Abstract:
A method is provided for characterizing fluid flow in a pipe where the fluid includes a drag reducing polymer of a particular type and particular concentration. A computational model is configured to model flow of a fluid in a pipe. The computational model utilizes an empirical parameter for a drag reducing polymer of the particular type and the particular concentration. The computational model can be used to derive information that characterizes the flow of the fluid in the pipe. The empirical parameter for the particular type and the particular concentration of the drag reducing polymer can be identified by solving another computational model that is configured to model turbulent Couette flow in a Couette device for a fluid that includes a drag reducing polymer of the particular type and the particular concentration. The empirical data needed for identification of the empirical parameter are obtained from Couette device experiments.

Description:
BACKGROUND 
       [0001]    1. Field 
         [0002]    The present application relates to laboratory analysis of fluids, particularly fluids with drag reducing additives added thereto. 
         [0003]    2. Related Art 
         [0004]    Drag reducers are chemical additives, which being added to a fluid, significantly reduce friction pressure losses on fluid transport in a turbulent regime through pipelines. Such chemical additives, usually polymers, may decrease pressure drop by up to 80 percent, and thus allows reducing the friction losses to the same extent. 
         [0005]    The efficiency of drag reducers is usually tested in a flow loop. For a given drag reducer type and concentration a pressure drop along a laboratory pipe is measured at the Reynolds number that is maximally close to that expected in an industrial pipeline. A relative reduction in the pressure drop, in comparison to that in a flow free of drag reducers, is a measure of the additive efficiency. 
         [0006]    Kalashnikov, V. N., “Dynamical Similarity and Dimensionless Relations for Turbulent Drag Reduction by Polymer Additives,”  Journal of Non - Newtonian Fluid Mechanics , Vol. 75, 1998, pp. 1209-1230, describes a Taylor-Couette device used for studies of turbulent drag reduction caused by polymer additives. The Taylor-Couette device includes a rotating outer cylinder and an immobile inner cylinder. An effect of a drag reducer was evaluated by the torque, applied to the inner cylinder. The greater reduction in torque resulting from the additive resulted in improved drag reducer performance. The drag reduction was investigated for a wide range of Reynolds numbers and the author suggests dimensionless criteria for drag reduction characterization. 
         [0007]    Koeltzsch et al., “Drag Reduction Using Surfactants in a Rotating Cylinder Geometry,”  Experiments in Fluids , Vol. 24, 2003, pp. 515-530, studies turbulent drag reduction in a device of a similar design. Note that measurement of the torque applied to the inner cylinder has a limited accuracy due to unavoidable friction in bearings. 
       SUMMARY 
       [0008]    The present application provides a method of characterizing fluid flow in a pipe where the fluid includes a drag reducing polymer of a particular type and particular concentration. A computational model is configured to model flow of a fluid in a pipe. The computational model utilizes an empirical parameter for a drag reducing polymer of the particular type and the particular concentration. The computational model can be used to derive information that characterizes the flow of the fluid in the pipe. 
         [0009]    In one embodiment, the empirical parameter for the particular type and concentration of the drag reducing polymer can be derived by solving another computational model that is configured to model turbulent flow in a Couette device for a fluid that includes a drag reducing polymer of the particular type and concentration. The solution of the empirical parameter for the particular type and concentration of the drag reducing polymer can calculated from experimental data derived from operation of the Couette device with a fluid that includes a drag reducing polymer of the particular type and concentration. 
         [0010]    In another embodiment, the computational model of the pipe flow includes a drag reduction parameter that is a function of the empirical parameter. The drag reduction parameter is a function of a dimensionless pipe radius R + . For example, the computational model of the pipe flow can be configured to relate the drag reduction parameter to the empirical parameter by an equation of the form: 
         [0000]        D   * =1+α *   R   + 
 
         [0011]    where
       D *  is the drag reduction parameter,   α *  is the empirical parameter, and   R +  is the dimensionless pipe radius.       
 
         [0015]    In yet another embodiment, the computational model of the pipe flow includes a friction factor that is a function of the drag reduction parameter, wherein the friction factor relates pressure loss due to friction along a given length of pipe to the mean flow velocity through the pipe. For example, the computational model of the pipe flow can be configured to relate the friction factor to the drag reduction parameter by an equation of the form: 
         [0000]    
       
         
           
             
               1 
               
                 f 
                 0.5 
               
             
             = 
             
               
                 4 
                  
                 
                   
                     log 
                     10 
                   
                    
                   
                     ( 
                     
                       Re 
                        
                       
                           
                       
                        
                       
                         f 
                         0.5 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 8.2 
                  
                 
                   D 
                   * 
                   2 
                 
               
               - 
               8.6 
               - 
               
                 12.2 
                  
                 
                   log 
                   10 
                 
                  
                 
                   D 
                   * 
                 
               
             
           
         
       
     
         [0016]    where
       f is the friction factor,   D *  is the drag reduction parameter, and   Re is the Reynolds number of the flow in the pipe.       
 
         [0020]    The computational model of the pipe flow can be further configured to relate the Reynolds number Re to a dimensionless pipe radius R + . The information derived from the computational model of the pipe flow can include a solution for the friction factor f for given flow conditions and possibly a pressure drop over a given length of pipe based on the solution for the friction factor f. 
         [0021]    The computational model of the pipe flow can be based upon a representation of the flow as two layers consisting of a viscous outer sublayer that surrounds a turbulent core. 
         [0022]    The computational model for the turbulent Couette flow can be based upon a representation of the turbulent Couette flow as three layers consisting of viscous outer and inner sublayers with a turbulent core therebetween. 
         [0023]    In one embodiment, the Couette device defines an annulus between first and second annular surfaces, and the computational model for the turbulent Couette flow includes a first drag reduction parameter associated with the first annular surface and a second drag reduction parameter associated with the second annular surface, wherein both the first and second drag reduction parameters are also functions of the empirical parameter specific to a drag reducing polymer of the particular type and the particular concentration. The first and second drag reduction parameters are also functions of a dimensionless torque G applied to the Couette device rotor. 
         [0024]    The computational model for the turbulent Couette flow can also be based on an equation that defines a fluid velocity at a boundary of a viscous sublayer adjacent one of the first and second annular surfaces. Such equation can be derived by momentum conservation for a turbulent core. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0025]      FIG. 1  is a schematic cross-sectional diagram of an exemplary cylindrical Couette device. 
           [0026]      FIG. 2  is a schematic diagram of the annular surfaces of the cylindrical Couette device of  FIG. 1 . 
           [0027]      FIG. 3  is a schematic diagram of a sensing apparatus that employs the Lenterra technique to measure shear stress of the fluid at the surface of the wall of the outer cylinder of the cylindrical Couette device of  FIG. 1 . 
           [0028]      FIG. 4  is a schematic diagram of the layers of the flow field used in a computational model of turbulent Couette fluid flow in the cylindrical Couette device of  FIG. 1 . 
       
    
    
     DETAILED DESCRIPTION 
       [0029]      FIG. 1  shows an exemplary cylindrical Couette device  100 , which includes a top wall  102 , outer cylinder  104 , inner cylinder  120 , and bottom wall  106  that define the boundary of an annulus  108  disposed between the inner surface  104 A of the outer cylinder  104  and the outer surface  120 A of the inner cylinder  120 . The Couette device  100  further has top and bottom retaining plates  110 ,  112  set apart by spacers  114  and mechanically secured, for example, by nuts and bolts  116 . A fluid is loaded into the annulus  108  preferably through one or more fluid paths (e.g., one shown as port  119  and passageway  118 ). 
         [0030]    The inner cylinder  120  is mounted on bearings and is coaxial with the outer cylinder  104 . The outer cylinder  104  is fixed in position and thus remains stationary. The inner cylinder  120  rotates independently of the outer cylinder  104 . A shaft  122  extends down from the bottom of the inner cylinder  120 . A motor  124  has an output shaft  124 A that is mechanically coupled to the shaft  122  by means of a coupling device  128 , which can be a magnetic coupler, a rigid coupler, a flexible coupler, or other suitable coupling mechanism. In the preferred embodiment, the motor  124  can operate over a wide range of rotational speeds (e.g., 100-20,000 rpm) for rotating the inner cylinder  120  at different angular velocities. 
         [0031]    Instrumentation can be added to the Couette device  100  as needed. For example, devices for heating and/or cooling the fluids within the annulus  108  of the Couette device  100  may be added. Such devices may be used in conjunction with loading fluid into the annulus  108  to achieve a predetermined pressure in the annulus  108 . Pumps are used to transfer the fluids into the annulus  108 . The pumps define and maintain the pressure of the system. One or more temperature sensors and one or more pressure sensors can be mounted adjacent the annulus  108  to measure fluid temperature and pressure therein. In one embodiment, the rotational speed of the inner cylinder  120  is measured through the use of a proximity sensor, which measures the rotational speed of the shaft  122  mechanically coupled to the inner cylinder  120 . 
         [0032]    A schematic diagram of the Couette device  100  is shown in  FIG. 2 , with the radius R denoting the radius of the inner wall surface  104 A of the outer cylinder  104  and the radius r 0  denoting the radius of the outer surface  120 A of the inner cylinder. The annulus or gap  108  between the inner and outer cylinders has a width H of (R−r 0 ) and a height of L. A fluid mixture that employs a drag reducer of a specific type and concentration is loaded into the annulus  108  via port  119  and passageway  118 . The motor  124  is operated at a sufficient speed to provide turbulent flow of the fluid in the annulus  108  such that the Reynolds number of the Couette flow Re c  exceeds 1.3×10 4 . The Couette flow circulates in the annulus  108  during such operations. 
         [0033]    The shear stress of the fluid at the inner surface  104 A of the wall of the outer cylinder  104  is measured using the Lenterra technique that combines a floating element  301  and a mechanical cantilever beam  303  with a micro-optical strain gauge (fiber Bragg grating or FBG)  305  as shown in  FIG. 3 . The shear stress is applied to the floating element  301  attached to the cantilever beam  303 . The floating element is installed flush with the inner wall surface  104 A of the outer cylinder  104  in a sensor enclosure  307 . Displacement of the floating element  301  leads to bending of cantilever beam  303 . When the cantilever beam bends, the FBG is strained in a manner that shifts its optical spectrum. By interrogating the FBG with a light source via optical fibers  309 , this strain (and therefore the shear stress) is measured by tracking the shift in the resonant wavelength. The shear stress is calculated as τ w =kΔλ, where k is the calibration coefficient and Δλ is the shift in the resonant wavelengths. This technique provides more accurate shear stress measurements (as compared to the measurement of the torque applied to the Couette device spindle as is commonplace in many Couette devices). The shear stress measuring sensor devices of  FIG. 3  are available from Lenterra, Inc. of Newark, N.J., USA. 
       A. Couette Device Computational Model 
       [0034]    The Couette fluid flow in the annulus  108  of the Couette device  100  can be studied in terms of the dimensionless torque G and the Reynolds number Re c  for such fluid flow. The dimensionless torque G is defined as a function of the torque T derived from shear stress τ w  measured at the inner wall surface  104 A of the outer cylinder  104  of the Couette device  100  as follows: 
         [0000]    
       
         
           
             
               
                 
                   G 
                   = 
                   
                     T 
                     
                       ρ 
                        
                       
                           
                       
                        
                       
                         v 
                         2 
                       
                        
                       L 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             where T=τ w 2πR 2 L, R is the radius of the inner wall of the outer cylinder of the Couette device, ρ is the density of the fluid, ν is the kinematic viscosity of the fluid, and L is the height of the gap of the Couette device. 
           
         
       
     
         [0036]    The Reynolds number Re c  for Couette fluid flow in the annulus  108  of the Couette device  100  can be calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     Re 
                     c 
                   
                   = 
                   
                     
                       ω 
                        
                       
                           
                       
                        
                       
                         
                           r 
                           0 
                         
                          
                         
                           ( 
                           
                             R 
                             - 
                             
                               r 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     v 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
       
         
           
             where r 0  is the outer radius of the inner cylinder  120  of the Couette device  100 , and
           ω is the rotor angular velocity of the inner cylinder  120  of the Couette device  100 .   
         
           
         
       
     
         [0039]    To model the Couette flow in the Couette device  100 , the flow field of turbulent Couette fluid flow in the Couette device  100  can be described by three layers including a relatively viscous outer sublayer  151  adjacent the inner surface  104 A of the outer cylinder  104 , a viscous inner sublayer  155  adjacent the outer surface  120 A of the inner cylinder  120 , and a turbulent layer or core  153  between the viscous inner and outer sublayers  151 ,  155  as shown in  FIG. 4 . The streamwise (tangential) velocity of the fluid flow in the Couette device  100  is shown as vector u in  FIG. 4 . Velocity fluctuations in the fluid flow can have a streamwise component u′ and a radial component v′ as shown. 
         [0040]    The model assumes a linear velocity distribution across the viscous outer sublayer  151 : 
         [0000]        u   +   =y   +   ,y   + ≦δ 0   +   (3)
       where u +  is the normalized fluid velocity given by u + =u/u 0*  where u is the streamwise velocity of the viscous outer sublayer  151  and u 0*  is the friction velocity of the viscous outer sublayer  151  given by u 0* =(τ w /τ) 0.5 ,
           y +  is given as y + =u 0* y/ν where y is the distance from the inner wall  104 A of the outer cylinder  104 , and   δ 0   +  is the dimensionless thickness of the viscous outer sublayer  151 , δ 0   +  is set to a predetermined value such as 11.6.   
           The velocity distribution across the turbulent core  153  for the region confined by the boundary of the viscous outer sublayer  151  and the gap centerline  154  at R m =0.5(r 0 +R) is described by the ordinary differential equation as:       
 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         u 
                         r 
                       
                     
                     
                        
                       r 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           u 
                           
                             0 
                             * 
                           
                         
                          
                         R 
                       
                       
                         
                           κ 
                            
                           
                             ( 
                             
                               R 
                               - 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           r 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
       
         
           
             where κ is the Von Karman constant, which can be set to a predetermined value such as 0.45; other suitable values of the Von Karman constant can be used; note that a decrease in the Von Karman constant can require a corresponding reduction in δ 0   + , while an increase in the Von Karman constant can require a corresponding increase in δ 0   + . 
           
         
       
     
         [0046]    The initial condition for Eq. (4) is the normalized velocity at the viscous outer sublayer surface boundary u +  (δ 0   + ) equal to a parameter λ (i.e., u + (δ 0   + )=λ). In one embodiment, the parameter λ is set to a predetermined value such as 11.6 assuming the dimensionless velocity distribution across the laminar sublayer in a Couette flow is identical to that in the pipe wall vicinity. 
         [0047]    The analytical solution of Eq. (4) is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         κ 
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               r 
                               R 
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     r 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       γ 
                        
                       
                         r 
                         R 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 where 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     γ 
                     = 
                     
                       
                         λ 
                         - 
                         
                           
                             1 
                             κ 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     a 
                                   
                                   ) 
                                 
                                  
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       a 
                                       
                                         1 
                                         - 
                                         a 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         a 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
       
         
           
             where α=δ 0   + /R +  where R +  is the dimensionless radius of the inner wall surface  104 A of the outer cylinder  104  of the Couette device  100 . 
           
         
       
     
         [0049]    R +  can be expressed through the dimensionless torque G as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     + 
                   
                   = 
                   
                     
                       G 
                       
                         2 
                          
                         π 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0050]    The momentum conservation equation for the turbulent core  153  for the region confined by the gap centerline  154  at R m =0.5(r 0 +R) and the outer surface of the viscous inner sublayer  155  can be given as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         u 
                         r 
                       
                     
                     
                        
                       r 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           u 
                           
                             0 
                             * 
                           
                         
                          
                         R 
                       
                       
                         
                           κ 
                            
                           
                             ( 
                             
                               r 
                               - 
                               
                                 r 
                                 0 
                               
                             
                             ) 
                           
                         
                          
                         
                           r 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0051]    The initial condition for Eq. (8) can derived from the normalized velocity at the gap centerline  154  at R m =0.5(r 0 +R) according to Eq. (5) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         
                           R 
                           m 
                         
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         κ 
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               
                                 1 
                                 + 
                                 η 
                               
                               2 
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     - 
                                     η 
                                   
                                   
                                     1 
                                     + 
                                     η 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       γ 
                        
                       
                         
                           1 
                           + 
                           η 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
       
         
           
             where η is the ratio r 0 /R. 
           
         
       
     
         [0053]    Then, the analytical solution of Eq. (8) is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           u 
                            
                           
                             ( 
                             
                               R 
                               m 
                             
                             ) 
                           
                         
                         
                           u 
                           
                             0 
                             * 
                           
                         
                       
                        
                       
                         r 
                         
                           R 
                           m 
                         
                       
                     
                     + 
                     
                       
                         1 
                         κ 
                       
                        
                       
                         R 
                         
                           r 
                           0 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             1 
                           
                           + 
                           
                             r 
                             
                               R 
                               m 
                             
                           
                           + 
                           
                             
                               r 
                               
                                 r 
                                 0 
                               
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     - 
                                     
                                       
                                         r 
                                         0 
                                       
                                       
                                         R 
                                         m 
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                       
                                         r 
                                         0 
                                       
                                       r 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0054]    The circumferential velocity U i  is the velocity of the rotating cylinder surface  120 A of the inner cylinder  120  and calculated as: 
         [0000]        U   i   =ωr   0 .  (11a)
 
         [0000]    The circumferential velocity U i  can also be calculated by: 
         [0000]        U   i   =u ( r   0 +δ i )+λ u   i*   (11b)
       where δ i  is the thickness of the viscous inner sublayer  155  adjacent the outer surface  120 A of the inner cylinder  120 ,
           u is the streamwise velocity for the viscous inner sublayer  155 ,   λ is a parameter set to a predetermined value such as 11.6, and   u i*  is the friction velocity of the viscous inner sublayer  155  at the outer surface  120 A of the inner cylinder  120 .   
               
 
         [0059]    Eq. (11b) can be rewritten as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       U 
                       i 
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         u 
                          
                         
                           ( 
                           
                             
                               r 
                               0 
                             
                             + 
                             
                               δ 
                               i 
                             
                           
                           ) 
                         
                       
                       
                         u 
                         
                           0 
                           * 
                         
                       
                     
                     + 
                     
                       λ 
                       η 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0060]    The left-hand side of Eq. (12) can be expressed through the dimensionless torque G and the Reynolds number Re c  to obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             2 
                              
                             π 
                           
                           ) 
                         
                         0.5 
                       
                       
                         1 
                         - 
                         η 
                       
                     
                      
                     
                       
                         Re 
                         c 
                       
                       
                         G 
                         0.5 
                       
                     
                   
                   = 
                   
                     
                       
                         u 
                          
                         
                           ( 
                           
                             
                               r 
                               0 
                             
                             + 
                             
                               δ 
                               i 
                             
                           
                           ) 
                         
                       
                       
                         u 
                         
                           0 
                           * 
                         
                       
                     
                     + 
                     
                       λ 
                       η 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0061]    For the right-hand side of Eq. (13), the velocity u (r 0 +δ i ) can be equated to u (r 0 +δ 0 η) and then calculated by Eq. (10) to give: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         
                           
                             r 
                             0 
                           
                           + 
                           
                             δ 
                             i 
                           
                         
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           u 
                            
                           
                             ( 
                             
                               R 
                               m 
                             
                             ) 
                           
                         
                         
                           u 
                           
                             0 
                             * 
                           
                         
                       
                        
                       
                         
                           2 
                            
                           
                               
                           
                            
                           η 
                         
                         
                           1 
                           + 
                           η 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           a 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         κ 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               1 
                               η 
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   a 
                                 
                                 ) 
                               
                             
                             
                               1 
                               + 
                               η 
                             
                           
                           + 
                           
                             
                               
                                 1 
                                 + 
                                 a 
                               
                               η 
                             
                              
                             ln 
                              
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   η 
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   η 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   a 
                                 
                                 ) 
                               
                               a 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0062]    Eqs. (9), (13) and (14) represent a computation model for Couette flow without a drag reducer that can be solved to calculate the relationship of the dimensionless torque G as a function of the Reynolds number Re e  for the Couette flow without a drag reducer. 
       B. Extension of Couette Device Computational Model to Account for Drag Reducer 
       [0063]    The computational model for the Couette flow without a drag reducer as described above can be extended by considering two distinct drag reduction parameters: the drag reduction parameter D 0*  for the viscous outer sublayer  151 , and the drag reduction parameter D i*  for the viscous inner sublayer  155 . 
         [0064]    The drag reduction parameter D 0*  for the viscous outer sublayer  151  can be related to the parameter α *  that is a function of the drag reducer agent type and its concentration as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       0 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         α 
                         * 
                       
                        
                       
                         
                           
                             u 
                             
                               0 
                               * 
                             
                           
                            
                           H 
                         
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             v 
                           
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0065]    where
       u 0*  is the friction velocity of the viscous outer sublayer  151  given by u 0* =(τ w /ρ) 0.5 ,   H is the gap width of the Couette device (H=r 0 −R), and   ν is the kinematic viscosity of the fluid.       
 
         [0069]    Similarly, the drag reduction parameter D i*  for the viscous inner sublayer  155  can be related to the parameter α *  that is a function of the drag reducer agent type and its concentration as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       i 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         α 
                         * 
                       
                        
                       
                         
                           
                             u 
                             
                               i 
                               * 
                             
                           
                            
                           H 
                         
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             v 
                           
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0070]    where
       u i*  is the friction velocity of the viscous inner sublayer  155 ,   H is the gap width of the Couette device (H=r 0 −R), and   ν is the kinematic viscosity of the fluid.       
 
         [0074]    The Reynolds number of the Couette flow Re c  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     Re 
                     c 
                   
                   = 
                   
                     
                       
                         U 
                         i 
                       
                        
                       H 
                     
                     v 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0075]    where
       U i  is the circumferential velocity of the inner cylinder  120 ,   H is the gap width of the Couette device (H=r 0 −R), and   ν is the kinematic viscosity of the fluid.       
 
         [0079]    Eq. (16) can be used to rewrite Eq. (15a) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       0 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         α 
                         * 
                       
                        
                       
                         
                           
                             u 
                             
                               0 
                               * 
                             
                           
                            
                           
                             Re 
                             c 
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             U 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     17 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0080]    Similarly, Eq. (16) can be used to rewrite Eq. (15b) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       i 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         α 
                         * 
                       
                        
                       
                         
                           
                             u 
                             
                               i 
                               * 
                             
                           
                            
                           
                             Re 
                             c 
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             U 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     17 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0081]    As described in Eqs. (11) and (12) above, the ratio 
         [0000]    
       
         
           
             
               
                 u 
                 
                   0 
                   * 
                 
               
                
               
                 Re 
                 c 
               
             
             
               
                   
               
                
               
                 U 
                 i 
               
             
           
         
       
     
         [0000]    of Eq. (17a) can be defined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         
                           0 
                           * 
                         
                       
                        
                       
                         Re 
                         c 
                       
                     
                     
                       
                           
                       
                        
                       
                         U 
                         i 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         - 
                         η 
                       
                       ) 
                     
                      
                     
                       
                         G 
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0082]    Eq. (18) can be used to rewrite Eq. (17a) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       0 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         
                           α 
                           * 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           η 
                         
                         ) 
                       
                        
                       
                         
                           G 
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0083]    Similarly, the ratio 
         [0000]    
       
         
           
             
               
                 u 
                 
                   i 
                   * 
                 
               
                
               
                 Re 
                 c 
               
             
             
               U 
               i 
             
           
         
       
     
         [0000]    of Eq. (17b) can be defined as: 
         [0000]    
       
         
           
             
               
                 
                   u 
                   
                     0 
                     * 
                   
                 
                  
                 
                   Re 
                   c 
                 
               
               
                 U 
                 i 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     - 
                     η 
                   
                   ) 
                 
                 η 
               
                
               
                 
                   G 
                   
                     2 
                      
                     
                         
                     
                      
                     π 
                   
                 
               
             
           
         
       
     
         [0084]    Eq. (20) can be used to rewrite Eq. (17b) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       i 
                       * 
                     
                   
                   = 
                   
                     1 
                     + 
                     
                       
                         
                           α 
                           * 
                         
                         2 
                       
                        
                       
                         
                           ( 
                           
                             1 
                             - 
                             η 
                           
                           ) 
                         
                         η 
                       
                        
                       
                         
                           G 
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0085]    By analogy with a pipe flow, the thickness δ 0   +  of the viscous outer sublayer  151  is related to the drag reduction parameter D 0*  for the viscous outer sublayer  151  as follows: 
         [0000]      δ 0   + =11.6 D   0*   3   (22)
 
         [0086]    Similarly, the thickness δ i   +  of the viscous inner sublayer  155  is related to the drag reduction parameter D i*  for the viscous inner sublayer  155  as follows: 
         [0000]      δ i   + =11.6 D   i*   3   (23)
 
         [0087]    The corresponding dimensionless velocity λ 0  at the boundary of the viscous outer sublayer  151  can be given as: 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     0 
                   
                   = 
                   
                     
                       
                         δ 
                         0 
                         + 
                       
                       
                         D 
                         
                           0 
                           * 
                         
                       
                     
                     = 
                     
                       11.6 
                        
                       
                           
                       
                        
                       
                         D 
                         
                           0 
                           * 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0088]    The corresponding dimensionless velocity λ i  at the boundary of the viscous inner sublayer  155  can be given as: 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     i 
                   
                   = 
                   
                     
                       
                         δ 
                         i 
                         + 
                       
                       
                         D 
                         
                           i 
                           * 
                         
                       
                     
                     = 
                     
                       11.6 
                        
                       
                           
                       
                        
                       
                         D 
                         
                           i 
                           * 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0089]    The normalized velocity at the boundary of the inner viscous sublayer  155  can be derived on the basis of Eq. (10) above to provide: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         
                           
                             r 
                             0 
                           
                           + 
                           
                             δ 
                             i 
                           
                         
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           u 
                            
                           
                             ( 
                             
                               R 
                               m 
                             
                             ) 
                           
                         
                         
                           u 
                           
                             0 
                             * 
                           
                         
                       
                        
                       
                         
                           2 
                            
                           
                               
                           
                            
                           η 
                         
                         
                           1 
                           + 
                           η 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           m 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         κ 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               1 
                               η 
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   m 
                                 
                                 ) 
                               
                             
                             
                               1 
                               + 
                               η 
                             
                           
                           + 
                           
                             
                               
                                 1 
                                 + 
                                 m 
                               
                               η 
                             
                              
                             ln 
                              
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   η 
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   η 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   m 
                                 
                                 ) 
                               
                               m 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where m=δ i   + /r 0   + , r 0   + =R +  where R +  is calculated by Eq. (7). 
         [0090]    The normalized velocity at the gap centerline  154  of Eq. (26) 
         [0000]    
       
         
           
             
               u 
                
               
                 ( 
                 
                   R 
                   m 
                 
                 ) 
               
             
             
               u 
               
                 0 
                 * 
               
             
           
         
       
     
         [0000]    is given by Eqs. (9) and (6) as repeated below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                        
                       
                         ( 
                         
                           R 
                           m 
                         
                         ) 
                       
                     
                     
                       u 
                       
                         0 
                         * 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         κ 
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               
                                 1 
                                 + 
                                 η 
                               
                               2 
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     - 
                                     η 
                                   
                                   
                                     1 
                                     + 
                                     η 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       γ 
                        
                       
                         
                           1 
                           + 
                           η 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       where 
                        
                       
                           
                       
                        
                       γ 
                     
                     = 
                     
                       
                         λ 
                         - 
                         
                           
                             1 
                             κ 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     a 
                                   
                                   ) 
                                 
                                  
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       a 
                                       
                                         1 
                                         - 
                                         a 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         a 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The parameter α=δ 0   + /R +  and λ=λ 0  needed for Eq. (28) can be determined for the outer cylinder by Eqs. (19), (22) and (24). 
         [0091]    The set of Eqs. (13), (19), (21)-(27) define a computational model for the Couette flow that accounts for the drag reducing effects of the drag reducer. The computational model is dependent on the dimensions of the Couette device  100 , including the radius R of the inner wall surface  104 A of the outer cylinder  104 , the ratio η (which is the ratio r 0 /R), and the height L of the gap of the Couette device  100 . The parameter α *  is the major model variable that is a function of the drag reducer agent type and its concentration. 
         [0092]    The fluid density ρ and the kinematic viscosity ν of the fluid are measured separately. 
         [0093]    The shear stress τ w  and corresponding rotor angular velocity ω of the Couette device  100  are measured during operation of the Couette device  100  for a given drag reducer agent type and its concentration. 
         [0094]    The value of the parameter α *  for the given drag reducer agent type and concentration of the test can be provided by statistical analysis of experimental data. Specifically, experiments can be carried out with the Couette device for a fluid solution that employs a given drag reducing additive at a particular concentration where the shear stress τ w  is measured for a set of different rotor angular velocities ω. The set of measurements of shear stress and corresponding rotor angular velocity as well as the measured fluid density ρ and the kinematic viscosity ν are input to the computation model based on Eqs. (13), (19), (21)-(27) to solve for the parameter α *  for the given drag reducer agent type and its concentration. 
         [0095]    An important aspect of drag reduction phenomenon, not accounted for by the drag reduction model presented, is the maximum drag reduction asymptote. This asymptote provides the minimum Fanning friction factors obtainable. The minimum friction factor obtainable in a pipe flow is described by the empirical equation of Virk (1971): 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       f 
                       0.5 
                     
                   
                   = 
                   
                     
                       19 
                        
                       
                         
                           log 
                           10 
                         
                          
                         
                           ( 
                           
                             Re 
                              
                             
                                 
                             
                              
                             
                               f 
                               0.5 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     32.4 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
       
         
           
             Equation (29) is applicable to a Couette flow, where the Reynolds number (Re c ) number calculated by Eq. 16 is used instead of Re and the Fanning friction factor is calculated for the inner cylinder wall. 
           
         
       
     
         [0097]    The Fanning friction factor f for the inner cylinder of the Couette device is calculated from the shear-stress equation applied to the inner cylinder using Eqs. (1) and (7) and takes the following form: 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       f 
                       0.5 
                     
                   
                   = 
                   
                     
                       Re 
                       c 
                     
                     
                       
                         ( 
                         
                           
                             1 
                             η 
                           
                           - 
                           1 
                         
                         ) 
                       
                        
                       
                         
                           G 
                           π 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0098]    The preferable radius ratio η for the Couette device is below 0.7. It follows from the dependences 1/f 0.5  versus Re f 0.5  calculated for different coefficients α *     0    at different η. The smaller the Couette device radius ratio, the stronger the curves 1/f 0.5  vs. Re f 0.5  are shifted downward from the drag reduction asymptote. The lower these curves are located, the wider range of the coefficient α *     0    can be identified during testing different drag reducing chemicals. 
       C. Application of Couette Device Computational Model Solution of Part B to Pipe Flow Modeling and Analysis 
       [0099]    For a fluid solution employing a drag reducing agent that flows in a pipe, the flow field for turbulent flow in the pipe can be described by two layers, which include a relatively viscous outer sublayer adjacent the pipe wall and a turbulent inner core surrounded by the viscous outer sublayer. 
         [0100]    Applying the approach proposed by Yang and Dou in “Turbulent Drag Reduction with Polymer Additive in Rough Pipes,”  Journal of Fluid Mechanics , Vol. 642, 2010, pp. 279-294, the velocity distribution across the viscous sublayer at the pipe wall may be obtained in the following form: 
         [0000]        u   + =2.5 ln  y   + +11.6 D   *   2 −7.5 ln  D   * −6.1  (31)
 
         [0101]    where
       u +  is the normalized fluid velocity given by u + =u/u 0* ,   y +  given as y + =u * y/ν where y is the distance from the pipe wall, u *  is the friction velocity given by u * =(τ w /ρ) 0.5 , τ w  is the shear stress at the pipe wall, ρ is the density of the fluid of the solution, and ν is the kinematic viscosity of the fluid of the solution; and   D *  is a drag reduction parameter.       
 
         [0105]    The drag reduction parameter D *  of Eq. (31) is related to the parameter α *  that is a function of the drag reducer agent type and its concentration as described above in the computational model of Part B as follows: 
         [0000]        D   * =1+α *   R   +   (32)
 
         [0000]    where R +  is the dimensionless pipe radius. 
         [0106]    The normalized mean flow velocity 
         [0000]    
       
         
           
             U 
             
               u 
               * 
             
           
         
       
     
         [0000]    can be calculated by averaging the velocity u +  of Eq. (31) over the pipe cross-section as: 
         [0000]    
       
         
           
             
               
                 
                   
                     U 
                     
                       u 
                       * 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           π 
                            
                           
                             ( 
                             
                               R 
                               + 
                             
                             ) 
                           
                         
                         2 
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         R 
                       
                        
                       
                         
                           u 
                           + 
                         
                          
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           
                             R 
                             + 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 + 
                               
                               - 
                               
                                 y 
                                 + 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           
                             y 
                             + 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0107]    Furthermore, the normalized mean flow velocity 
         [0000]    
       
         
           
             U 
             
               u 
               * 
             
           
         
       
     
         [0000]    is related to the friction factor f by: 
         [0000]    
       
         
           
             
               
                 
                   
                     U 
                     
                       u 
                       * 
                     
                   
                   = 
                   
                     
                       2 
                       f 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
       
         
           
             where U is the mean flow velocity through the pipe, and
           u *  is the friction velocity given by u * =(τ w /ρ) 0.5 .
 
The friction factor f relates pressure loss due to friction along a given length of the pipe to the mean flow velocity through the pipe.
   
         
           
         
       
     
         [0110]    The integration of Eq. (33) can be performed analytically to obtain an equation for the friction factor f as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       f 
                       0.5 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         
                           log 
                           10 
                         
                          
                         
                           ( 
                           
                             Re 
                              
                             
                                 
                             
                              
                             
                               f 
                               0.5 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       8.2 
                        
                       
                         D 
                         * 
                         2 
                       
                     
                     - 
                     8.6 
                     - 
                     
                       12.2 
                        
                       
                         log 
                         10 
                       
                        
                       
                         D 
                         * 
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0111]    As given by Eq. (32), the drag reduction parameter D *  of Eq. (35) is a function of the dimensionless pipe radius R + , which can be related to the Reynolds number Re of the flow in the pipe and the friction factor f by: 
         [0000]        R   + =0.5 Re √{square root over ( f/ 2)}  (36)
 
         [0112]    The Reynolds number Re of Eq. (36) is Oven by: 
         [0000]    
       
         
           
             
               
                 
                   Re 
                   = 
                   
                     
                       U 
                        
                       
                           
                       
                        
                       D 
                     
                     v 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0113]    The dimensionless pipe radius is calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     + 
                   
                   = 
                   
                     
                       0.5 
                        
                       
                         
                           
                             u 
                             * 
                           
                            
                           
                               
                           
                            
                           D 
                         
                         v 
                       
                     
                     = 
                     
                       0.5 
                        
                       
                         
                           
                             
                               ( 
                               
                                 
                                   τ 
                                   w 
                                 
                                 / 
                                 ρ 
                               
                               ) 
                             
                             0.5 
                           
                            
                           D 
                         
                         v 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
       
         
           
             where
           u *  is the friction velocity given by u * =(τ w /ρ) 0.5 ,   D is the diameter of the pipe, and   ν is the kinematic viscosity of the fluid.   
         
           
         
       
     
         [0118]    Equation (35) is the major model equation for calculating the friction factor f *  The friction factor can be used for engineering calculations of the pipe flow, such as the predicted pressure drop in the pipe (over the length L) for the flow employing a particular drag reducer agent type and concentration, which is given as 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               p 
             
             = 
             
               2 
                
               
                   
               
                
               ρ 
                
               
                   
               
                
               f 
                
               
                 
                   U 
                   2 
                 
                 D 
               
                
               
                 L 
                 . 
               
             
           
         
       
     
         [0119]    The fluid density ρ and the kinematic viscosity ν of the fluid are measured separately. 
         [0120]    The Reynolds number Re of the fluid flow is determined by the mean flow velocity, the pipe diameter and the fluid kinematic viscosity, which are known. 
         [0121]    The parameter α *  that is a function of the drag reducer agent type and its concentration is given by the solution of the computational model in Part B for the given drag reducer agent type and its concentration. 
         [0122]    These operations can be carried out for a number of different concentrations of a particular drag reducer agent type or over different drag reducer agents to characterize the expected pipe flow for these different scenarios. It can also be carried out for a number of fluid flows with different Reynolds number Re to characterize the expected pipe flow for these different scenarios. 
         [0123]    The computational models of Parts A, B, and C of the present application can be realized by one or more computer programs (instructions and data) that are stored in the persistent memory (such as hard disk drive or solid state drive) of a suitable data processing system and executed on the data processing system. The data processing system can be a realized by a computer (such as a personal computer or workstation) or a network of computers. 
         [0124]    Advantageously, the computational model of Part C characterizes the pipe flow of a dilute drag reducer polymer solution through the use of an empirical parameter that is a function of the drag reducer polymer type and concentration. This empirical parameter can be derived from the solution of a computation model for turbulent Couette flow of such drag reducer polymer solutions based upon experiments that generate and measure properties of the turbulent Couette flow for such dilute drag reducer polymer solutions as described in Part B above. 
         [0125]    Furthermore, the computational models of the present application employ a two layer representation of the boundary layer interfaces for both the turbulent Couette flow and the pipe flow in order to simplify the equations for such fluid flow. Specifically, the turbulent Couette flow is represented by three layers including viscous inner and outer sublayers with a turbulent core therebetween, and the pipe flow is represented by a viscous outer sublayer that surrounds a turbulent core. The computation model of the turbulent Couette flow also provides for computation of the dimensionless torque applied to the Couette device as a function of the rotation speed for a given drag reducer polymer type and its concentration. 
         [0126]    There have been described and illustrated herein embodiments of computational models that characterize the pipe flow of a dilute drag reducer polymer solution with the use of an empirical parameter that is a function of the drag reducer polymer type and its concentration. This empirical parameter can be derived from the solution of a computation model for turbulent Couette flow of such drag reducer polymer solutions based upon experiments that generate and measure properties of the turbulent Couette flow for such dilute drag reducer polymer solutions. While particular embodiments have been described, it is not intended that the embodiments be limited thereto. It will therefore be appreciated by those skilled in the art that yet other modifications could be made to the provided embodiments without deviating from its scope as claimed.