) may be from about 1 minute to about 10 minutes, such as from about 2 minutes to about 8 minutes, such as from about 3 minutes to about 5 minutes. The cooling period  606  may be at least about 2 minutes, such as at least about 5 minutes, or such as at least about 8 minutes. The cooling period  606  may be less than about 9 minutes, such as less than about 7 minutes, such as less than about 5 minutes, or such as at least about 3 minutes. 
     In some embodiments, a number of cooling periods  606  ( FIGS.  6 A- 6 D ) performed may be from 1 to about 5, such as from about 2 to about 5, or such as from about 3 to about 5. The number of cooling periods  606  may be at least 3, such as at least 4, or such as at least 5. 
     In some embodiments, a total cooling time, determined from a sum of the cooling periods  606  ( FIGS.  6 A- 6 D ) may be from about 3 minutes to about 20 minutes, such as from about 5 minutes to about 15 minutes, or such as from about 8 minutes to about 12 minutes. The total cooling time may be at least about 3 minutes, such as at least about 5 minutes, such as at least about 10 minutes, or such as at least about 15 minutes. The total cooling time may be less than about 15 minutes, such as less than about 10 minutes, or such as less than about 5 minutes. 
     In some embodiments, an engine start windows  614  ( FIGS.  6 A- 6 C ) may be from about 1 minute to about 10 minutes, such as from about 2 minutes to about 8 minutes, such as from about 3 minutes to about 5 minutes. The cooling period  606  may be at least about 2 minutes, such as at least about 5 minutes, or such as at least about 8 minutes. 
     In some embodiments, without cooling, a bowed rotor condition may exist for a period of from about 30 minutes to about 180 minutes, such as from about 60 minutes to about 160 minutes, such as from about 90 minutes to about 120 minutes. Without cooling, a bowed rotor condition may exist for at least about 30 minutes, such as at least about 60 minutes, such as at least about 90 minutes, such as at least about 120 minutes, or such as at least about 160 minutes. 
     Referring now to  FIG.  7   , an exemplary engine control system  110  is further described. An engine control system  110  may be configured to perform one or more control operations associated with an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . The control operations may include, one or more control commands associated with a cooling treatment, such as in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. 
     As shown in  FIG.  7   , an exemplary engine control system  110  may include a controller  700 , such as an electronic engine controller, a full-authority digital engine control (FADEC) device, or the like. The controller  700  may include one or more control modules  702  configured to cause the controller  700  to perform one or more control operations. The one or more control modules  702  may include control logic executable to provide control commands configured to control one or more controllable components associated with an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . For example, a control module  702  may be configured to provide one or more control commands executable to control operation of one or more components of an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . The control commands may be configured to perform a method of controlling thermal bias in accordance with the present disclosure. 
     The controller  700  may be communicatively coupled with one or more components of an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . The controller  700  may be communicatively coupled with one or more sensors  120 , such as temperature sensors, of a turbomachine  106 . For example, the controller  700  may be communicatively coupled with one or more high pressure compressor-sensors  270 , one or more high pressure turbine-sensors  272 , and/or one or more low pressure turbine-sensors  274 . Additionally, or in the alternative, the controller  700  may be communicatively coupled with a tachometer  276  of a motoring system  254 . The controller  700  may also be communicatively coupled with a management system  116  and/or a user interface  118 . 
     The controller  700  may include one or more computing devices  704 , which may be located locally or remotely relative to a turbomachine  106 , a motoring system  254 , and/or an auxiliary power unit  108 . The one or more computing devices  704  may include one or more processors  706  and one or more memory devices  708 . The one or more processors  706  may include any suitable processing device, such as a microprocessor, microcontroller, integrated circuit, logic device, and/or other suitable processing device. The one or more memory devices  708  may include one or more computer-readable media, including but not limited to non-transitory computer-readable media, RAM, ROM, hard drives, flash drives, and/or other memory devices  708 . 
     As used herein, the terms “processor” and “computer” and related terms, such as “processing device” and “computing device”, are not limited to just those integrated circuits referred to in the art as a computer, but broadly refers to a microcontroller, a microcomputer, a programmable logic controller (PLC), an application specific integrated circuit, and other programmable circuits, and these terms are used interchangeably herein. A memory device  708  may include, but is not limited to, a non-transitory computer-readable medium, such as a random access memory (RAM), and computer-readable nonvolatile media, such as hard drives, flash memory, and other memory devices. Alternatively, a floppy disk, a compact disc-read only memory (CD-ROM), a magneto-optical disk (MOD), and/or a digital versatile disc (DVD) may also be used. 
     As used herein, the term “non-transitory computer-readable medium” is intended to be representative of any tangible computer-based device implemented in any method or technology for short-term and long-term storage of information, such as, computer-readable instructions, data structures, program modules and sub-modules, or other data in any device. The methods described herein may be encoded as executable instructions embodied in a tangible, non-transitory, computer readable media, including, without limitation, a storage device and/or a memory device. Such instructions, when executed by a processor, cause the processor to perform at least a portion of the methods described herein. Moreover, as used herein, the term “non-transitory computer-readable medium” includes all tangible, computer-readable media, including, without limitation, non-transitory computer storage devices, including, without limitation, volatile and nonvolatile media, and removable and non-removable media such as a firmware, physical and virtual storage, CD-ROMs, DVDs, and any other digital source such as a network or the Internet, as well as yet to be developed digital means, with the sole exception being a transitory, propagating signal. 
     The one or more memory devices  708  may store information accessible by the one or more processors  706 , including computer-executable instructions  710  that can be executed by the one or more processors  706 . The instructions  710  may include any set of instructions which when executed by the one or more processors  706  cause the one or more processors  706  to perform operations, including operations associated with cooling treatments configured to control thermal bias, such as in connection with an on-ground cooling method, an engine shut-down method, and/or an engine start-up method. 
     The memory devices  708  may store data  712  accessible by the one or more processors  706 . The data  712  can include current or real-time data  712 , past data  712 , or a combination thereof. The data  712  may be stored in a data library  714 . As examples, the data  712  may include data  712  associated with or generated by an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . Additionally, or in the alternative, the data  712  may include data  712  associated with or generated by one or more sensors  120 . Additionally, or in the alternative, the data  712  may include data  712  associated with or generated by one or more control modules  702 , such as thermal bias control module  400 . Such data  712  may include model outputs  406 , such as cooling commands  416  and/or engine protection control commands  418 . The data  712  may also include other data sets, parameters, outputs, information, associated with controlling thermal bias, control modules  702 , an aircraft  100 , a turbomachine  106 , a motoring system  254 , an auxiliary power unit  108 , and/or a ground start unit  266 . 
     The one or more computing devices  704  may also include a communication interface  716 , which may be used for communications with a communication network  114  via wired or wireless communication lines  718 . The communication interface  716  may include any suitable components for interfacing with one or more network(s), including for example, transmitters, receivers, ports, controllers, antennas, and/or other suitable components. The communication interface  716  may allow the computing device  704  to communicate with various nodes on the communication network  114 , such as nodes associated with the control modules  702 , the aircraft  100 , the turbomachine  106 , the motoring system  254 , the auxiliary power unit  108 , the ground start unit  266 , the management system  116 , and/or a user interface  118 . The communication network  114  may include, for example, a local area network (LAN), a wide area network (WAN), SATCOM network, VHF network, a HF network, a Wi-Fi network, a WiMAX network, a gatelink network, and/or any other suitable communication network  114  for transmitting messages to and/or from the controller  700  across the communication lines  718 . The communication lines  718  of communication network  114  may include a data bus or a combination of wired and/or wireless communication links. 
     The communication interface  716  may allow the computing device  704  to communicate with various components of an aircraft  100  and/or various components of a turbomachine  106  communicatively coupled with the communication interface  716  and/or communicatively coupled with one another. The communication interface  716  may additionally or alternatively allow the computing device  704  to communicate with the management system  116  and/or the user interface  118 . The management system  116  may include a server  720  and/or a data warehouse  722 . As an example, at least a portion of the data  712  may be stored in the data warehouse  722 , and the server  720  may be configured to transmit data  712  from the data warehouse  722  to the computing device  704 , and/or to receive data  712  from the computing device  704  and to store the received data  712  in the data warehouse  722  for further purposes. The server  720  and/or the data warehouse  722  may be implemented as part of an engine control system  110  and/or as part of the management system  116 . 
     Now turning to  FIGS.  8 A- 8 D  exemplary methods  800  of controlling thermal bias in a turbomachine  106  will be described. Exemplary methods may be performed at least in part by an engine control system  110 , and/or one or more controllers  700 , such as an electronic engine controller, a full-authority digital engine control (FADEC) device, or the like. Additionally, or in the alternative, exemplary methods may be performed at least in part by one or more control modules  702 , such as a thermal bias control module  400 . 
     As shown in  FIG.  8 A , an exemplary method  800  may include, at block  802 , determining a thermal bias-value  500 . The thermal bias value may be determined when the turbomachine  106  is in a non-started state. The non-started state may include an engine-operating parameter having been disabled for the turbomachine  106 . The thermal bias-value  500  may include a difference between an upward temperature-value  304  and a downward temperature-value  306 . The upward temperature-value  304  may correspond to a first one or more temperature measurements  308  of an upward portion of the turbomachine  106 . The downward temperature-value  306  corresponding to a second one or more temperature measurements of a downward portion of the turbomachine  106 . At block  804 , the exemplary method  800  may include performing a cooling treatment based at least in part on the thermal bias-value  500 . The cooling treatment may include at least one of: circulating air through at least a portion of the turbomachine  106 , and rotating a shaft of the turbomachine  106  with a motoring system  254 . In some embodiments, rotating the shaft of the turbomachine  106  causes the air to circulate through the at least a portion of the turbomachine  106 . In some embodiments, performing the cooling treatment may include modulating the cooling treatment based at least in part on the thermal bias-value. Additionally, or in the alternative, performing the cooling treatment may include activating a cooling treatment when the thermal bias-value  500  exceeds an upper control limit  508 . The cooling treatment may be activated when the turbomachine is in a non-started state. The non-started state may include an engine-operating parameter having been disabled for the turbomachine  106 . The cooling treatment may include rotating a shaft  201  of the turbomachine  106  with a motoring system  254 . Rotating the shaft  201  of the turbomachine  106  may cause air to circulate through at least a portion of the turbomachine  106 . 
     In some embodiments, the exemplary method  800  may include, at block  806 , determining the thermal bias-value  500  at one or more times after activating the cooling treatment. The cooling treatment may lower the thermal bias-value  500 . Block  806  of the exemplary method  800  may additionally or alternatively include deactivating the cooling treatment when the thermal bias-value  500  is less than a lower control limit  510  at a time after activating the cooling treatment. At block  808 , an exemplary method  800  may include determining the thermal bias-value  500  at one or more times after deactivating the cooling treatment. The thermal bias-value  500  may increase after deactivating the cooling treatment. The exemplary method  800  may include, at block  810 , setting an engine start status to a disabled operating state and activating the cooling treatment, when the thermal bias-value  500  exceeds the upper control limit  508  at a time after deactivating the cooling treatment. Additionally, or in the alternative, at block  812 , the exemplary method  800  may include determining the upward temperature-value  304  based at least in part on a first one or more temperature measurements  308  from one or more sensors  120 . Additionally, or in the alternative, at block  812 , the exemplary method  800  may include determining the downward temperature-value  306  based at least in part on a second one or more temperature measurements  308  from the one or more sensors  120 . Additionally, or in the alternative, at block  814 , the exemplary method  800  may include determining an operating state of the turbomachine  106 . The operating state may include a non-started state. The non-started state may include an engine-operating parameter having been disabled for the turbomachine  106 . 
     As shown in  FIG.  8 B , the exemplary method  800  may include, at block  816 , determining an operating state for the cooling treatment. The operating state for the cooling treatment may be determined from a group that includes an active state and an inactive state. At block  818 , the exemplary method  800  may include comparing the thermal bias-value  500  to the lower control limit  510  when the operating state for the cooling treatment includes the active state, and deactivating the cooling treatment when the thermal bias-value  500  is less than the lower control limit  510 . Additionally, or in the alternative, the exemplary method  800  may include, at block  820 , comparing the thermal bias-value  500  to the upper control limit  508  when the operating state for the cooling treatment includes the inactive state, and deactivating the cooling treatment when the thermal bias-value  500  is less than the upper control limit  508 . 
     As shown in  FIG.  8 C , the exemplary method  800  may include, at block  822 , performing one or more engine protection control commands  418  prior to activating the cooling treatment. The one or more engine protection control commands  418  may include setting an engine start status to a disabled operating state. Additionally, or in the alternative, as shown in  FIG.  8 D , the exemplary method  800  may include, at block  824 , setting an engine start status to an enabled operating state, with the enabled operating state of the engine start status becoming effective when the cooling treatment has an inactive operating state, and the inactive operating state becoming effective at least in part by deactivating the cooling treatment. At block  826 , the exemplary method  800  may include initiating an engine start-up method for the turbomachine  106  when the thermal bias-value is between the lower control limit and the upper control limit. With the engine start-up method initiated, the motoring system  254  may be utilized to start up the turbomachine  106 . Upon having started the turbomachine  106 , the operating status of the turbomachine  106  may transition from a non-started state to a started state. 
     Thus, the present disclosure provides systems and methods that may mitigate rotor bow or bowed rotor conditions in a turbomachine. The presently disclosed systems and methods may include activating and deactivating cooling treatments one or more times, for example, as part of an on-ground cooling method, when an initial/current thermal bias is less than a lower control limit, thereby providing for more efficient cooling treatments and/or a more efficient reduction in thermal bias while still providing good protection from bowed rotor conditions. Additionally, or in the alternative, the presently disclosed systems and methods may include activating a cooling treatment when either an initial or current value, or a projection value, for thermal bias exceeds an upper control limit, thereby providing good control of thermal bias when the thermal bias is sufficiently high or projected to be sufficiently high. Additionally, or in the alternative, the presently disclosed systems and methods may include starting a turbomachine even when a current value for thermal bias is above a lower control limit, such as even when a thermal bias-value is increasing, for example, during a time period prior to a time when a projection value for the thermal bias exceeds the upper control limit. In this way, the presently disclosed systems and methods for thermal bias control may provide a series of time periods when the turbomachine may be started even when the turbomachine has a sufficient level of residual heat that may lead to a thermal bias value that exceeds the upper control limit. As such, the presently disclosed systems and methods may provide increased flexibility in scheduling start-up times for turbomachines, for example, to accommodate schedule changes to a flight itinerary for an aircraft powered by one or more turbomachines that include engine control systems and motoring systems configured to control thermal bias in accordance with the present disclosure. 
     Further aspects of the present disclosure are provided by the subject matter of the following clauses: 
     An engine control system comprising a memory device and a processor, wherein the memory device comprises computer-executable instructions, which when executed by the processor, cause the engine control system to perform a method of controlling thermal bias in a turbomachine during a non-started state, the method comprising: determining a thermal bias-value, wherein the thermal bias-value comprises a difference between an upward temperature-value and a downward temperature-value, the upward temperature-value corresponding to a first one or more temperature measurements of an upward portion of the turbomachine and the downward temperature-value corresponding to a second one or more temperature measurements of a downward portion of the turbomachine; and performing a cooling treatment based at least in part on the thermal bias-value, wherein the cooling treatment comprises at least one of: circulating air through at least a portion of the turbomachine, and rotating a shaft of the turbomachine with a motoring system. 
     The engine control system of any clause herein, wherein the thermal bias-value comprises at least one of a thermal bias-initial value and a thermal bias-projection value. 
     The engine control system of any clause herein, wherein performing the cooling treatment comprises: modulating the cooling treatment based at least in part on the thermal bias-value. 
     The engine control system of any clause herein, wherein performing the cooling treatment comprises: activating the cooling treatment when the thermal bias-value exceeds an upper control limit. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining the thermal bias-value at one or more times after activating the cooling treatment, wherein the cooling treatment decreases the thermal bias-value; and deactivating the cooling treatment when the thermal bias-value is less than a lower control limit at a time after activating the cooling treatment. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining the thermal bias-value at one or more times after deactivating the cooling treatment, wherein the thermal bias-value increases after deactivating the cooling treatment; and setting an engine start status to an enabled operating state, wherein the enabled operating state of the engine start status becomes effective when the cooling treatment has an inactive operating state, the inactive operating state becoming effective at least in part by the deactivating the cooling treatment. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: initiating an engine start-up method for the turbomachine when the thermal bias-value is between the lower control limit and the upper control limit. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: when the thermal bias-value exceeds the upper control limit at a time after deactivating the cooling treatment, setting an engine start status to a disabled operating state, and activating the cooling treatment. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: performing one or more engine protection control commands prior to activating the cooling treatment, wherein the one or more engine protection control commands comprises setting an engine start status to a disabled operating state. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining an operating state for the cooling treatment, wherein the operating state for cooling treatment is determined from a group that includes an active state and an inactive state; and when the operating state for cooling treatment comprises the active state, comparing the thermal bias-value to a lower control limit, and deactivating the cooling treatment when the thermal bias-value is less than the lower control limit; and when the operating state for the cooling treatment comprises the inactive state, comparing the thermal bias-value to the upper control limit, and deactivating the cooling treatment when the thermal bias-value is less than the upper control limit. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining an operating state of the turbomachine, wherein the operating state comprises a non-started state. 
     The engine control system of any clause herein, wherein the cooling treatment comprises: rotating the shaft of the turbomachine with a motoring system, wherein rotating the shaft of the turbomachine causes the air to circulate through the at least a portion of the turbomachine. 
     The engine control system of any clause herein, wherein rotating the shaft of the turbomachine with the motoring system comprises at least one of: rotating a high pressure shaft and rotating a low pressure shaft. 
     The engine control system of any clause herein, wherein rotating the shaft of the turbomachine with the motoring system comprises: rotating the high pressure shaft without rotating the low pressure shaft; or rotating the low pressure shaft without rotating the high pressure shaft. 
     The engine control system of any clause herein, wherein rotating the high pressure shaft comprises rotating at least one of: a high pressure compressor, a combustion chamber, and a high pressure turbine. 
     The engine control system of any clause herein, wherein rotating the low pressure shaft comprises rotating at least one of: a low pressure compressor and a low pressure turbine. 
     The engine control system of any clause herein, wherein rotating the shaft of the turbomachine with the motoring system comprises: supplying motoring air to a motoring system. 
     The engine control system of any clause herein, wherein the motoring air causing the motoring system to rotate a radial drive shaft coupled to the shaft of the turbomachine. 
     The engine control system of any clause herein, wherein supplying motoring air to a motoring system comprises supplying motoring air to a motor configured to rotate the high pressure shaft and/or the low pressure shaft of the turbomachine. 
     The engine control system of any clause herein, wherein supplying motoring air to a motoring system comprises supplying motoring air to one or more regions of the turbomachine, the motoring air circulating within and/or flowing through the one or more regions of the turbomachine. 
     The engine control system of any clause herein, wherein supplying motoring air to one or more regions of the turbomachine comprises supplying motoring air to the high pressure compressor. 
     The engine control system of any clause herein, wherein supplying motoring air to one or more regions of the turbomachine causes the shaft of the turbomachine to rotate. 
     The engine control system of any clause herein, wherein supplying motoring air comprises at least partially opening a motoring air supply valve, wherein at least partially opening the motoring air supply valve allows the motoring air to flow from at least one of: an auxiliary power unit, a ground start unit, and an additional turbomachine. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining the upward temperature-value based at least in part on a first one or more temperature measurements from one or more sensors; and determining the downward temperature-value based at least in part on a second one or more temperature measurements from the one or more sensors. 
     The engine control system of any clause herein, wherein determining the upward temperature-value comprises determining the first one or more temperature measurements with an upward temperature sensor located at an upward portion of a core engine of the turbomachine; and wherein determining the downward temperature-value comprises determining the second one or more temperature measurements with a downward temperature sensor located at a downward portion of the core engine of the turbomachine. 
     The engine control system of any clause herein, wherein determining the upward temperature-value comprises determining the first one or more temperature measurements with the shaft of the turbomachine oriented at (2π)-radians, +/−(π/3)-radians, on a circumferential axis; and wherein determining the downward temperature-value comprises determining the second one or more temperature measurements with the shaft of the turbomachine oriented at (π)-radians, +/−(π/3)-radians, on the circumferential axis. 
     The engine control system of any clause herein, wherein the instructions, when executed, cause the engine control system to further perform the method, the method comprising: determining at least one of the upward temperature-value and the downward temperature-value based at least in part on one or more temperature measurements obtained during or after an initial period of rotating the shaft of the turbomachine with the motoring system, the initial period commencing prior to activating the cooling treatment. 
     A turbomachine, comprising: core engine comprising a shaft, and a compressor stage and a turbine stage coupled to the shaft; one or more sensors respectively configured to determine temperature measurements from at least one of an upward portion of the turbomachine and a downward portion of the turbomachine; a motoring system coupled to the shaft, the motoring system configured to rotate the shaft; and an engine control system comprising a memory device and a processor, wherein the memory device comprises computer-executable instructions, which when executed by the processor, cause the engine control system to perform a method of controlling thermal bias in the turbomachine, the method comprising: determining a thermal bias-value, wherein the thermal bias-value comprises a difference between an upward temperature-value and a downward temperature-value, the upward temperature-value corresponding to a first one or more temperature measurements of the upward portion of the turbomachine and the downward temperature-value corresponding to a second one or more temperature measurements of the downward portion of the turbomachine; and performing a cooling treatment based at least in part on the thermal bias-value, wherein the cooling treatment comprises at least one of: circulating air through at least a portion of the turbomachine, and rotating a shaft of the turbomachine with the motoring system. 
     The turbomachine of any clause herein, wherein the turbomachine comprises an engine control system configured according to any clause herein. 
     A non-transitory computer-readable medium comprising computer-executable instructions, which, when executed by a processor operably coupled to an engine control system, cause the engine control system to: determine a thermal bias-value, wherein the thermal bias-value comprises a difference between an upward temperature-value and a downward temperature-value, the upward temperature-value corresponding to a first one or more temperature measurements of an upward portion of the turbomachine and the downward temperature-value corresponding to a second one or more temperature measurements of a downward portion of the turbomachine; and perform a cooling treatment based at least in part on the thermal bias-value, wherein the cooling treatment comprises at least one of: circulating air through at least a portion of the turbomachine, and rotating a shaft of the turbomachine with a motoring system. 
     The computer-readable medium of any clause herein, comprising computer-executable instructions, which, when executed by a processor operably coupled to an engine control system, cause the engine control system to perform a method of controlling thermal bias in accordance with the engine control system of any clause herein. 
     A method of controlling thermal bias in a turbomachine during a non-started state, the method comprising: determining a thermal bias-value, wherein the thermal bias-value comprises a difference between an upward temperature-value and a downward temperature-value, the upward temperature-value corresponding to a first one or more temperature measurements of an upward portion of the turbomachine and the downward temperature-value corresponding to a second one or more temperature measurements of a downward portion of the turbomachine; and performing a cooling treatment based at least in part on the thermal bias-value, wherein the cooling treatment comprises at least one of: circulating air through at least a portion of the turbomachine, and rotating a shaft of the turbomachine with a motoring system. 
     The method of any clause herein, wherein the method is performed using an engine control system and/or a turbomachine of any clause herein. 
     A method of controlling thermal bias in a turbomachine during a non-started state, wherein the method is performed using an engine control system and/or a turbomachine of any clause herein. 
     This written description uses exemplary embodiments to describe the presently disclosed subject matter, including the best mode, and also to enable any person skilled in the art to practice such subject matter, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the presently disclosed subject matter is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they include structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.