Chemical Engineering November 2012 - 40

Pump curves with all static head
Feature Report
but not the same pump whose curves
are shown in Figure 4. Pumps are usually
selected such that the rated point
falls at a lower flowrate than the best
efficiency point of the pump. Accordingly,
when the operating point backs
up on the pump curve, the efficiency
decreases. The operating point for the
pump with the VFD is a little closer to
the best efficiency point than that of
the pump operating at fixed speed, so
the pump with the VFD is predicted to
have a slightly higher efficiency. The
formula for the hydraulic power of a
pump is given [13] as follows with the
numerical constant replaced by C1 to
make the equation independent of a
specific set of units:
W =
()∆
1
PQ
C
(4)
Where W is power (typically kW or
hp) and ∆P is pressure drop (typically
kPa, kg/cm² or psi). Equation (4) is restated
with efficiencies introduced to
give electrical power.
W =
C εε εPM V
()∆
1
PQ
(5)
Where P is pump efficiency, M is
motor efficiency and V is VFD efficiency.
The VFD loss has been restated
as an efficiency for consistency with
the other terms. Equation (5) shows
why the advantage for the VFD is
small. The difference in pressure drop
between the two cases is small. Note
that the zero in Figure 4 has been suppressed
to show the differences clearly.
The pump with the VFD has a slightly
higher efficiency. However, the constant
loss of the VFD translates into a
decreasing efficiency when the power
is reduced. This works against the two
advantages of the VFD case. In any
event, the differences are small.
Now, consider the case of all-frictional
pressure drop and small differences
are most emphatically not the
case. The Darcy equation for frictional
pressure drop in pipe [14] is used to
calculate the system curve and is
stated as follows:
∆P =
CfLQ
d
2 ρ 2
5
(6)
Where ∆P is pressure drop (usually
120
110
100
90
80
70
60
1
2
Rate, m3/h
3
4
5
2
1
FIGURE 4.
Pump curves for
the case of all
static pressure
drop show the
limited opportunity
for energy
savings (violet
is fi xed speed
pump, green and
blue are with a
VFD and red is
the system curve)
kPa, kg/cm², or psi); C2 replaces the
numerical constant to make the equation
independent of a set of units. The
subscript merely emphasizes that the
constant is numerically different from
other constants; f is the friction factor;
L is the length or the equivalent
length, (usually m or ft);  is density
(usually kg/m³ or lb/ft³); and the internal
diameter is d (usually mm or in.).
The formula for converting pressure
drop to head loss, which will be used
in pump calculations, is:
∆P =
H
C
ρ
3
(7)
Where H is the head loss (usually m
or ft). Combining to state the pressure
drop as head loss gives:
H =
CfLQ
d
4
5
(8)
The friction factor is constant for welldeveloped
turbulent flow, which is the
usual situation for pump circuits. The
length is used as the equivalent length
of all pipe, fittings and equipment. It
is constant for a given circuit and so is
the diameter of the pipe. Equation (8)
reduces to:
HC5Q=
2
(9)
Which is a familiar relationship.
When the flow is reduced, the head
loss, or ∆P, is reduced by the square of
the flow. This provides an excellent opportunity
for saving power as shown
by Equation (5). Also, if Equation (9)
is applied at two points and the equations
divided, the result is:
H
H
2
1
=
Q
Q
2
2
2
1
(10)
Where the subscripts indicate the
points. Equations (9) and (10) apply
34 CHEMICAL ENGINEERING WWW.CHE.COM NOVEMBER 2012
2
to the system curve. Equation (10) is
identical to Equation (3), the affinity
law; consequently the same equation
also connects corresponding points on
the pump curves for different speeds.
The pump curves and system curve,
shown in Figure 6, illustrate the
greater potential for energy savings
when the head loss is all frictional.
As with the curves for the situation
with all static loss, the violet pump
curve represents the case with a fixed
pump speed and a control valve. The
green pump curve represents the case
with a VFD reducing the speed of the
pump. The red curve represents both
the system curve and the curve that
connects corresponding points on the
pump curves. Point 1 is an operation
at a reduced flowrate, where the double-headed
arrow indicates the head
loss being consumed by the control
valve as in the case with the control
valve. The operating point has
backed up on the pump curve, like
the case with the all-static pressure
drop, and the pump efficiency has
declined similarly. In the case with
the VFD, the operating point at the
lower speed corresponds to the rated
point, therefore the pump efficiency
is about the same.
An examination of the efficiencies
of a few API 610 pumps at standard
fixed-speeds showed that the efficiency
declined by less than 2% for a
50% reduction in speed and less than
3% for a 75% reduction. Given the few
points considered and the considerable
scatter, a linear relation through the
points stated is as good of a representation
as is justified in this range, but
it seems obvious that the relationship
could not be approximately linear over
a longer range. The percent reduction
stated is a percentage of the percent
efficiency. The reduction in efficiency
agrees roughly with Shukla and others
[6] who cite a 3-4% reduction for
Head, m
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Chemical Engineering November 2012

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