Page 195 - Mechanical Engineers' Handbook (Volume 2)

184 Temperature and Flow Transducers

propagation path intersects the centerline, a simple Reynolds-number-dependent calibration
factor could be used. Such flows are rare, however, in industry. More realistic flows present
more complex nonuniformities. Pipe bends, elbows, and tees lying in one plane generate
centrifugal forces that move the highest velocity fluid from the centerline of the pipe toward
the outside of each turn. This movement, in turn, causes secondary flows that roll up into
pairs of vortices near the inside walls of the turns. If the elbows and bends are ‘‘out of
plane,’’ the resulting velocity distributions may have several local maxima, not just one. Such
effects have been discussed by Ruppel and Peters, 77 who experimentally investigated the
effects of several upstream piping configurations on the accuracy of an ultrasonic flowmeter.
One approach to this problem has been to use several acoustic paths disbursed across the
cross section of the pipe. Flowmeters with up to five paths have been tested. There are also
different ways of ‘‘weighting’’ the data from the individual paths in forming the average.
A complex set of velocity distributions has been studied by Moore et al. 78 who used
closed-form descriptions of representative multimodal velocity distributions (constant den-
sity) to challenge a family of multipath transit time flowmeter designs. These distributions
were used as inputs to a program that calculated the ‘‘reading’’ of the flowmeter. These
readings were then compared to the ‘‘real’’ flow deduced by integrating the velocity distri-
bution over the flow area. Three different mathematical schemes were compared for weight-
ing the readings from different paths as well as different numbers of acoustic paths from 2
to 5. For each flowmeter configuration, 90 trials were made, with the velocity profile being
rotated 2 degrees each time. The range of calibration factors (maximum to minimum) for
these 90 trials was recorded as a measure of the orientation sensitivity of the configuration.
The authors offer this approach as a good tool for optimizing the design of multipath flow-
meters.
Yeh et al. 79 combined a four-path transit time ultrasonic flowmeter with pattern recog-
nition software and a ‘‘learning algorithm.’’ The system was ‘‘educated’’ on a set of exper-
imental and computational flow fields and ‘‘learned’’ to recognize and accommodate flow
fields and adjust its calibration according to the flow field it encountered.

Temperature Gradients
If there is heat transfer to or from the fluid, the resulting density gradient is capable of
distorting the flow calibration even in a straight run of pipe with fully developed flow. In
situations with bends, the distortions may be worse with heat transfer than the isothermal
cases.
80
Willatzen looked at the effect of the radial density gradient resulting from heat transfer
out of a water flow with a fully developed parabolic velocity distribution. The author con-
sidered a system in which idealized (permeable) transducers of different diameters centered
within the pipe generated ultrasonic waves. The smallest diameter transducers generated
waves containing up to eight modes; the largest (R/R 1) produced only the fundamental
0
mode. In the absence of density gradients, all modes except the fundamental overestimated
the mean flow while excitation of the fundamental mode alone gave error-free measurements.
The predicted error was zero when the transducer disks occupied the entire pipe cross section
(fundamental mode only). With a radial temperature gradient in the water (heat transfer out
of the fluid into the surroundings) the indicated flow was low by about 2% even with only
the fundamental mode excited.
Pulsing Flow
Pulsations in the flow can cause errors if the pulsation frequency aliases with the meter pulse
frequency. This topic has been treated by Berrebi et al. 81 and by Carlander and Delsing, 82
who reported experimentally measured errors of more than 2–4% induced by unsteady pres-