The present invention relates to ultrasonic flow measurement systems for measuring the flow of fluid in a conduit, and has particular utility in systems such as clamp-on transducer systems, or customized spool pieces having defined transducer locations that determine signal paths through the fluid.  In particular, the invention is applicable to situations wherein parameters such as Reynolds number or frictional properties of the conduit interior that affect the actual flow profile, may be unknown, obscuring the relationship between the ultrasonic signal data and the actual flow rate. 
In such circumstances, it is customary to perform a number of flow simulations, or to test the actual system in which the flow meter is to be installed, in order to determine a meter factor for converting transit time measurements to flow measurements in use.  However this procedure can be effectively carried out only for certain relatively common flow situations and materials.  As a practical matter, it is generally not possible to directly experiment on, measure and correlate various flow and transit time readings occurring during actual operation in which fluids of different types at various temperatures, pressures or flow velocities are being passed through conduits during the course of an industrial process, such as a refining or chemical reaction process.  Since it is generally not feasible to directly measure either the flow, or system parameters such as the friction factor, ultrasonic measurements are often carried out by a process of informed guesswork to set an appropriate meter factor or other relationship which converts the ultrasonic measurements to a flow velocity, average flow rate or other quantity which is to be measured. 
When the system output is to be used as an input to a process controller, it is only necessary to detect some functional flow input parameter, and this suffices since the control laws are modified on an heuristic basis to control the process conditions.  However, when it is necessary to accurately measure the actual flow, whether for economic or technical reasons, this approach leaves much to be desired. 
It would therefore be desirable to provide an ultrasonic flow measurement system having improved accuracy. 
It would also be desirable to provide a flow measurement system capable of determining friction factor. 
It would also be desirable to provide an ultrasonic measurement system configured to produce stable and repeatable results in diverse flow situations. 
These and other desirable traits are provided in a method and system for ultrasonic flow measurement which operates by transmitting a signal along one or more paths through a fluid and determining the transit time of the signal along the paths.  Flow in the conduit is described by first variables related to a coordinate system, and by second variables not related to a coordinate system, wherein the first variables are separable or substantially separable from the second variables.  The method includes the step of transmitting signals along a first path and gong a second path which differs from the first to determine the average velocities of the fluid on the paths, and combining the first path measurement and the second path measurement to form an expression independent of the variables of the second type.  In one embodiment, the first and second path measurements are combined in accordance with the velocity relationships given by the Pao equation relating velocity profile to friction, to produce a flow measurement which is independent of the friction factor.  In alternative embodiments, a power law flow distribution is assumed.  A single calibration then allows the meter to be used on diverse conduits with different wall roughness or friction factor.  Once the two paths have been selected to produce a measure free of dependence on friction, the flow may be accurately determined from these two paths, and a measurement may be taken along a single path to yield a measure of the friction factor for a given conduit, thus allowing the non-invasive measurement of this parameter.  Preferably, this technique is then used to set up a conventional flow meter by solving for flow profile and analytically determining a meter factor or appropriate measurement path. 
The measured transit times along the first and second paths may be combined in accordance with either a compiled, or a theoretical, model of the velocity distribution to determine a chord on which the measured velocity equals the mean flow velocity across the conduit cross-sectional area.  Alternatively, the transit times may combined in accordance with the model of velocity profile to determine a meter factor, or the profile relation may be used to determine a single chord along which the measurements are independent of the second variable.  Preferably the two paths used for eliminating the second variable differ substantially in their path average velocity. 
In accordance with another aspect of the invention, the variables of the second type may be eliminated by taking a diametral transit time measurement, together with one or more point velocity measurements, or by taking a reflected path measurement together with one or more point measurements.  Each point velocity measurement may be effected using a pitot tube situated at the point. 
A detailed example follows for the analytic solution assuming the Pao relationship.