Abstract:
A method for determining a velocity of a flowing fluid includes estimating a Reynolds number for the flowing fluid; comparing the estimated Reynolds number with a selected range; and determining the velocity of the flowing fluid based on a flow model selected from a laminar flow model, a turbulent flow model, and a partial laminar flow model. An ultrasonic flow meter includes a plurality pairs of transducers configured to form a plurality of measurement paths in a pipe, wherein the plurality of measurement paths are arranged asymmetrically relative to a centerline of the pipe.

Description:
BACKGROUND OF INVENTION  
       [0001]     1. Field of the Invention  
         [0002]     The invention relates generally to flow rate measurement methods and systems. In particular, embodiments relate to multi-path ultrasonic measurement of partially developed flow profiles.  
         [0003]     2. Background Art  
         [0004]     In industries where a flow of fluid is involved, accurate measurements of flow rates are often required. For example, in the oil and gas industry, accurate flow rate measurements are needed in custody transfer (the transfer of ownership, for example at a crude oil loading and off-loading station), leak detection, and process control applications. Conventional flow measurement technologies include turbine flow meters and positive displacement flow meters. Recently, multi-channel ultrasonic meters are gaining shares in this market due to their advantages over conventional technologies. These advantages include: excellent long-term repeatability, less sensitivity to fluid properties such as viscosity and pressure, better open-box accuracy, wider range of linearity, and lower cost of maintenance due to the fact that no moving parts are used in these ultrasonic meters.  
         [0005]     In a typical operation, an ultrasonic flow meter uses a transducer to transmit an ultrasonic beam into the flow stream, and the ultrasonic energy is received by a second transducer. The flow carrying the ultrasonic wave alters the wave&#39;s frequency (Doppler effect) and transit time (velocity superposition), and these two quantities can be measured to determine the flow rate. Based on these principles, two major ultrasonic flow measurement technologies exist: Doppler and transit-time. In some configurations of flow meters, transducers are clamped on the outside wall of a pipe. To achieve better measurement accuracy, transducers may be alternately placed inside the pipe wall, and such transducers are referred to as “wetted” transducers. Some methods to measure flow profiles have been developed, mostly based on Doppler technology (e.g., U.S. Pat. No. 6,067,861, U.S. Pat. No. 6,378,357). However, Doppler signals rely heavily on particle size and concentration that sometimes may vary and lead to poor repeatability. It is widely accepted in the industry that only multi-path transit-time meters combined with wetted transducers are capable of high accuracy applications mentioned above.  
         [0006]     The principles of transit-time ultrasonic measurements are well established. According to the an America Petroleum Institute (API) standard (API H00008, Manual of Petroleum Measurement Standards, Measurement of Liquid Hrdrocarbons by Ultrasonic Flowmeters Using Transit Time), an average velocity along an ultrasonic path can be derived from:  
               V   i     =       L     2   ⁢           ⁢   cos   ⁢           ⁢   θ       *         t   2     -     t   1           t   1     *     t   2                   (   1   )             
 
 where V i  is a path-average flow velocity (i.e., an average of velocities along a particular ultrasonic path) for the path i, L is the ultrasonic path length, θ is the angle between the ultrasonic path and the fluid velocity vector, and t 1  and t 2  are the ultrasonic travel times in and against the flow direction, respectively. 
 
         [0007]     It should be noted that the measured path-average velocity V i  is different from the flow-average velocity V avg , the latter being the velocity averaged over the flow cross section. V i  is directly measured from an ultrasonic transit-time flow meter using Equation (1), while V avg  gives the flow rate that is important in applications such as custody transfer. A ratio K i  between these two velocities can be defined as:  
                 K   i     =       V   avg       V   i         ,           (   2   )             
 
 and is referred to as the channel factor. In the following description, V i  is referred to as the path velocity, and V avg  is referred to as the average velocity. 
 
         [0008]     Pipe flows mostly run in one of the two modes: the laminar mode and the turbulent mode. Widely accepted mathematical models for these flow modes are:  
                 V   ⁡     (   r   )       =       V   c     *     (     1   -       r   2       R   2         )         ,     for   ⁢           ⁢   laminar   ⁢           ⁢   flow             (   3   )                   V   ⁡     (   r   )       =       V   c     *       (     1   -     r   R       )       1   N           ,     for   ⁢           ⁢   turbulent   ⁢           ⁢   flow   ⁢           ⁢   in   ⁢           ⁢   smooth   ⁢     -     ⁢   wall   ⁢           ⁢   pipes             (   4   )             
 
 where V(r) is the velocity at a distance r from the pipe centerline, R is the pipe radius, V c  is the flow velocity along the pipe centerline, and N is a power-law factor. 
 
         [0009]     The power-law factor N is a characteristic value of a turbulent flow. For a fully developed turbulent flow, the power-law factor N can be estimated using an empirical equation described in the literature (e.g., L. Lynnworth, “Ultrasonic Measurement for Process Control”, Academic Press, San Diego, 1989): 
 
 N= 1.66*log  Re   (5) 
 
 where Re is the Reynolds number, which is a function of the flow velocity V and the fluid viscosity μ:  
               Re   =       DV   ⁢           ⁢   ρ     μ       ,           (   6   )             
 
 where D is the pipe diameter, and p is the fluid density. 
 
         [0010]     In real-world applications, the pipe and the fluid conditions often cannot be precisely quantified, and Equations (5) and (6) generally cannot be used to obtain N in high-accuracy measurements. Hence, for a given turbulent flow, at least two measurements at two flow paths are needed in order to solve Equation (4) for the two unknowns, N and V c . This is why a multi-path ultrasonic technology is often needed to resolve flow profile variations.  
         [0011]     Referring to  FIG. 1A , a pipe  1  is shown with three pairs of transducers, Ha and  11   b ,  12   a  and  12   b ,  13   a  and  13   b . An arrow  2  shows the flow direction. The lines between transducer pairs show their ultrasonic paths. In this configuration, the ultrasonic path between  11   a  and  11   b  crosses the pipe centerline, and is referred to as a diagonal path. The shortest distance from a path to the pipe centerline is referred to as channel level. A diagonal path has a channel level of 0. The path between  12   a  and  12   b  and the path between  13   a  and  13   b  have the same channel levels h even though the two transducer pairs are at different locations.  FIG. 1A  also illustrates an exemplary flow profile  3 .  
         [0012]     The channel factor K i  depends on flow profiles and the position of the ultrasonic path. For a diagonal ultrasonic path the channel factor is:  
               0.75   ,     
     ⁢       for   ⁢           ⁢   a   ⁢           ⁢   laminar   ⁢           ⁢   flow     ;   and       ⁢     
     ⁢           ∫   0   R     ⁢         (     1   -     r   R       )       1   N       *   2   *   r   *           ⁢     ⅆ   r           R   *       ∫   0   R     ⁢         (     1   -     r   R       )       1   N       *           ⁢     ⅆ   r             ,     for   ⁢           ⁢   a   ⁢           ⁢   turbulent   ⁢           ⁢     flow   .                 (   7   )             
 
         [0013]     Referring to  FIG. 1B , the channel factor K (vertical axis) is shown as a function of Reynolds number Re (horizontal axis) in a case of a diagonal ultrasonic path. For non-diagonal ultrasonic paths, the K values can also be derived in a similar way. The relation between K value and path positions has been well studied for the laminar flow and the turbulent flow profiles, dating back to a 1978 U.S. Pat. No. 4,078,428. Hence, for both laminar flow and turbulent flow profiles, the relations between measured velocities and the actual average velocities are well defined.  
         [0014]     A major challenge for ultrasonic meters is to detect a flow profile promptly based on information from a limited number of paths. For a turbulent flow, the randomness of measured path velocities in selected paths can result in instant deviation as high as 10 percent from the average values, and the small number of paths makes it difficult to obtain a satisfactory statistical average flow velocity. Unlike turbine meters that inherently average out the whole cross section of a flow, transit-time ultrasonic flow meters only measure a limited number of selected paths of flow velocity. To average out randomness of the measured velocities, ultrasonic meters need to have either a large damping on the raw data or more paths distributed across a flow profile. Using a large damping will adversely affect the system response time and result in poor repeatability when measuring a small volume. On the other hand, adding more channels to measure more paths may substantially increase the system cost.  
         [0015]     A more challenging problem for ultrasonic flow meters is to detect a partially developed flow profile. A fully developed flow profile, by definition, is a flow velocity distribution pattern that does not change along a pipe. Any other flow profiles that have symmetric velocity distributions around the pipe centerline, but with a evolving flow velocity distribution along the pipe, are in this description referred to as partially developed flow profiles. There are two possible causes for partially developed flow profiles. One is a transitional profile between turbulent and laminar flows that can happen in high-viscosity fluids. Another is due to the presence of a flow conditioning device that does not have enough downstream length for the profile to fully develop.  
         [0016]     The transition between a turbulent flow and a laminar flow normally takes place when the Reynolds number is around 2300 and has been demonstrated by numerous experiments. However, as illustrated in  FIG. 1B , this transition can happen in a wide range of Reynolds numbers and can have memory effects, depending on fluid and pipe conditions. As a result, it is not accurate to use the Reynolds number as a sole indicator of a flow profile mode. A partially developed profile near the transition range may cause an unacceptably poor repeatability in flow measurements, as neither the laminar nor the turbulent model fits the profile well.  
         [0017]     As a known fact, a steady profile needs some straight, obstruction-free distance in the pipe to fully develop. Referring to  FIG. 2 , a plug flow  21 , which has a constant velocity throughout the cross section of the pipe  23  having a diameter D, enters the pipe  23  from a much larger pipe  22 . The flow initially has a partially developed profile  24 . After an entrance length  25 , the flow has a fully developed profile  26 . Both theories and experiments indicate that the entrance length  25  needs to be as long as 100 times the pipe diameter D for a laminar profile to fully develop, and 80 times the pipe diameter D for a turbulent profile to fully develop (see, R. W. Fox and A. T. McDonald, “Introduction to Fluid Mechanics”, 3 rd  ed., John Wiley and Sons, New York, 1992). In practice, entrance flow rarely has a form of a plug profile, and a straight length of 10 to 15 times the pipe diameter D is commonly recommended by meter manufactures to have a predictable measurement of flow profiles.  
         [0018]     For a multi-path custody-transfer flow meter, it is a common practice to have a flow conditioning device installed upstream of the flow meter. The main purpose of the flow conditioning device is to reduce swirls and to reduce asymmetric profile distortion. A shorter flow conditioning installation distance is always beneficial for manufacturers and customers. U.S. Pat. No. 6,647,806 also suggests that a shorter distance between a flow conditioning device and a flow meter can improve the repeatability of measurements.  
         [0019]     Referring to  FIG. 3 , which shows an API recommended flow conditioning device  31 , made of a bundle of small tubes having a length B, is installed in the pipe  32  at a distance A from the pipe entrance. The distance C downstream of the flow conditioning device  31 , is recommended to be at least 5 times the diameter D of the pipe  32 . The device  31  tends to smooth out a flow profile. As a result, the presence of the device  31  will either disturb a laminar profile, or flatten a turbulent profile. In either case, within a limited entrance distance, a typical flow profile (i.e., laminar flow or turbulent flow) may not fully develop. In particular, downstream of the conditioning device, turbulent flow may have a much higher N number than that estimated using equation (5), and turbulent-laminar transition can happen at a Reynolds number much lower than 2300. In either case, the flow profile will be unpredictable based on the Reynolds number. What are still needed are improved systems and methods for monitoring partially developed flow profiles in real time.  
       SUMMARY OF INVENTION  
       [0020]     In one aspect, embodiments disclosed herein relate to a method for determining a velocity of a flowing fluid, comprising: estimating a Reynolds number for the flowing fluid; comparing the estimated Reynolds number with a selected range; and determining the velocity of the flowing fluid based on a flow model selected from a laminar flow model, a turbulent flow model, and a partial laminar flow model.  
         [0021]     In another aspect, embodiments disclosed herein relate to an ultrasonic flow meter comprising a plurality pairs of transducers configured to form a plurality of measurement paths in a pipe. The plurality of measurement paths are arranged asymmetrically relative to a pipe centerline.  
         [0022]     Other aspects and advantages of the invention will become apparent from the following description and the attached claims. 
     
    
     BRIEF SUMMARY OF THE DRAWINGS  
       [0023]      FIG. 1A  shows a configuration of a prior art multi-path ultrasonic transit-time flow meter.  
         [0024]      FIG. 1B  shows a chart illustrating K values for a diagonal ultrasonic path at different Reynolds numbers Re.  
         [0025]      FIG. 2  shows a process where a certain entrance length is required for a flow velocity profile to fully develop.  
         [0026]      FIG. 3  illustrates an API recommended flow conditioning device.  
         [0027]      FIGS. 4A and 4B  show a partial laminar flow profile model in accordance with one embodiment of the present invention.  
         [0028]      FIG. 5  shows a pre-computed two-dimensional K value curve for turbulence flows.  
         [0029]      FIG. 6  shows a pre-computed two-dimensional K value curve for partial laminar flows.  
         [0030]      FIG. 7  shows a set of derived one-dimensional turbulent flow K curves in accordance with one embodiment of the present invention.  
         [0031]      FIG. 8  shows a set of derived one-dimensional partial laminar flow K curves in accordance with one embodiment of the present invention.  
         [0032]      FIG. 9  shows a flow chart of a method for evaluating a flow profile in accordance with one embodiment of the present invention.  
         [0033]      FIG. 10  illustrates an exemplary flow meter spool with an asymmetric distribution of ultrasonic transducers. 
     
    
       [0034]     It is to be understood that the drawings are to be used for the purpose of illustration only, and not as a definition of the metes and bounds of the invention, or as a basis for reading non-existent or un-recited limitations into the claims.  
       DETAILED DESCRIPTION  
       [0035]     In one aspect, some embodiments disclosed herein relate to methods that can be used to measure flow rates even in the presence of partially-developed flow profiles, caused either by a flow conditioning device or by a laminar-turbulent profile transition. Embodiments of the invention may include some or all of the following components: (1) a partial laminar flow profile model that addresses the uncertain phase during the laminar-turbulent transition; (2) a profile searching and fitting algorithm that covers a large range of profiles; (3) a combination of ultrasonic paths having different channel levels to a pipe centerline.  
         [0036]     In another aspect, some embodiments disclosed herein relate to methods and multi-path ultrasonic flow meter systems that can accurately measure partially developed flow profiles. A system in accordance with one embodiment of the invention may include a cylindrical ultrasonic spool piece that has two or more ultrasonic paths having distinctive channel levels. Some embodiments may also include a flow conditioning device that removes most of the swirls and asymmetric flow components. A method of the invention may use a partial laminar model that better represents a flow profile partially developed after a flow conditioning device, especially, at a low Reynolds number. A method of the invention may also use an algorithm that searches for best fit of a detected profile to a laminar flow, a partial laminar flow, or a turbulent flow profile.  
         [0037]     A partial laminar flow profile model in accordance to one embodiment of the invention is illustrated in  FIGS. 4A and 4B .  FIG. 4A  shows a view of a cross section along the pipe centerline. The pipe has a diameter D. Dashed lines  41 - 44  represent the projected positions of four ultrasonic paths. The flow velocity profile may be divided into two regions: a laminar region  46  and a flat region  45 . V c  is the centerline velocity if the laminar flow profile is fully developed. The flattening of the central flow profile that deviates from a laminar flow (the dashed curve in  FIG. 4A ) may be a result of the laminar flow passing through a flow conditioning device. Consequently, the set of measured path velocities in this situation will not fit a pure laminar flow profile model very well.  
         [0038]     Ideally, after traveling a sufficient distance downstream of the flow conditioning device, the laminar flow will rebuild from the pipe wall vicinity and the flat region will shrink and gradually disappear. However, in real-world applications, the distance between the flow conditioning device and the flow meter may not be long enough for the laminar profile to fully develop, and the ultrasonic beams would have to go through a partially-developed laminar profile similar to the ones presented in  FIG. 4A .  
         [0039]      FIG. 4B  shows the cross sectional view in a plane perpendicular to the flow direction. In this model of a partially-developed laminar flow profile, the flat region  45  has a radius r from the pipe center line. A partial laminar factor d is defined as the percentage of the flat region in the pipe having a radius R, i.e., d=r/R. The partial laminar factor is an unknown and needs to be solved during flow measurements when the Reynolds number is close to a transition point.  
         [0040]     The position of a specific path is characterized by a channel level h, which is the shortest distance from the center point of a path to the pipe centerline. Referring back to  FIG. 4A , the channel level of the path  41  does not equal the channel level of the path  43  nor the channel level of the path  44 , indicating that paths  41 ,  43 , and  44  are asymmetrically distributed around the pipe centerline.  
         [0041]     Similarly, when a turbulent flow passes through a conditioning device the downstream length may not be enough for the turbulent profile to fully re-develop. The flattened profile will still be a turbulent profile but has a higher power law factor N than that estimated from the Reynolds number. Again, a fast and wide-range profile searching method is needed to solve the two unknowns: the center velocity V c  and the power law factor N.  
         [0042]     Some embodiments of the invention relate to a profile searching and fitting algorithm. An algorithm in accordance with embodiments of the invention is suitable for all profiles, including partially developed profiles. As mentioned above, laminar flows and turbulent flows occur in certain Reynolds number ranges. An algorithm in accordance with embodiments of the present invention has little dependence on particular Reynolds numbers. Instead, the algorithm has three large search ranges: pure laminar, transitional, and turbulent. In the transitional range, laminar, partially-developed laminar, and turbulent flow profiles are all test-fitted and the best fit is used as the profile model. This algorithm requires intensive computation, especially when a lot of iterations of integral calculations are involved.  
         [0043]     As mentioned above, for a known ultrasonic path and a known flow profile, a channel factor K relates the path velocity and the average velocity. In order to derive the average velocity from the measured path velocity, it is necessary to first derive the channel factor K for the particular path. In accordance with some embodiments of the invention, a program may be used to generate a database of channel factor K as functions of channel levels h, power-law factors N, and partial laminar factor r/R. The database can then be used as a lookup table to quickly identify, for a known channel level h, relationships between the channel factor K and the power law factor N or the partial laminar factor r/R.  
         [0044]     Referring to  FIG. 5 , for a turbulent flow or partially developed turbulent flow, channel factors K for all possible combinations of channel levels h (in the form of a percentage of the pipe radius R) and power-law factors N are pre-computed and put into two-dimensional data arrays. Similarly,  FIG. 6  shows, for a partially developed laminar flow, two-dimensional arrays of the K factors as functions of r/R and h/R. It can be seen that a pure laminar flow is a special case of a partial laminar flow when the radius of the flat region r is zero. For example, referring to  FIG. 6 , when the flat region does not exist (r/R=0), and when the ultrasonic path is diagonal (h/R=0), the K factor of the data point  61  has a value of 0.75.  
         [0045]     The two-dimensional channel factor arrays shown in  FIG. 5  and  FIG. 6  universally characterize flow measurements using ultrasonic paths. These K values may be saved as lookup tables in a database as part of a flow meter computer program. During a meter configuration stage, a set of channel levels h are derived from the positions of the transducer pairs. The program then looks up K i  function curves from the lookup tables for each ultrasonic path i.  
         [0046]     For illustration purposes, two sets of one-dimensional arrays sampled from the two-dimensional arrays with known channel levels are shown in  FIG. 7  and  FIG. 8 .  FIG. 7  shows turbulent flow K factors as functions of N. The four curves  721 - 724  correspond to the four different paths of channels  1 - 4 , and are obtained by looking up the two-dimensional arrays in  FIG. 5  for the four specific h/R values. For example, the curve  722  is the K factors for the channel with a diagonal path, and is derived from  FIG. 5  by setting an h/R=0.  
         [0047]      FIG. 8  shows the K factors as functions of d, for a partially developed laminar flow. The four curves  821 - 824  correspond to the four different paths of channels  1 - 4 , and are obtained by looking up  FIG. 6  for curves having the specified channel levels h. When d=0, i.e., when the flow is pure laminar, the K value measured at the diagonal path  822  is 0.75. On the other hand, when d=1, i.e., when the flow is a plug flow, all K values converge at a value of 1.  
         [0048]     In a typical straight pipe, transitional flow occurs when the Reynolds number is around 2300. This value may change depending on pipe conditions as well as the flow history. The presence of a flow conditioning device makes the flow profile modes even more unpredictable. Results from experiments have shown that a laminar flow can fully develop after passing through a flow conditioning device and before reaching the meter spool, if the Reynolds number is less than 1000. On the other hand, it is unlikely that a flow can re-establish a laminar profile at a Reynolds number greater than 5000 when a flow conditioning device is present. Based on these results, in accordance with one embodiment of the present invention, when the Reynolds number is between a lower limit, for example, 1000, and an upper limit, for example, 5000, a partial laminar model is used for fitting the flow profile. The lower limit of 1000 and upper limit of 5000 values will be used for illustration purposes in this description. One of ordinary skill in the art would appreciate that other limits may also be used without departing from the scope of the invention. For example, a wider range of Reynolds numbers, for example, from 500 to 10000, may be used to search for possible transitional profiles, if an accurate fluid viscosity is not available when performing measurements.  
         [0049]     During a flow measurement, in accordance with one embodiment of the invention, an initial estimate of an average flow velocity, and consequently an estimate of a Reynolds number is made. If the Reynolds number Re&gt;5000, it is likely that the flow is turbulent. The program then calculates an approximate value of N using Equation (5), based on the estimated Reynolds number Re. Within a range of estimated N, the program then searches the four curves in  FIG. 7  for an N that gives the best set of K factors, that is, the four derived V avg =V i *K i  values, based on the four measured V i &#39;s and the four K i &#39;s obtain from  FIG. 7  using the N value, having the least statistical variance. This newly derived N value is more accurate than that calculated using Equation (5). The searched N value may be fed back to the program for iterations in order to have a even better accuracy. The curve fitting may be based on best fit with the least statistical deviation. Those of ordinary skill in the art will recognize that many types of curve fitting methods can be used without departing from the scope of the present invention. Also, the number of paths can be more than 4, or fewer than 4.  
         [0050]     If on the other hand Re&lt;1000, the program fits the measured data to a laminar profile, using appropriate equations. If Re is between 1000 and 5000, and it is not certain that if the flow is laminar, partial laminar, or turbulent, the program tries to fit the measured data using the three different profiles, and finds the best fit that in turn determines the true flow profile. If, for example, a partial laminar model best fits the data, the program determines a d value from  FIG. 8 . From the pipe centerline to a distance r=dR, the flow is “flat.” Those skilled in the art will recognize that although the flow profile in the flat region is modeled using a constant velocity profile, more sophisticated models may be used to model a flow profile in the flat region that is different from a laminar flow profile.  
         [0051]     During the run time of the flow meter, because accessing lookup tables is much faster than doing real-time integration calculations, the data processing time would be dramatically reduced and the flow meter can be configured on site with different path arrangements. However, those of ordinary skill in the art would appreciate that embodiments of the invention may use either pre-computed lookup tables or a program for real-time computation.  
         [0052]     An method in accordance with one embodiment of the invention is illustrated in  FIG. 9 . The algorithm includes two states: the configuration state  91  and the runtime state  92 . During the configuration state  91 , in step  911 , a user determines a unique combination of paths and inputs this information into a computer program. Using two-dimensional channel factor arrays pre-computed in step  912 , the computer program derives, in step  913 , the channel factor curves for the specific path configuration, as described above with reference to  FIG. 7  and  FIG. 8 .  
         [0053]     During the runtime state  92 , the flow profile calculation component  921  first calculates a Reynolds number using Equation (6) and compares, in step  922 , the Reynolds number with a pre-determined range. If the Reynolds number is less than a lower limit of the predetermined range, the lower limit in this exemplary case being 1000, the program does a laminar flow calculation in step  923 . If the Reynolds number is greater than an upper limit of the predetermined range, the higher limit in this exemplary case being 5000, the program does a turbulent flow calculation in step  924 . If the Reynolds number is within the predetermined range, in this exemplary case being between 1000 and 5000, the program searches for a best fit (step  925 ), among laminar profiles, turbulent profiles, and partial laminar profiles, and then executes steps  923 ,  924 , or  926 , accordingly. The above steps may be iterated until the desired accuracy of the results is satisfied.  
         [0054]     Methods of the invention, as illustrated above, may be embodied in one or more computer programs suitable for field configuration over a large range of multi-path designs and flow conditioning device installations.  
         [0055]     Some embodiments of the invention relate to a multi-path ultrasonic flow meter. Referring to  FIG. 10 , where an exemplary multi-path ultrasonic spool piece with an asymmetric path layout design is illustrated. In this embodiment, a 4-path meter spool is shown on a cross sectional view. Four pairs of ultrasonic transducers,  101   a  and  101   b ,  102   a  and  102   b ,  103   a  and  103   b ,  104   a  and  104   b , are attached to the wall  105  of the pipe, each pair measuring the flow velocity of a specific path, such as the paths  41 - 44  in  FIG. 4A . The transducer pairs are connected to a control unit  106 , which may include an input/output control circuit  107 , an electronic digital computer or a central processing unit (CPU)  108 , and a computer readable medium  109 . The computer readable medium (e.g., memory)  109  may store a program containing instructions to execute a method in accordance with embodiments of the invention. In some embodiments, the computer readable medium  109  may also store a database including the pre-computed results in the form of lookup tables. Those of ordinary skill in the art will recognize that many possible embodiments exist for the control unit  106 , and the programs and/or databases may be stored or transferred using many different types of computer readable media, including, but not limited to, ROM, RAM, hard drive, floppy disk, CD, and flash drive.  
         [0056]     As mentioned above, at lease two paths, each having a different distance from the pipe centerline, are required to solve a common flow profile. In practice, however, more than two paths are often necessary to average out the randomness in measurements and to improve the resolution of profiles buried in random processes. On the other hand, the number of ultrasonic paths is always limited by dimensional restrictions and by cost.  
         [0057]     It is noted that in accordance with a preferred embodiment of the present invention, the multiple paths have an asymmetric distribution around the pipe centerline. Each of these paths has its distinctive channel level from the pipe centerline and provides unique information about the flow profile. In contrast, a meter spool with the same number of paths but using a symmetric path distribution would provide less information.  
         [0058]     Advantages of the present invention include one or more of the following: (a) accurate and fast flow measurements covering a wide range of flow profiles, including laminar flow, turbulent flow, and partial laminar flow; (b) less dependence on the selection of a conditioning device, and a shorter-than-standard distance between the conditioning device and the ultrasonic measuring area may be sufficient, benefiting users in the installation and the cost; (c) faster response as a result of using pre-computed lookup tables; (d) fewer ultrasonic paths and transducers are needed as a result of the asymmetric configuration of paths.  
         [0059]     While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be advised which do not depart from the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the attached claims.