Page 194 - Mechanical Engineers' Handbook (Volume 2)

9 Flow Rate 183

Yamanaka et al. 72 describe a velocity profile measuring technique using a time-domain
correlation method which they claim has two advantages over the conventional pulse Dopp-
ler: higher time resolution and avoidance of the Nyquist limit on maximum velocity.
Transit Time Meter Principles
Transit time ultrasonic flowmeters are discussed by Lynnworth, 73 who traces the history of
such meters back to the 1930s, and by Yoder 74 as part of a survey article. A transit time
ultrasonic flowmeter compares the transit times of pulses transmitted upstream and down-
stream and infers the average fluid velocity along the beam path from the difference of the
two transit times. Typically this is accomplished using pairs of transmitter–receiver units.
Each unit serves alternately as a transmitter and a receiver, sending out a burst and then
receiving a burst from the other unit. Commercial units are available. 75
The time required for the pulse to travel from the upstream transmitter to the downstream
receiver is shorter than the time required for the pulse originating at the downstream unit to
move upstream, against the flow velocity. This time difference, t, is the basis for the flow
76
measurement, as described in Efunda :
L L
t (t t ) (33)
u
d
a V cos a V cos
2VX
2V cos a 2
L (34)
2
2
2
2
2
a V cos 1 (V /a ) cos
2
If V/a 1 (flow Mach number 1), then
2VX
t
(35)
a 2
2
a t
V
(36)
2X
where a acoustic velocity in fluid
t upwind transit time
u
t downwind transit time
d
t t t d
u
L line-of-sight distance between transponders
angle between line of sight and pipe centerline
V fluid velocity (assumed uniform along line of sight)
X axial distance between transponders, L cos
Flood 71 introduces an additional term to account for the transmission time of the pulse
through the body of the transducer and the wall of the pipe, so that the calibration equation
can be based on the overall time difference. Eq. (36) deals only with the transit times within
the fluid and does not include the wall delay.
Both types of ultrasonic flowmeters use information gathered along the propagation path
(or paths) to infer the average flow over the entire flow area, as though the velocity were
uniform. Considerable effort has been put into the development of methods for dealing with
nonuniform flow.
Nonuniform Velocity
The simplest nonuniformities to deal with are those found in isothermal, fully developed
laminar or turbulent flows. These distributions are symmetric about the centerline and, if the