Water with
a density of 1,000kg/m3 flow
through a 2.00cm diameter pipe with
velocity of 24 onls. What
volume
flow rate?
Answers
Answer:
Flow rate Q is defined to be the volume of fluid passing by some location through an area during a period of time, as seen in Figure 1. In symbols, this can be written as
Q
=
V
t
,
where V is the volume and t is the elapsed time. The SI unit for flow rate is m3/s, but a number of other units for Q are in common use. For example, the heart of a resting adult pumps blood at a rate of 5.00 liters per minute (L/min). Note that a liter (L) is 1/1000 of a cubic meter or 1000 cubic centimeters (10-3 m3 or 103 cm3). In this text we shall use whatever metric units are most convenient for a given situation.
The figure shows a fluid flowing through a cylindrical pipe open at both ends. A portion of the cylindrical pipe with the fluid is shaded for a length d. The velocity of the fluid in the shaded region is shown by v toward the right. The cross sections of the shaded cylinder are marked as A. This cylinder of fluid flows past a point P on the cylindrical pipe. The velocity v is equal to d over t.
Figure 1. Flow rate is the volume of fluid per unit time flowing past a point through the area A. Here the shaded cylinder of fluid flows past point P in a uniform pipe in time t. The volume of the cylinder is Ad and the average velocity is
¯¯¯
v
=
d
/
t
so that the flow rate is
Q
=
Ad
/
t
=
A
¯¯¯
v
.
EXAMPLE 1. CALCULATING VOLUME FROM FLOW RATE: THE HEART PUMPS A LOT OF BLOOD IN A LIFETIME
How many cubic meters of blood does the heart pump in a 75-year lifetime, assuming the average flow rate is 5.00 L/min?
Strategy
Time and flow rate Q are given, and so the volume V can be calculated from the definition of flow rate.
Solution
Solving Q = V/t for volume gives
V = Qt.
Substituting known values yields
V
=
(
5.00
L
1 min
)
(
75
y
)
(
1
m
3
10
3
L
)
(
5.26
×
10
5
min
y
)
=
2.0
×
10
5
m
3
.
Discussion
This amount is about 200,000 tons of blood. For comparison, this value is equivalent to about 200 times the volume of water contained in a 6-lane 50-m lap pool.