Tokyo Keiki UFP-20, Installation & Operation Manual

Page: 198 / 218

3-13
( )
τ
θ θ
+
⋅ + ⋅
=
cos sin V C
d
td
(2)
Conversely, the following formula can be written when the propagation time of an
ultrasonic pulse from P2 to P1 (in other words the reverse direction of the fluid) is
defined as tu.
( )
τ
θ θ
+
⋅ − ⋅
=
cos sin V C
d
tu
(3)
In these formulas d is the internal diameter of the pipe,
θ
is the angle between the
ultrasonic pulse advance and the flow direction, and
τ
is the fixed delay time (sum of
time required for the pulse to travel through the wedge, pipe wall, and lining and the
electronic delay time of the flowmeter).
Since the velocity of sound in water C is much larger than flow rate V, the following
assumption can be made: C
2
>V
2
cos
2
θ
Therefore, when the propagation time difference ∆t = tu-td is calculated, the following
formula can be deduced from formulas (2) and (3).
( )
2
cos sin 2
C
V d
td tu t
θ θ ⋅ ⋅ ⋅
= − = ∆
(4)
If the propagation time in still water is defined as to, formula (5) can be deduced from
formulas (2) and (3).
τ
θ
+ =
+
=
C
d
td tu
to
sin
2
(5)
The following is then obtained by substituting formula (4) into the above formula.
( )
θ
θ τ
sin
cos 2
2
d
V to
t
⋅ ⋅ − ⋅
= ∆
(6)
Finally, the following is obtained by solving for V in formula (6).
( ) ( )
t
to
d
t
to
d
V ∆ ⋅
− ⋅ ⋅ ⋅
= ∆ ⋅
⋅ − ⋅
=
2 2
cos sin 2 cos 2
sin
τ θ θ θ τ
θ
(7)
Since the flow velocity V obtained by the ultrasonic flowmeter is an average velocity
through the diameter between the transducers, the actual average velocity
V
is
different. The ratio between these 2 velocities is expressed using flow volume correction
coefficient k, as shown below.