Abstract:
Various methods and systems are provided for the control of radiator cooling fans. In one example, among others, a method includes determining heat input to a cooling system and a ram air flow velocity through a radiator of the cooling system, selecting one or more radiator cooling fans, and adjusting operation of the fans in response to the selection. The selection of the fans can be based at least in part upon the heat input from the heat source and the ram air flow velocity. In another example, a system includes a plurality of cooling fans distributed across a cooling surface of a radiator of a cooling system and a radiator fan control system. The radiator fan control system can determine the heat input to the cooling system and a ram air flow velocity through the radiator, select one or more fans, and adjust operation of the selected fans.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority to, and the benefit of, co-pending U.S. provisional application entitled “CONTROL OF RADIATOR COOLING FANS” having Ser. No. 61/982,050, filed Apr. 21, 2014, which is hereby incorporated by reference in its entirety. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0002]    This invention was made with government support under agreement W56HZV-04-2-001 awarded by the U.S. Army. The Government has certain rights in the invention. 
     
    
     BACKGROUND 
       [0003]    The majority of ground transportation platforms rely heavily on internal combustion engines which use either gasoline or diesel fuel. Cooling systems for internal combustion engines remove waste heat to ensure a normal in-cylinder combustion process. To accomplish this task, cooling liquid is circulated by a water pump through the engine block and a radiator to reject heat to the local environment. A thermostat valve can be used to mechanically adjust the cooling liquid circulation based upon cooling liquid temperature. In addition, a radiator fan can provide air flow over the radiator, based upon the cooling liquid temperature, to aid in the heat rejection to the local environment through forced convection heat transfer. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0004]    Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views. 
           [0005]      FIG. 1  is a graphical representation of an example of a cooling system for an internal combustion engine in accordance with various embodiments of the present disclosure. 
           [0006]      FIG. 2  is a graphical representation illustrating sources of air flow through a radiator of the cooling system of  FIG. 1  in accordance with various embodiments of the present disclosure. 
           [0007]      FIG. 3  is a graphical representation of an example of a radiator fan control system for the cooling system of  FIG. 1  in accordance with various embodiments of the present disclosure. 
           [0008]      FIGS. 4A and 4B  are graphical representations of an example of a setup for testing the radiator fan control system of  FIG. 3  in accordance with various embodiments of the present disclosure. 
           [0009]      FIG. 5A  illustrates examples of fan configurations for six test sets in accordance with various embodiments of the present disclosure. 
           [0010]      FIG. 5B  is a table listing the operating conditions during the test sets of  FIG. 5A  in accordance with various embodiments of the present disclosure. 
           [0011]      FIG. 6  is a table listing the system model parameters used during the test sets of  FIG. 5A  in accordance with various embodiments of the present disclosure. 
           [0012]      FIGS. 7-9  show experimentally measured data for the test sets of  FIG. 5A  in accordance with various embodiments of the present disclosure. 
           [0013]      FIGS. 10A-10C  illustrate the estimation of the heat transfer efficiency, ε, and efficiency, η, using the experimentally measured data in accordance with various embodiments of the present disclosure. 
           [0014]      FIG. 11  is a table illustrating an example of a rule of thumb control scheme for radiator fans in accordance with various embodiments of the present disclosure. 
           [0015]      FIG. 12  is a plot of an example of a theoretical relationship between the fan power and heat rejection in accordance with various embodiments of the present disclosure. 
           [0016]      FIGS. 13A and 13B  illustrate an example of an optimization control scheme for radiator fans in accordance with various embodiments of the present disclosure. 
           [0017]      FIG. 14  is a table showing fan power consumptions for experimental and theoretical results in accordance with various embodiments of the present disclosure. 
           [0018]      FIG. 15  is a table illustrating an example of model validation test profiles for selected radiator fans in accordance with various embodiments of the present disclosure. 
           [0019]      FIG. 16A  includes an image and graphical representation of a radiator illustrating an example of air velocity measurement points in accordance with various embodiments of the present disclosure. 
           [0020]      FIG. 16B  is a table showing air speed recordings at the measurement points of  FIG. 16A  in accordance with various embodiments of the present disclosure. 
           [0021]      FIG. 16C  is a graphical representation illustrating an example of air mass flow rate profiles at various fan speeds in accordance with various embodiments of the present disclosure. 
           [0022]      FIG. 16D  is a plot illustrating an example of modeled and measured air mass flow rate versus fan speed in accordance with various embodiments of the present disclosure. 
           [0023]      FIGS. 17A and 17B  are plots illustrating an example of radiator coolant temperatures in accordance with various embodiments of the present disclosure. 
           [0024]      FIG. 17C  is a table illustrating radiator coolant temperatures and heat rejected at various fan speeds in accordance with various embodiments of the present disclosure. 
           [0025]      FIG. 17D  is a plot illustrating an example of modeled and measured heat rejected in accordance with various embodiments of the present disclosure. 
           [0026]      FIG. 18A  is a table illustrating cooling fan power consumption at various fan speeds in accordance with various embodiments of the present disclosure. 
           [0027]      FIG. 18B  is a plot illustrating an example of modeled and measured cooling fan power consumption in accordance with various embodiments of the present disclosure. 
           [0028]      FIG. 19  is flowchart illustrating an example of cooling control that can be implemented by the radiator fan control system of  FIG. 3  in accordance with various embodiments of the present disclosure. 
           [0029]      FIG. 20  is a schematic diagram of an example of a computing device used to implement the radiator fan control system of  FIG. 3  in accordance with various embodiments of the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0030]    Disclosed herein are various examples related to the control of radiator cooling fans. Reference will now be made in detail to the description of the embodiments as illustrated in the drawings, wherein like reference numbers indicate like parts throughout the several views. 
         [0031]    Internal combustion engines are utilized to provide power in a number of applications such as, but not limited to, stationary power generators, marine vehicles, and ground vehicles such as passenger, commercial, off-road and military vehicles, etc. A thermal management system is a subsystem which consumes accessory torque of the internal combustion engine. A cooling system of the thermal management system can include a radiator, a fluid pump, a thermostat valve, one or more radiator fan(s), hoses, a coolant (or cooling fluid) reservoir, a temperature sensor, and/or an engine water jacket. The cooling system is similar for both ground vehicles and stationary power generators. On average, in a gasoline engine 25% (38% for diesel engines) of the total energy generated from combustion is lost to the cooling system, 46% (27%) to exhaust gases and about 26% (35%) gets converted to useful power. Thus, the cooling system accommodates a significant amount of heat while maintaining the system temperature within a prescribed range to accommodate normal combustion while operating under ambient conditions. If the operational temperature of the engine is too hot, then fuel economy and tailpipe emissions can be degraded by abnormal in-cylinder combustion. The cooling system can also impact the engine warm up time, general thermal transients, and peak operating temperatures, which also affect tailpipe emissions and/or fuel efficiency. 
         [0032]    Forced heat transfer convection, which depends on the size and type of the cooling system, can be used in addition to natural (or free) radiator convection. Multiple radiator fans in, e.g., a matrix or an array can be controlled with different fan and speed combinations to cool a thermal loaded engine while reducing energy usage for subsequent efficiency and performance gains. Increased control of the fan motors is possible with electric actuators, which have the potential of making the decoupled fan matrix cooling system energy efficient. Since the cooling system consumes a portion of the engine&#39;s crankshaft power, using minimal input energy during operation can improve performance. Power consumption by the cooling system can be reduced or minimized using thermal system actuators. In addition, a highly controllable cooling system running on feedback from dynamically acquired sensor data using a control algorithm can help reduce tailpipe emissions. A mathematical model for the radiator fan(s) and the forced convection heat transfer process can be used in the control scheme. Based on test data and accompanying mathematical analysis, optimization of the control strategy can reduce the fan matrix power consumption for specified thermal loads. Analysis of test data has indicated that reductions in power consumption are possible up to a range of about 55% to about 67%. Other cooling system configurations may provide even greater reductions. 
         [0033]    Referring to  FIG. 1 , shown is an example of a cooling system  100  for an internal combustion engine with cooling fluid flow indicated by the arrows. Cooling fluid (or coolant) is supplied by a coolant pump  106  to an engine block  109  (arrow  103   a ), where a portion of the heat (q in ) generated by the engine is removed by the cooling fluid. The heated cooling fluid (arrow  103   b ) is directed to a smart bypass valve  112 , which controls cooling fluid flow to the radiator  115 . The smart bypass valve  112  can direct some or all of the heated cooling fluid from the engine block  109  to the radiator  115  (arrow  103   c ), where a portion of the heat (q out ) is removed from the cooling fluid and dissipated to the local environment. In the example of  FIG. 1 , a matrix (or array) of radiator fans  118 , which can be individually controlled, provides forced convection heat transfer to aid in heat removal from the cooling fluid. The cooled cooling fluid returns to the inlet of the coolant pump  106  (arrow  103   d ). In some cases, a portion of the heated cooling fluid can be diverted to the inlet of the coolant pump  106  (arrow  103   e ). 
         [0034]    An analytical model for the combustion process generated heat that is expelled through forced air convection by the radiator fans  118  can be calculated. In the analysis, seven assumptions may be imposed on the thermal system during the evaluation.
       No heat losses in the cooling system  100  occur other than forced convection through the radiator  115 .   The heat output from the radiator  115  equals the heat input from the engine block  109  during steady-state operation (i.e., q in =q out ).   Ram air effects may need to be considered due to vehicle motion and/or wind.   System temperatures and fluid flows are measured using available sensors.   Coolant flows entirely through the radiator  115  and not the bypass circuit based on the thermostat.   Coolant pump  106  operation is fixed so that system heat reflection is based on fan speeds.   Air temperature drop, ΔT (°K), across the radiator  115  is a known constant.       
 
         [0042]    The heat rejection rate from the engine combustion cylinders to the coolant (or cooling fluid) flowing through the water jackets of the engine block  109 , q c  (kW), may be stated as: 
         [0000]        q   c   =UA   p ( T   g   −T   c ),   EQN. (1)
 
         [0000]    where U (kW/m 2 ·°K) is the overall heat transfer coefficient, A p  (m 2 ) is the piston head surface area, T g  (°K) is the mean effective gas temperature, and T c  (°K) is the coolant (or cooling fluid) temperature. For a given engine application, the variable T g  can either be directly measured by an in-cylinder sensor or obtained from a table look-up based upon the air/fuel ratio, spark angle, and load per calibration. 
         [0043]    The heat supplied to the cooling system  100  can be expressed as: 
         [0000]        q   in   ={dot over (m)}   c   Cp   cool ( T   HI   −T   HO ),   EQN. (2)
 
         [0000]    where {dot over (m)} c  (kg/sec) is the coolant (or cooling fluid) mass flow rate due to the coolant pump  106 , Cp cool  (kJ/kg·°K) is the specific heat of the coolant (or cooling fluid) at constant pressure, and T HI  and T HO  (°K) are the engine block  109  (or heat exchanger) inlet and outlet temperatures, respectively. 
         [0044]    To minimize the power consumed by the radiator fan(s)  118 , the heat rejected by forced convection can be adjusted through fan motor speed control. The rate of heat rejection from the radiator  115 , q out  (kW), can be expressed as: 
         [0000]        q   out   =ε{dot over (m)}   air   Cp   air ( T   aout   −T   ain ),   EQN. (3)
 
         [0000]    where T aout  (°K) is the air temperature exiting the radiator, T ain  (°K) is the air temperature at the radiator inlet, and Cp air  (kJ/kg·°K) is the specific heat of the air. The radiator heat transfer efficiency, 0&lt;ε&lt;1, depends on the air mass flow rate, {dot over (m)} air . In this analysis, a quadratic relationship for the efficiency ε=a{dot over (m)} air   2 +b{dot over (m)} air +c was selected, where coefficients a, b and c are constants. An example of the determination of the values of a, b and c will be discussed in more detail below. 
         [0045]    The variable {dot over (m)} air  (kg/sec) denotes the air mass flow rate through the radiator  115  and may be calculated as: 
         [0000]        {dot over (m)}   air   =Q   air ρ air =υ air   A   r ρ air ,   EQN. (4)
 
         [0000]    where Q air ={dot over (m)} air /ρ air  (m 3 /sec) is the volume flow rate, A r  (m 2 ) is the radiator area, ρ air  (kg/m 3 ) is the air density, and ν air  (m/sec) is the air flow velocity through the radiator  115 . This air flow velocity is a combination of two different sources: the radiator fan(s)  118 ; and a ram air effect produced by movement of the radiator  115  through the air.  FIG. 2  shows an example of the combination of the air flow sources. The total air flow velocity through the radiator  115 , although complex, can be simplified as: 
         [0000]        K   r ν air   2   =K   I ν ram   2   +K   f ν f   2 ,   EQN. (5)
 
         [0000]    where ν f  and ν ram  (m/sec) are the air flow velocities associated with the radiator fans  118  and the ram air effect, respectively, and K r , K I , and K f  are the pressure coefficients at the radiator  115 , air inlet  203 , and radiator fans  118 , respectively, as indicated in  FIG. 2 . EQN. (5) may be rewritten to calculate the air velocity through the radiator  115  as: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     air 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               K 
                               I 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             v 
                             ram 
                             2 
                           
                         
                         + 
                         
                           
                             
                               K 
                               f 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             v 
                             f 
                             2 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0046]    With the air flow velocity ν air  as defined in EQN. 6, the volume flow rate, Q air , and mass flow rate, {dot over (m)} air , in EQN. (4) can now be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             m 
                             . 
                           
                           air 
                         
                         = 
                           
                          
                         
                           
                             ρ 
                             air 
                           
                            
                           
                             
                               
                                 
                                   
                                     K 
                                     I 
                                   
                                   
                                     K 
                                     r 
                                   
                                 
                                  
                                 
                                   v 
                                   ram 
                                   2 
                                 
                                  
                                 
                                   A 
                                   r 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   
                                     K 
                                     f 
                                   
                                   
                                     K 
                                     r 
                                   
                                 
                                  
                                 
                                   v 
                                   f 
                                   2 
                                 
                                  
                                 
                                   A 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 ρ 
                                 air 
                               
                                
                               
                                 
                                   
                                     
                                       
                                         K 
                                         I 
                                       
                                       
                                         K 
                                         r 
                                       
                                     
                                      
                                     
                                       Q 
                                       ram 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         K 
                                         f 
                                       
                                       
                                         K 
                                         r 
                                       
                                     
                                      
                                     
                                       Q 
                                       f 
                                       2 
                                     
                                   
                                 
                               
                             
                             = 
                             
                               
                                 
                                   
                                     
                                       K 
                                       I 
                                     
                                     
                                       K 
                                       r 
                                     
                                   
                                    
                                   
                                     
                                       m 
                                       . 
                                     
                                     ram 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       K 
                                       f 
                                     
                                     
                                       K 
                                       r 
                                     
                                   
                                    
                                   
                                     
                                       m 
                                       . 
                                     
                                     f 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where Q ram =ν ram A r  and Q f =ν f A r  are the volume flow rates caused by the ram air effect and radiator fans  118 , respectively. The variables {dot over (m)} ram =Q ram ρ air  (kg/sec) and {dot over (m)} f =Q f ρ air  (kg/sec) denote the ram air and fan air mass flow rates. To obtain the general function for system heat rejection, EQN. (3) may now be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     out 
                   
                   = 
                   
                     ɛ 
                      
                     
                       
                         
                           
                             
                               K 
                               I 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             
                               m 
                               . 
                             
                             ram 
                             2 
                           
                         
                         + 
                         
                           
                             
                               K 
                               f 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             
                               m 
                               . 
                             
                             f 
                             2 
                           
                         
                       
                     
                      
                     
                       
                         Cp 
                         air 
                       
                        
                       
                         ( 
                         
                           
                             T 
                             aout 
                           
                           - 
                           
                             T 
                             ain 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0047]    The relationship between the electrical power and fan matrix configuration (e.g., fan speed, fan number, and/or fan position) can be derived as an objective equation. The nonlinear function between the fan torque, τ (N·m), and the motor speed, N (RPM), may be expressed as: 
         [0000]      τ=kN 2 ,   EQN. (9)
 
         [0000]    where the factor k for fans  118  depends on the blade design and its characteristic curve. However, the factor k tends to be typically independent of the motor. As a result, the electrical power, P e  (kW), of a fan  118  with speed N may be determined by introducing the efficiency, η, which is the ratio between the mechanical output power and the electrical input power so that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       e 
                     
                     = 
                     
                       
                         
                           P 
                           m 
                         
                         η 
                       
                       = 
                       
                         
                           τω 
                           η 
                         
                         = 
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                             
                             
                               60 
                                
                               η 
                             
                           
                            
                           
                             N 
                             3 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where P m  (kW) is the mechanical output power. As an alternative, the electrical power can also be determined as P e =iV s , where i (A) is the current and V s  (V) is the supply voltage. 
         [0048]    Further justification for this relationship is based on the fan laws given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       1 
                     
                     
                       P 
                       2 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             N 
                             1 
                           
                           
                             N 
                             2 
                           
                         
                         ) 
                       
                       3 
                     
                     · 
                     
                       
                         ( 
                         
                           
                             d 
                             1 
                           
                           
                             d 
                             2 
                           
                         
                         ) 
                       
                       5 
                     
                     · 
                     
                       
                         ( 
                         
                           
                             ρ 
                             1 
                           
                           
                             ρ 
                             2 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    In other words, the mechanical power can be directly related to the cube of the speed when the diameter and density are constant. 
         [0049]    The efficiency, η, of the mechanical output to electrical input power may be represented as a polynomial function of the motor speed, N, assuming an uniform load. For example, a quadratic expression that can be considered may be expressed as: 
         [0000]      η= dN   2   +eN+f ,   EQN. (12)
 
         [0000]    where coefficients d, e and f are constants. An example of the determination of the values of d, e and f will be discussed in more detail below. Using this relationship, EQN. (10) may now be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     e 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         k 
                       
                       
                         60 
                          
                         
                           ( 
                           
                             
                               dN 
                               2 
                             
                             + 
                             eN 
                             + 
                             f 
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         N 
                         3 
                       
                       . 
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0050]    The volume flow rate, Q f  (m 3 /sec), through a single fan  118  can also determine the fan motor speed. The speed for a single axial fan  118  to generate a target volume flow rate may be stated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     N 
                     = 
                     
                       
                         
                           60 
                            
                           
                             Q 
                             f 
                           
                         
                         
                           4 
                            
                           
                             π 
                             2 
                           
                            
                           
                             r 
                             m 
                             3 
                           
                            
                           
                             Φ 
                             m 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             + 
                             
                               v 
                               2 
                             
                           
                           
                             1 
                             - 
                             
                               v 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where r m  (m) is the mean radius, ν is the hub ratio, and Φ m  is the flow rate coefficient. 
         [0051]    The fan air mass flow rate, {dot over (m)} f  (kg/sec), can be determined by rewriting this expression as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       m 
                       . 
                     
                     f 
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           π 
                           2 
                         
                          
                         
                           ρ 
                           air 
                         
                          
                         
                           r 
                           m 
                           3 
                         
                          
                         
                           Φ 
                           m 
                         
                          
                         Nn 
                       
                       60 
                     
                      
                     
                       ( 
                       
                         
                           1 
                           - 
                           
                             v 
                             2 
                           
                         
                         
                           1 
                           + 
                           
                             v 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
         [0052]    The integer variable n was introduced to denote the number of fans  118  operating in parallel in the radiator fan array (or matrix) configuration. The mean radius, r m , is dependent on the fan tip and hub radius, r t  and r h  (m), such that: 
         [0000]        r   m =√{square root over ( 1/2 ( r   t   2   +r   h   2 ))}.   EQN. (16)
 
         [0000]    Similarly, the hub ratio, ν, may be calculated as: 
         [0000]      ν= r   h   /r   t .   EQN. (17)
 
         [0000]    Finally, the flow rate coefficient, Φ m , may be obtained for axial flow fans  118  using specific speed and pitch cord ratios. 
         [0053]    Based on the analysis above, substitution of EQN. (15) into EQN. (8) allows the general function for system heat rejection to be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     out 
                   
                   = 
                   
                     ɛ 
                      
                     
                       
                         
                           
                             
                               K 
                               I 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             
                               m 
                               . 
                             
                             ram 
                             2 
                           
                         
                         + 
                         
                           
                             
                               K 
                               f 
                             
                             
                               K 
                               r 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     4 
                                      
                                     
                                       π 
                                       2 
                                     
                                      
                                     
                                       ρ 
                                       air 
                                     
                                      
                                     
                                       r 
                                       m 
                                       3 
                                     
                                      
                                     
                                       Φ 
                                       m 
                                     
                                      
                                     Nn 
                                   
                                   60 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       1 
                                       - 
                                       
                                         v 
                                         2 
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         v 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                      
                     
                       
                         Cp 
                         air 
                       
                        
                       
                         ( 
                         
                           
                             T 
                             aout 
                           
                           - 
                           
                             T 
                             ain 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     18 
                     ) 
                   
                 
               
             
           
         
       
     
         [0054]    When no ram air effects are considered, the ram air mass flow rate {dot over (m)} ram =0, which corresponds to a stationery engine scenario. The application for this case may be a parked vehicle without blowing wind (e.g., silent sentry mode) or an engine-based generator. Thus, EQN. (18) can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     out 
                   
                   = 
                   
                     ɛ 
                      
                     
                       
                         
                             
                         
                          
                         
                           4 
                            
                           
                             π 
                             2 
                           
                            
                           
                             ρ 
                             air 
                           
                            
                           
                             r 
                             m 
                             3 
                           
                            
                           
                             Φ 
                             m 
                           
                            
                           Nn 
                         
                       
                       60 
                     
                      
                     
                       ( 
                       
                         
                           1 
                           - 
                           
                             v 
                             2 
                           
                         
                         
                           1 
                           + 
                           
                             v 
                             2 
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         
                           K 
                           f 
                         
                         
                           K 
                           r 
                         
                       
                     
                      
                     
                       
                         Cp 
                         air 
                       
                        
                       
                         ( 
                         
                           
                             T 
                             aout 
                           
                           - 
                           
                             T 
                             ain 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     19 
                     ) 
                   
                 
               
             
           
         
       
     
         [0055]    Now, the relationship between the fan power consumption and the heat rejected for various fan configurations and fan(s) speeds can be derived. An optimization problem can be formulated and applied to solve this smart cooling system challenge for power minimization which meets the heat transfer demands of the cooling system  100 . Referring to  FIG. 3 , shown is a block diagram of an example of a radiator fan control system  300 . The cooling system  100  includes the coolant pump  106  for supplying cooling fluid to the engine block  109  being cooled, the bypass valve  112 , the radiator  115 , a radiator fan array  303  including one or more fans  118  ( FIGS. 1 and 2 ), and a motor driver  306  that provides power to the fans  118  of the radiator fan array  303 . In one embodiment, the radiator fan array  303  can include a matrix of six fans  118  positioned about the radiator surface as illustrated in  FIG. 1 . The radiator fan control system  300  is configured to control fan operation of the radiator fan array  303  via the motor driver  306 . Operational conditions of the cooling system  100  can be monitored and utilized by the radiator fan control system  300  to adjust the operation of the radiator fan array  303 . The radiator fan control system  300  may be implemented using a computer system or other appropriate processing circuitry. 
         [0056]    The governing system dynamics stated in EQNS. (13) and (19) are well suited for mixed integer nonlinear programming (MINP), which may be expressed as: 
         [0000]      min f(N,n)   EQN. (20a)
 
         [0000]      subject to:  h ( N,n )=0, and   EQN. (20b)
 
         [0000]        g ( N,n )≦0,   EQN. (20c)
 
         [0000]    where N and integer n are programming variables. The objective function, f (.,.), may be stated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       
                         N 
                         , 
                         n 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       e 
                     
                     = 
                     
                       
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           k 
                         
                         
                           60 
                            
                           
                             ( 
                             
                               
                                 dN 
                                 2 
                               
                               + 
                               eN 
                               + 
                               f 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                           nN 
                           3 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     21 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    For EQNS. (20b) and (20c), the equality and inequality constraint functions h(N, n) and g(N,n) can be described as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           h 
                            
                           
                             ( 
                             
                               N 
                               , 
                               n 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             q 
                             out 
                           
                           - 
                           
                             q 
                             
                               i 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           ɛ 
                            
                           
                               
                           
                            
                           
                             
                               4 
                                
                               
                                 π 
                                 2 
                               
                                
                               
                                 ρ 
                                 air 
                               
                                
                               
                                 r 
                                 m 
                                 3 
                               
                                
                               
                                 Φ 
                                 m 
                               
                                
                               Nn 
                             
                             60 
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 - 
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   v 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 
                                   K 
                                   f 
                                 
                                 
                                   K 
                                   r 
                                 
                               
                             
                              
                             
                               
                                 Cp 
                                 air 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     aout 
                                   
                                   - 
                                   
                                     T 
                                     ain 
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             q 
                             
                               i 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     22 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     g 
                      
                     
                       ( 
                       
                         N 
                         , 
                         n 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 N 
                                 low 
                               
                               ≤ 
                               N 
                               ≤ 
                               
                                 N 
                                 high 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               ≤ 
                               n 
                               ≤ 
                               
                                 n 
                                 
                                   ma 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     23 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where N low  and N high  (RPM) denote the lower and upper fan speeds and n max  (RPM) denotes the maximum number of operating fans  118 . 
         [0057]    The notation in EQNS. (21) and (22) may be simplified by defining two constants g and h as: 
         [0000]    
       
         
           
             g 
             = 
             
               
                 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     k 
                   
                   60 
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 h 
               
               = 
               
                 ɛ 
                  
                 
                     
                 
                  
                 
                   
                     4 
                      
                     
                       π 
                       2 
                     
                      
                     
                       ρ 
                       air 
                     
                      
                     
                       r 
                       m 
                       3 
                     
                      
                     
                       Φ 
                       m 
                     
                   
                   60 
                 
                  
                 
                   ( 
                   
                     
                       1 
                       - 
                       
                         v 
                         2 
                       
                     
                     
                       1 
                       + 
                       
                         v 
                         2 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     
                       K 
                       f 
                     
                     
                       K 
                       r 
                     
                   
                 
                  
                 
                   
                     Cp 
                     air 
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    In addition, let ΔT=T aout −T ain  correspond to the air temperature drop across the radiator  115 . 
         [0058]    The objective function and equality constraint of EQNS. (21) and (22) may now be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           N 
                           , 
                           n 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         g 
                         
                           ( 
                           
                             
                               dN 
                               2 
                             
                             + 
                             eN 
                             + 
                             f 
                           
                           ) 
                         
                       
                        
                       
                         nN 
                         3 
                       
                     
                   
                   , 
                   and 
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     24 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         N 
                         , 
                         n 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ɛ 
                        
                       
                           
                       
                        
                       hNn 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       T 
                     
                     - 
                     
                       
                         q 
                         
                           i 
                            
                           
                               
                           
                            
                           n 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     25 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The value of ΔT (approximately 10° C.→20° C.) does not significantly affect the optimization results if assumed to be a user-specified constant. In this analysis, all the fans were assumed to have uniform rotational speeds. 
         [0059]    To solve this MINP problem, a related nonlinear constrained programming (NCP) problem can be introduced to determine successive solutions, (N s ,n s ). With an Matlab optimization package, the successive solutions may be first calculated. The function “fmincon” provides various types of algorithms that can be utilized; the interior-point approach to constrained minimization was selected to solve a sequence of nonlinear programming minimization problems. 
         [0060]    The gradient of the objective function, f (.,.), in EQN. (24) may be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∇ 
                     f 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             ∂ 
                             f 
                           
                           
                             ∂ 
                             N 
                           
                         
                         , 
                         
                           
                             ∂ 
                             f 
                           
                           
                             ∂ 
                             n 
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       dN 
                                       2 
                                     
                                     + 
                                     
                                       2 
                                        
                                       eN 
                                     
                                     + 
                                     
                                       3 
                                        
                                       f 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   gnN 
                                   2 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       dN 
                                       2 
                                     
                                     + 
                                     eN 
                                     + 
                                     f 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             , 
                             
                               
                                 gN 
                                 3 
                               
                               
                                 ( 
                                 
                                   
                                     dN 
                                     2 
                                   
                                   + 
                                   eN 
                                   + 
                                   f 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   EQN 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     26 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    which establishes the search direction for the optimization algorithm. To obtain an integer value for the number of fans, n, per given heat load, q out , the radiator fan control system  300  calculates the two nearest points around the successive solution, (N s , n s ). The point that does not satisfy the stated constraints is dropped. If both of the points satisfy the constraints, then the objective function is calculated based on each point. The point which leads to the smallest objective function value is selected as the best solution. 
         [0061]    Referring next to  FIG. 4A , shown is an example of a setup for testing the radiator fan control system  300  of  FIG. 3 . An experimental setup was created to provide a safe and repeatable approach for study of internal combustion engine cooling systems. The test setup of  FIG. 4A  includes a coolant pump  106  for supplying cooling fluid from the radiator  115  to the engine block  109 . A steam driven heat exchanger  403  was used to mimic the internal combustion engine in-cylinder thermal source while creating a test environment for exact heating control. The cooling fluid was heated by the heat exchanger  403  and supplied to the cooling jacket of a 6.8 L International Truck V8 diesel engine to simulate heat dissipation from the engine block  109 . The radiator fan array  303  included a 3×2 matrix of fans  118 , as illustrated in  FIG. 1 , which were controlled via the motor driver  306 . The test setup also featured a wind tunnel and a set of thermal actuators (e.g., variable speed pump, smart bypass valve, and radiator with electric fans) and assorted sensors for monitoring system operation. 
         [0062]    In the experimental setup, a low pressure steam source simulated the heat generated by the internal combustion engine. The steam transferred heat through a multi-pass heat exchanger  403  to the coolant (or cooling fluid), which was circulated through the conventional automotive cooling system. Safety equipment such as a pressure regulator, pressure gauge, and safety valve were inserted into the steam system to avoid injury. During testing, the flow rate of the low pressure steam was constant. 
         [0063]    The coolant flow circuit included the low pressure heat exchanger  403 , engine block  109 , bypass valve  112 , variable speed coolant pump  106 , and radiator  115  with fan array  303 . The cooling fluid received heat from the steam in the multi-pass heat exchanger  403  and then was pumped into the cooling jacket of the engine block  109 . From the outlet of the engine block  109 , the cooling fluid flowed into the directional control bypass valve  112  which, based upon a defined set point temperature, can be controlled to direct the flow to the radiator  115  or divert the cooling fluid back into the engine heat exchanger  403 . The variable speed coolant pump  106  was driven by an AC motor controlled by a computer algorithm. For this investigation, the bypass valve  112  directed all cooling fluid through the radiator  115 . 
         [0064]    A DC motor thermostat may be used to direct fluid through the radiator  115 . The cooling fluid (or coolant) entered the radiator  115  from a top inlet port and flowed through a mesh of tubes with attached fins, expelling its heat energy to the ambient surroundings. The cooling fluid exited through an outlet at the bottom portion of the radiator  115 , where it was supplied to the centrifugal coolant pump  106 . The coolant pump  106  circulated the cooling fluid back to the steam heat exchanger  403 , thus completing the cycle. Insulated galvanized pipes and pipe connections were used to transport the cooling fluid between the various system components. 
         [0065]    A wind tunnel was constructed to measure the air flow and pressure drop across the radiator  115  in the experimental setup. The wind tunnel had a rectangular matrix arrangement with six fans  118  arranged in three rows and two columns (3×2). Each of the fans  118  were connected to a controller area network (CAN) based DC motor controller in the motor driver  306 . The controllers were connected via a CAN bus card plugged in the computer PCI port. The fan motors are EMP Fil-11 electric fans with 24V, 600 W brushless DC motors. 
         [0066]    The experimental setup of  FIG. 4A  contained electrical, electronic, and computer subsystems to acquire, process, record, and display data generated during the test runs.  FIG. 4B  shows the data flow path for the subsystems. The sensors used in the experiments included an air speed sensor, a turbine type flow meter, a linear variable differential transducer, five thermocouples, and an ammeter. The continuous analog data from the sensors were supplied to a dSPACE data acquisition system. A computer interface included the software programs MATLAB/Simulink with the Vehicle Network Tool (VNT) box, and dSPACE. The signal inputs were received through the dSPACE hardware board, while the output control signals were transmitted via the dSPACE digital-to-analog converter (DAC) and the CAN bus controller. 
         [0067]    A series of six test sets (#I -#VI), under various fan configurations and operating scenarios, were investigated using the experimental setup of  FIGS. 4A and 4B  to evaluate heat rejection of the cooling system and radiator fan power consumption.  FIG. 5A  graphically illustrates the six test configurations using the 3×2 matrix of fans  118  of  FIG. 1 . The six fans are identified as  1  through  6 , with the shaded portions indicating which fan(s)  118  are operating during that test. The locations of the inlet and outlet connections of the radiator  115  are indicated in test configuration #I. The shaded arrows indicate the trends in coolant temperature as the cooling fluid passes through the radiator  115 . As can be understood, the hottest area of the radiator  115  is right behind fan No.  2 , which corresponds to the radiator inlet location. Similarly, the region behind fans No.  2 ,  4 , and  6  is hotter than that behind fans No.  1 ,  3 , and  5 . The connection locations and temperature trends are consistent with the other test configurations #II through #VI. 
         [0068]    The six fan combinations were designed to evaluate the fan effects on heat dissipation. The table of  FIG. 5B  indicates the operating conditions during the each test set (#I-#VI) and the respective operational speeds. The fan(s)  118  were connected and controlled via the CAN bus based on the control strategy. The system model parameters are listed in the table of  FIG. 6  for the optimization problem. 
         [0069]    The voltage, V s , and the current, i, of the fan motors were used to calculate the power consumption, P e =iV s . The steady state DC power consumption values were measured and calculated for the given fan number (No.  1 - 6 ) and operating fan(s) speeds. This fan power consumption data was used to determine the energy usage for the cooling system  100  for a given engine heat load. Referring to  FIG. 7 , shown is a table summarizing the experimentally measured data including the fan speed, N, air flow rate, {dot over (m)} air , and the calculated heat rejected, q out , based on the relationship: 
         [0000]        q   out   ={dot over (m)}   c   Cp   cool ( T   RI   −T   RO ) 
         [0000]    where T RI  and T RO  are the radiator coolant inlet and outlet temperatures (as shown in  FIG. 4A ), respectively, which can be measured by thermocouple sensors. The variable {dot over (m)} c  is the coolant mass flow rate listed in the table of  FIG. 6 . The heat transfer efficiency, ε, is calculated based on EQN. (3) with: 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   q 
                   out 
                 
                 
                   
                     
                       m 
                       . 
                     
                     air 
                   
                    
                   
                     
                       Cp 
                       air 
                     
                      
                     
                       ( 
                       
                         
                           T 
                           aout 
                         
                         - 
                         
                           T 
                           ain 
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the air specific heat, Cp air , and ΔT=T aout −T ain  are summarized in the table of  FIG. 6 . 
         [0070]    Referring to  FIG. 8 , shown is a plot of the electrical power, P e , consumed by the fan motors for the six test sets (#I-#VI) at the various motor speeds, N. The graph of  FIG. 8  illustrates the general trend that the increased power consumption is dependent upon the number of fans  118  and their operating speeds. However, there were exceptions observed at 3,000 RPM for test set #I with the single fan configuration and for test set #V with the five fan configuration. In these two cases, the power consumption increased at 3,000 RPM. The power consumption was observed to be almost equal for speeds up to 3,000 RPM when either five (test set #V) or six (test set #VI) fan motors were operated. However, all the fan motor combinations have a comparatively smaller variation in power consumption for motor speeds up to 2,000 RPM. Mechanical fans consume 24 kW to 50 kW of engine power to operate while radiator electric fans utilize approximately 0.7 kW, which offers a significant energy saving. 
         [0071]    Referring to  FIG. 9 , shown is a plot of the heat rejected by the cooling system  100  at various fan powers for the six test sets (#I-#VI) over a 0&lt;P e &lt;4 kW operating range. The plots represent four parameters: heat rejected, q out ; fan speed, N; fan number, n; and the fan electric power consumption, P e . The rejected heat can be considered an objective system input, with both the fan number and the fan(s) speeds being selected to achieve the heat rejection needed for the cooling system  100 . The plot of  FIG. 9  can offer guidance in selecting the best energy efficient fan number and speed for a given heat load. The variation in the data of test set #I at a heat rejection of 20 kW and fan power at 0.28 may be attributed to diversity of the different fan models. 
         [0072]    The data obtained from the experiments indicated that reduced energy consumption can be achieved for heat rejection rates above q out &gt;56 kW using all six fan operating configuration. However, when heat rejection need below 56 kW, other fan configurations were observed to be more energy efficient. For example, a heat rejection rate of 48 kW shows test set #V with a fan matrix power consumption of P e =0.30 kW while test set #VI was 0.33 kW. For heat rejection in the zone 0&lt;q out &lt;16 kW, test set #I operated in the range of 1,000≦N≦2,000 RPM. Next, the zone of 16&lt;q out &lt;24 kW and 24&lt;q out &lt;35 kW corresponds to test sets #II and #III at N=1,000 RPM, respectively. If 35&lt;q out &lt;43 kW, then the test set #IV should operate at 1,000≦N≦2,000 RPM. Finally, 43&lt;q out &lt;56 kW and q out &gt;56 kW results in test set #V and test set #VI at N=2,000 RPM, respectively. 
         [0073]    The determination of the coefficients (a, b and c) for the heat transfer efficiency, ε, and the coefficients (d, e and f) for the efficiency, η, can also be based upon the measured data. The table of  FIG. 7  summarizes the experimentally measured data with the heat was calculated based on q out ={dot over (m)} c Cp cool (T RI −T RO ) and the heat transfer efficiency (ε) calculated based on EQN. (3) as previously described. Referring to  FIG. 10A , shown is a plot (curve  1003 ) of the heat transfer efficiency, ε, for different mass air flow rates, {dot over (m)} air . Using a second order curve fitting tool such as that provided by MATLAB, the coefficients for the equation ε=a{dot over (m)} air   2 +b{dot over (m)} air +c can be determined. Curve  1106  is a plot of the second order curve fit based upon the measured data. 
         [0074]    The efficiency, η, was calculated based on EQN. (10) with: 
         [0000]    
       
         
           
             
               η 
               = 
               
                 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     k 
                   
                   
                     60 
                      
                     
                       P 
                       e 
                     
                   
                 
                  
                 
                   N 
                   3 
                 
               
             
             , 
           
         
       
     
         [0000]    where the factor k is listed in the table of  FIG. 6 .  FIG. 10B  shows a plot (curve  1009 ) of the efficiency, η, for different fan speeds, N. The recorded fan speeds, N, and power consumptions, P e , are summarized in the table in  FIG. 10C . The fan power consumption, P e , was calculated as the average value for test set #VI. Using the second order curve fitting tool, the coefficients for the equation η=dN 2 +eN+f can also be determined. Curve  1012  shows the fitted trend curve of the fan speed verses efficiency. 
         [0075]    The corresponding numerical optimization results were compared with the test results to evaluate an operating strategy for the radiator fan control system  300  of  FIG. 3 . According to the experimental results show on  FIG. 9 , a general rule of thumb can be formulated. For heat rejection between 0-56 kW, begin with a single fan (test set #I) operating over a range from 1,000 RPM to 2,000 RPM. If more heat rejection is needed, then bring another fan online (test sets #II-#V) until all the six fans (test set #VI) operating at =1,000 RPM (up to perhaps 2,000 RPM, if needed). Above a heat rejection threshold of 56 kW, all six fans can be used while increasing the fan speed as needed to 5,000 RPM. Referring to  FIG. 11 , shown is a table illustrating the rule of thumb control combination for experimental heat rejection and power configuration with the heat rejection below 56 kW. 
         [0076]    The numerical optimization strategy offers an alternative approach to the rule of thumb.  FIG. 12  shows the theoretical relationship between the fan power and heat rejection for various configurations and speeds based on the mathematical model described with respect to EQNS. (1)-(19). Comparison of the experimental curves in  FIG. 9  and the theoretical curves in  FIG. 12  suggests that the trends demonstrated by the mathematical model are valid. 
         [0077]    Using an interior-point approach to solve the nonlinear optimization problem, the optimization results are calculated for the selection of fan number, n, and fan(s) speeds, N, to achieve specific heat rejection.  FIG. 13A  displays the relationship between the number of fans and the heat rejected. Representative results are listed in the table of  FIG. 13B . According to  FIGS. 13A and 13B , an optimization control strategy can be concluded. First, turn on a single fan with its speed at N=1,000 RPM, and increase the speed to achieve the increasing heat rejection requirement. Second, turn on the other fans one by one and adjust the corresponding fan(s) speeds. It can be demonstrated that this method is similar to the rule of thumb based on the experimental results in the table of  FIG. 11 . 
         [0078]    To illustrate the concept, two specific case studies were considered. In case 1, q out =30 kW was selected as the heat rejection. The power fan consumption was P e =0.6 kW for the experimental results and P e =0.51 kW for the theoretical results when a single fan was operated. For the optimization control method, the minimum consumption was reduced to P e =0.2 kW for experimental result (P e =0.16 kW for the theoretical result) when n=3 fans were operated. In case 1, the energy saving was up to 67%. 
         [0079]    In case 2, the heat rejection selected was q out =60 kW. A fan number of n=3 was selected for the comparison with the optimization result. It indicates that the power consumption was P e =0.9 kW for the experimental results (P e =0.81 kW for the theoretical results) when n=3 fans were operating. On the other hand, based on the optimization result, the minimum consumption was approximately P e =0.4 kW for the experimental result (P e =0.38 kW for the theoretical result) when n=6 fans were engaged. The total energy saving corresponds to 55%. For these studies (cases 1 and 2), the corresponding fan power consumptions for both experimental and theoretical results are shown in the table of  FIG. 14 . 
         [0080]    A series of experimental tests were performed on the test setup of  FIG. 4A  to validate the models. Specific test results that were considered include the radiator air mass flow rate ({dot over (m)} air ), the heat rejected from the radiator (q out ), and the fan matrix power consumption (P e ), for different fan speeds, N, and number of fans, n. To demonstrate the versatility of the dynamic models, two fans (No.  1  and  2 ) were selected for operation. The test profile and the corresponding results are listed in the table in  FIG. 15 . As demonstrated through the testing, the modeling can adequately estimate the system behavior. 
         [0081]    To begin, the radiator air mass flow rate model was validated. The relationship between the fan configuration (e.g., fan speed, N, and number of fans, n) and the air mass flow rate ({dot over (m)} air ) through the radiator can be expressed as 
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         [0000]    when there is this no ram air effect, {dot over (m)} ram =0, from EQNS. (7) and (15). Alternatively, the radiator air mass flow rate, {dot over (m)} air , can be experimentally determined by measuring the air speed exiting the radiator, ν* air *, and introducing some system parameters into the corresponding basic engineering calculations. 
         [0082]      FIG. 16A  includes an image and graphical representation illustrating the points on the frontal area of the radiator  115  ( FIG. 1 ) where the air velocity was measured. Thirty points were selected on the radiator  115  to establish a grid as shown in  FIG. 16A , and data was collected for different fan rotational speeds, N. The average air flow speed,  ν * air , was calculated based on the collected wind speed data to obtain the total air mass flow rate, {dot over (m)}* air , through the radiator area. For example, the recorded air flow speed, ν* air   i , at each point (i=1, 2, . . . , 30) with fans No.  1  and  2  operating at a speed of N=2000 r/min is listed in the table of  FIG. 16B . In these tests, the average air speed across the radiator  115  was  ν * air =1.33 m/s, and the testing mass flow rate, {dot over (m)}* air , can be calculated using EQN. (4) as {dot over (m)}* air =  ν * air A r ρ air =1.61 kg/s. 
         [0083]    Referring to  FIG. 16C , shown are examples of air mass flow rate profiles, {dot over (m)}* air , for increasing fan speed, 1000&lt;N&lt;5000 r/min, with fans No.  1  and  2  operating. The three-dimensional surfaces  1603  correspond to the measured air mass flow rate profiles, {dot over (m)}* air , for increasing fan speeds. The calculated air mass flow rates using the mathematical model (curve  1606 ) and the experimental test data (points  1609 ) were plotted against the fan speed, N, of fans No.  1  and  2  in  FIG. 16D , and summarized in the table of  FIG. 15 . The average absolute error between the mathematical model and the experimental results was 6.2%. 
         [0084]    Also, the model for the heat rejected from the radiator was validated. The radiator heat rejection, q out , can be described by EQN. (19). Based on the relationship q out ={dot over (m)} c Cp cool (T RI −T RO ), the coolant&#39;s thermal response can be used as an alternative representation of the heat rejection, q out , from the radiator  115  provided that the system has reached equilibrium. Each test was operated for a 500 second time period and the temperatures at steady state were selected to calculate the heat output. The test results for all cases are presented in the table of  FIG. 17A , which lists the coolant temperature and heat rejected calculation results for various speeds of fans No.  1  and  2 . 
         [0085]    Referring to  FIGS. 17B and 17C , shown are examples of the coolant temperatures at the radiator inlet, T RI , and outlet, T RO , when the twin radiator fans are operated at 2000 r/min and 5000 r/min, respectively. For example, when the fans operate at 2000 r/min, the temperature at the radiator inlet was T RI =79.9° C. while the temperature at the outlet was T RO =74.7° C. According to the relationship above, the heat rejection from the radiator  115  is q out =25.0 kW. In comparison, the numerical result from the mathematical model of EQN. (19) was calculated to be q out =27.8 kW. The calculated heat rejected from the radiator, q out , using the mathematical model (curve  1703 ) and the experimental test data (points  1706 ) were plotted against the fan speed, N, of fans No.  1  and  2  in  FIG. 17D , and summarized in the table of  FIG. 15 . The average absolute error between the mathematical model and the experimental results was 8.9%. 
         [0086]    Additionally, the radiator fan power consumption model was validated. To investigate the accuracy of EQN. (13), the operating current of the radiator fans was measured during the test matrix. The supply voltage was fixed at V s =30 V. The recorded current, i, and the calculated power consumption, P e =iV s , are listed in the table of  FIG. 18B . The calculated fan power consumption, P e , using the mathematical model (curve  1803 ) and the experimental test data (points  1806 ) were plotted against the fan speed, N, of fans No.  1  and  2  in  FIG. 18B , and summarized in the table of  FIG. 15 . The average absolute error between the mathematical model and the experimental results was 11.9%. 
         [0087]    An optimization control method for automotive (or other internal combustion) thermal-management systems can have a positive impact on the cooling system. A multiple electric fan radiator cooling configuration using an experimental bench and offline mathematic analysis was studied and analyzed. According to the experiment and simulation results, rule of thumb and optimization control strategies were introduced which can reduce the fan matrix power consumption for the specified cooling load. In the examined cases, the power consumption was reduced by approximately 67%. 
         [0088]    The radiator fan control system  300  ( FIG. 3 ) may be utilized in applications having a heat source such as, but not limited to, a combustion engine, an electric motor, a battery or any combination thereof. For example, the cooling system  100  ( FIG. 3 ) may be used for heat removal with hybrid electric vehicles (HEV) and/or electric vehicles (EV) that may feature a combustion engine and/or battery packs with electric motor(s) for propulsion. In a HEV or EV configuration, heat generated by the battery pack and/or electric motors can be removed by the cooling system  100  to ensure system reliability and longevity. In a fluid based cooling system design that features an array of radiator fans, the radiator fan control system  300  can be extended based on the heat rejection needs of these components. 
         [0089]    Cooling fluids can include, but are not limited to, liquids such as water or other coolant mixtures (e.g., ethyl-glycol and water). In some implementations, cooling liquids can include gases (e.g., air, nitrogen, or other gases or gas mixtures) that can be circulated by the coolant pump  106  or a coolant fan. A cooling chamber can enclose at least a portion of the heat source to allow for heat transfer to the gaseous cooling fluid. Fluid temperatures can be measured using installed sensors. 
         [0090]    For applications such as “green” vehicles that do not rely solely on an internal combustion engine, the battery pack and/or electric motors can be cooled using liquid and/or gas fluids. If the fluid-based cooling system  100  is utilized, then the electric radiator fans can be controlled in a similar manner as previously described. EQN. (1) can be replaced with an equation estimating the heat generated by the batteries and/or electric motors. For example, the heat generated in a lithium-ion battery pack includes the resistive dissipation, reversible entropic heat, and chemical reaction. In many thermal management cases, the heat generated in a battery can be estimated by only the resistive dissipation, which is also known as Joule losses from the battery, and which can be expressed as Q=I 2 R, where I is the input current and R is the battery internal resistance. The heat generated in an electric motor can also be determined by calculating the resistive dissipation and measuring the current. The magnitudes can be measured and/or estimated on-board the vehicle using installed sensors. 
         [0091]    Referring to  FIG. 19 , shown is a flowchart illustrating an example of fan control that can be implemented by the radiator fan control system of  FIG. 3 . Beginning with  1903 , system parameters are initialized for the working mode. For example, coolant mass flow rate, smart valve position, and other variables listed in the table of  FIG. 6 . The system parameters may be based upon manufacturer specifications and/or measurements taken during testing of the internal combustion engine and the cooling system  100 . 
         [0092]    In  1906 , sensor readings are taken for the operating system. The sensor readings include cooling fluid (or coolant) temperature T c , the mean effective gas temperature T g , air temperature at the radiator inlet (or ambient air temperature) T ain , air temperature exiting the radiator T aout , engine block inlet and outlet temperatures T HI  and T HO , cooling fluid (or coolant) mass flow rate {dot over (m)} c , and ram air flow velocity ν ram . Thermal sensors positioned within the cooling system  100  can be used to measure T c , T HI , T HO , T ain , and T aout . The mean effective gas temperature T g  can be measured directly using an in-cylinder sensor or can be obtained from a table look-up based upon the air/fuel ratio, spark angle, and load, which may be determined based on the operation of the internal combustion engine. The coolant mass flow rate {dot over (m)} c  can be measured using a flow meter in the line from the radiator  115  to the fluid pump  106  of  FIG. 3  and the ram air flow velocity ν ram  can be measured using an air flow meter mounted at the air inlet  203  to the matrix of fans  118  as illustrated in  FIG. 2 . The air flow meter can be offset to avoid interference from the air flow from operation of the fans  118 . 
         [0093]    At  1909 , the heat input (q in ) to the cooling system  100  can be determined using the initialized system parameters of  1903  and the measured data from  1906  based upon EQNS. (1) and (2). The ram air effect is checked at  1912  and, in response to the determined q in , it can be determined whether operation of one or more fans is needed to remove the heat input. If the ram air flow velocity ν ram  is sufficient to remove the heat from the internal combustion engine, then the fan control returns to  1906  where the sensor readings are taken again for the next control interval. 
         [0094]    If operation of one or more fans in the fan array is needed to remove the q out , the heat output from the radiator  115  ( FIG. 3 ) is determined at  1915  using, e.g., EQNS. (18) or (19). Based upon the heat input and heat output, the number (n) of fans  118  in the fan matrix and the speed (N) of the fans  118  ( FIG. 3 ) to remove the heat input can be determined at  1918 . The location of the fans  118  in the fan matrix may be predetermined or may be evaluated based upon cooling characteristics associated with each fan. As discussed with respect to EQNS. (20a)-(20c), this can be accomplished in real time using, e.g., mixed integer nonlinear programming (MINP) by minimizing the objective function, f(.,.), of EQN. 21 subject to the constraints to EQNS. (22) and (23). The gradient of the objective function, f(.,.) of EQN. (26) can be used to provide search direction for the optimization. To obtain an integer value for the number of fans, n, per given heat load, q out , the two nearest points around the successive solution, (N s , n s ) can be calculated. The point that does not satisfy the stated constraints of EQNS. (20b) and (20c) is dropped. If both of the points satisfy the constraints, then the objective function can be calculated for each point and the point which leads to the smallest objective function value is selected as the best solution. The objective function, f(.,.), may be minimized based upon a predefined threshold value for the objective function or a predefined threshold value for the change in the objective function between iterative steps. In this way, the fan operation adapts to the conditions of the internal combustion engine and the cooling system  100  ( FIG. 3 ). 
         [0095]    The number (n) of fans  118  and the speed (N) of the fans  118  can also be determined at  1918  by using an offline lookup table, where the speed and number is defined as a function of heat output or N, n=F(q out ). FIGS.  11  and  13 A- 13 B provide examples of lookup tables to determine the fan configuration. The heat input (q in ) can be compared to the heat rejections q out  indicated in the table to determine the number and/or the matrix configuration of the fans  118 . The fan motor speed can then be determined to provide the needed heat rejection. The identified fans  118  can all operate at a common motor speed or may be individually controlled for different motor speeds. 
         [0096]    Fan operation can be adjusted at  1921  based upon the determined number (n) and the speed (N) of the fans  118 . If fans  118  are currently operating with the appropriate matrix configuration and speed, then no adjustment may be needed and the fan control returns to  1906  where the sensor readings are taken again for the next control interval. Otherwise, commands can be sent to the appropriate radiator fan controllers of the motor driver  306  ( FIG. 3 ). Fans  118  may be started or stopped and/or motor speed may be adjusted to provide the appropriate cooling system operation. The fan control then returns to  1906  where the sensor readings are taken again for the next control interval. Control of the radiator fans  118  continues until the internal combustion engine stops. In some cases, control of the fans  118  may continue to provide a controlled cool down of the engine block  109 . 
         [0097]    With reference to  FIG. 20 , shown is a schematic block diagram of a computing device  2000  that can be used to implement the radiator fan control system of  FIG. 3  according to various embodiments of the present disclosure. The computing device  2000  includes at least one processor circuit, for example, having a processor  2003  and a memory  2006 , both of which are coupled to a local interface  2009 . To this end, the computing device  2000  may comprise, for example, at least one server computer or like device. The local interface  2009  may comprise, for example, a data bus with an accompanying address/control bus or other bus structure as can be appreciated. 
         [0098]    Stored in the memory  2006  are both data and several components that are executable by the processor  2003 . In particular, stored in the memory  2006  and executable by the processor  2003  may be a radiator fan control application  2015 , an operating system  2018 , and/or other applications  2021 . Also stored in the memory  2006  may be a data store  2012  and other data. The computing device  2000  can also include one or more analog-to-digital converter(s) (ADC)  2024  and/or one or more digital-to-analog converter(s) (DAC)  2027  to interface with system sensors and/or system controls. 
         [0099]    It is understood that there may be other applications that are stored in the memory  2006  and are executable by the processor  2003  as can be appreciated. Where any component discussed herein is implemented in the form of software, any one of a number of programming languages may be employed such as, for example, C, C++, C#, Objective C, Java®, JavaScript®, Perl, PHP, Visual Basic®, Python®, Ruby, Delphi®, Flash®, or other programming languages. 
         [0100]    A number of software components are stored in the memory  2006  and are executable by the processor  2003 . In this respect, the term “executable” means a program file that is in a form that can ultimately be run by the processor  2003 . Examples of executable programs may be, for example, a compiled program that can be translated into machine code in a format that can be loaded into a random access portion of the memory  2006  and run by the processor  2003 , source code that may be expressed in proper format such as object code that is capable of being loaded into a random access portion of the memory  2006  and executed by the processor  2003 , or source code that may be interpreted by another executable program to generate instructions in a random access portion of the memory  2006  to be executed by the processor  2003 , etc. An executable program may be stored in any portion or component of the memory  2006  including, for example, random access memory (RAM), read-only memory (ROM), hard drive, solid-state drive, USB flash drive, memory card, optical disc such as compact disc (CD) or digital versatile disc (DVD), floppy disk, magnetic tape, or other memory components. 
         [0101]    The memory  2006  is defined herein as including both volatile and nonvolatile memory and data storage components. Volatile components are those that do not retain data values upon loss of power. Nonvolatile components are those that retain data upon a loss of power. Thus, the memory  2006  may comprise, for example, random access memory (RAM), read-only memory (ROM), hard disk drives, solid-state drives, USB flash drives, memory cards accessed via a memory card reader, floppy disks accessed via an associated floppy disk drive, optical discs accessed via an optical disc drive, magnetic tapes accessed via an appropriate tape drive, and/or other memory components, or a combination of any two or more of these memory components. In addition, the RAM may comprise, for example, static random access memory (SRAM), dynamic random access memory (DRAM), or magnetic random access memory (MRAM) and other such devices. The ROM may comprise, for example, a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or other like memory device. 
         [0102]    Also, the processor  2003  may represent multiple processors  2003  and the memory  2006  may represent multiple memories  2006  that operate in parallel processing circuits, respectively. In such a case, the local interface  2009  may be an appropriate network that facilitates communication between any two of the multiple processors  2003 , between any processor  2003  and any of the memories  2006 , or between any two of the memories  2006 , etc. The local interface  2009  may comprise additional systems designed to coordinate this communication, including, for example, performing load balancing. The processor  2003  may be of electrical or of some other available construction. 
         [0103]    Although the radiator fan control application  2015 , application(s)  2021 , and other various systems described herein may be embodied in software or code executed by general purpose hardware as discussed above, as an alternative the same may also be embodied in dedicated hardware or a combination of software/general purpose hardware and dedicated hardware. If embodied in dedicated hardware, each can be implemented as a circuit or state machine that employs any one of or a combination of a number of technologies. These technologies may include, but are not limited to, discrete logic circuits having logic gates for implementing various logic functions upon an application of one or more data signals, application specific integrated circuits having appropriate logic gates, or other components, etc. Such technologies are generally well known by those skilled in the art and, consequently, are not described in detail herein. 
         [0104]    Although the flowchart of  FIG. 19  shows a specific order of execution, it is understood that the order of execution may differ from that which is depicted. For example, the order of execution of two or more blocks may be scrambled relative to the order shown. Also, two or more blocks shown in succession in  FIG. 19  may be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks shown in  FIG. 19  may be skipped or omitted (in favor, e.g., measured travel times). In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids, etc. It is understood that all such variations are within the scope of the present disclosure. 
         [0105]    Also, any logic or application described herein, including the radiator fan control application  2015  and/or application(s)  2021 , that comprises software or code can be embodied in any non-transitory computer-readable medium for use by or in connection with an instruction execution system such as, for example, a processor  2003  in a computer system or other system. In this sense, the logic may comprise, for example, statements including instructions and declarations that can be fetched from the computer-readable medium and executed by the instruction execution system. In the context of the present disclosure, a “computer-readable medium” can be any medium that can contain, store, or maintain the logic or application described herein for use by or in connection with the instruction execution system. The computer-readable medium can comprise any one of many physical media such as, for example, magnetic, optical, or semiconductor media. More specific examples of a suitable computer-readable medium would include, but are not limited to, magnetic tapes, magnetic floppy diskettes, magnetic hard drives, memory cards, solid-state drives, USB flash drives, or optical discs. Also, the computer-readable medium may be a random access memory (RAM) including, for example, static random access memory (SRAM) and dynamic random access memory (DRAM), or magnetic random access memory (MRAM). In addition, the computer-readable medium may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or other type of memory device. 
         [0106]    It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. 
         [0107]    It should be noted that ratios, concentrations, amounts, and other numerical data may be expressed herein in a range format. It is to be understood that such a range format is used for convenience and brevity, and thus, should be interpreted in a flexible manner to include not only the numerical values explicitly recited as the limits of the range, but also to include all the individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly recited. To illustrate, a concentration range of “about 0.1% to about 5%” should be interpreted to include not only the explicitly recited concentration of about 0.1 wt % to about 5 wt %, but also include individual concentrations (e.g., 1%, 2%, 3%, and 4%) and the sub-ranges (e.g., 0.5%, 1.1%, 2.2%, 3.3%, and 4.4%) within the indicated range. The term “about” can include traditional rounding according to significant figures of numerical values. In addition, the phrase “about ‘x’ to ‘y’” includes “about ‘x’ to about ‘y’”.