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13:4:37 Page 424
The early Romans developed elaborate water systems to supply public baths and private homes. In
fact, Sextus Frontinius (A.D. 40–103), commissioner of water works for Rome, authored a treatise on
design methods for urban water distribution systems. Evidence suggests that Roman designers
understood correlation between volume flow rate and pipe flow area. Weirs were used to regulate
bulk flow through aqueducts, and the cross-sectional area of terra-cotta pipe was used to regulate
fresh running water supplies to individual buildings.
Following a number of experiments conducted using olive oil and water, Leonardo da Vinci
(1452–1519) first formally proposed the modern continuity principle: that duct area and fluid
velocity were related to flow rate. However, most of his writings were lost until centuries later, and
Benedetto Castelli (ca. 1577–1644), a student of Galileo, has been credited in some texts with
developing the same steady, incompressible continuity concepts in his day. Isaac Newton (1642–
1727), Daniel Bernoulli (1700–1782), and Leonhard Euler (1707–1783) built the mathematical and
physical bases on which modern flow meters would later be developed. By the nineteenth century,
the concepts of continuity, energy, and momentum were sufficiently understood for practical
exploitation. Relations between flow rate and pressure losses were developed that would permit the
tabulation of the hydraulic coefficients necessary for the quantitative engineering design of many
modern flow meters.
10.3 FLOW RATE CONCEPTS
The flow rate through a conduit, be it a pipeline, duct, or channel, depends on fluid density, average
fluid velocity, and conduit cross-sectional area. Consider fluid flow through a circular pipe of radius
r 1 and having a velocity profile at some axial pipe cross section given by uðr; uÞ. The mass flow rate
depends on the average mass flux flowing through a cross-sectional area; that is, it is the average
product of fluid density times fluid velocity and area, as given by
_
m ¼
ðð
A
ruðr; uÞdA ¼ rUA
ð10:1Þ
The pipe area is simply A ¼ pr
2
1 . We see that to directly make a mass flow rate measurement, the
device must be sensitive to the area-averaged mass flux per unit volume, rU , or to the fluid mass
passing through it per unit time. Mass flow rate has the dimensions of mass per unit of time (e.g.,
units of kg/s, lb m /s, etc.).
The volume flow rate depends only on the area-averaged velocity over a cross section of flow as
given by
Q ¼
ðð
A
udA ¼ UA
ð10:2Þ
So to directly measure volume flow rate requires a device that is sensitive either to the average
velocity, U , or to the fluid volume passing through it per unit time. Volume flow rate has dimensions
of volume per unit time (e.g., units of m
3 /s, ft
3 /s, etc.).
The difference between Equations 10.1 and 10.2 is quite significant, in that either requires a
very different approach to its measurement: one sensitive to the product of density and velocity or to
mass rate, and the other sensitive only to the average velocity or to volume rate. In the simplest case,
where density is a constant, the mass flow rate can be inferred by multiplying the measured volume
flow rate by the density. But in the metering of many fluids, this assumption may not be good enough
424 Chapter 10 Flow Measurements
13:4:37 Page 424
The early Romans developed elaborate water systems to supply public baths and private homes. In
fact, Sextus Frontinius (A.D. 40–103), commissioner of water works for Rome, authored a treatise on
design methods for urban water distribution systems. Evidence suggests that Roman designers
understood correlation between volume flow rate and pipe flow area. Weirs were used to regulate
bulk flow through aqueducts, and the cross-sectional area of terra-cotta pipe was used to regulate
fresh running water supplies to individual buildings.
Following a number of experiments conducted using olive oil and water, Leonardo da Vinci
(1452–1519) first formally proposed the modern continuity principle: that duct area and fluid
velocity were related to flow rate. However, most of his writings were lost until centuries later, and
Benedetto Castelli (ca. 1577–1644), a student of Galileo, has been credited in some texts with
developing the same steady, incompressible continuity concepts in his day. Isaac Newton (1642–
1727), Daniel Bernoulli (1700–1782), and Leonhard Euler (1707–1783) built the mathematical and
physical bases on which modern flow meters would later be developed. By the nineteenth century,
the concepts of continuity, energy, and momentum were sufficiently understood for practical
exploitation. Relations between flow rate and pressure losses were developed that would permit the
tabulation of the hydraulic coefficients necessary for the quantitative engineering design of many
modern flow meters.
10.3 FLOW RATE CONCEPTS
The flow rate through a conduit, be it a pipeline, duct, or channel, depends on fluid density, average
fluid velocity, and conduit cross-sectional area. Consider fluid flow through a circular pipe of radius
r 1 and having a velocity profile at some axial pipe cross section given by uðr; uÞ. The mass flow rate
depends on the average mass flux flowing through a cross-sectional area; that is, it is the average
product of fluid density times fluid velocity and area, as given by
_
m ¼
ðð
A
ruðr; uÞdA ¼ rUA
ð10:1Þ
The pipe area is simply A ¼ pr
2
1 . We see that to directly make a mass flow rate measurement, the
device must be sensitive to the area-averaged mass flux per unit volume, rU , or to the fluid mass
passing through it per unit time. Mass flow rate has the dimensions of mass per unit of time (e.g.,
units of kg/s, lb m /s, etc.).
The volume flow rate depends only on the area-averaged velocity over a cross section of flow as
given by
Q ¼
ðð
A
udA ¼ UA
ð10:2Þ
So to directly measure volume flow rate requires a device that is sensitive either to the average
velocity, U , or to the fluid volume passing through it per unit time. Volume flow rate has dimensions
of volume per unit time (e.g., units of m
3 /s, ft
3 /s, etc.).
The difference between Equations 10.1 and 10.2 is quite significant, in that either requires a
very different approach to its measurement: one sensitive to the product of density and velocity or to
mass rate, and the other sensitive only to the average velocity or to volume rate. In the simplest case,
where density is a constant, the mass flow rate can be inferred by multiplying the measured volume
flow rate by the density. But in the metering of many fluids, this assumption may not be good enough
424 Chapter 10 Flow Measurements