|
Water Velocity: Its Impact on the Accuracy of Hydraulic
Calculations
September 1996
By: Roland Huggins, P.E.
Does excessive water velocity cause inaccuracies that are
sufficient enough to invalidate hydraulic calculations based on the Hazen-Williams
equation? I wouldn't be surprised if you've never had a problem with
this issue since it is often ignored by the industry but the NFPA sprinkler
Handbook does identify it as a concern. While the handbook also identifies
that NFPA 13 does not stipulate a maximum limit on velocities, it does
state:
"Twenty feet per second (ft/s) maximum velocities may result in
more conservative designs."
At velocities beyond this, the Hazen-Williams formula is not as conservative
as other methods for certain pipe types, such as Darcy-Weisbach.Even
though NFPA does not specify a value, other agencies, such as Factory
Mutual and Wausau, both use 20 ft/s and some governmental agencies, such
as Naval Facilities, use 32 ft/s.The primary objective of this evaluation
is to quantify the actual impact water velocity has on the accuracy of
the calculations, thus providing a basis for those AHJs who intend to
implement a maximum water velocity. I've also been curious for some time
about the actual impact velocity has on the accuracy of hydraulic calculations.It
should be noted that this evaluation does not address the impact of velocity
on the use of total pressure versus normal pressure in hydraulic calculations.
Background
This evaluation is based on comparing
calculated friction losses.As we all know, as water
flows through a pipe, there is a loss of energy due
to friction. This loss occurs because the water particles
rub against the pipe as well as each other.Technically,
this loss is not constant, but changes with the water
velocity. There are three primary phases of water velocity
that affect friction loss. These phases are laminar,
transition, and turbulent flows. In the laminar and
transition phases the friction loss factor changes
drastically with water velocity. These phases occur
at very low flow rates, with laminar flow occurring
in a 1 inch steel pipe at rates less than 1 gpm and
transition flow at less than 2 gpm. Fortunately, the
flows encountered in the sprinkler industry are only
in the turbulent phase, where the impact on the friction
loss factor is relatively linear.In estimating friction
losses, some equations account for this change, while
others do not.
It is worth noting that even though we
talk about the calculated loss, which implies an
absolute accuracy, the calculation is only a mathematical
estimation of the real losses.
The friction loss has been calculated by the Hazen-Williams
equation and then compared to that obtained by the Darcy-Weisbach equation
for a range of flow rates and pipe sizes.
These equations are:
Darcy-Weisbach: hL = f x v2
2
x D x g
hL: friction loss (ft)
f: friction factor
v: water velocity (ft/s)
D: pipe diameter (ft)
G: gravitational constant (ft/s2)
Hazen-Williams:
p = 4.52 x Q1.85
C1.85 x
D1.85
p: friction loss (psi)
Q: flow rate (gpm)
C: friction factor
D: pipe diameter (in)
The Darcy-Weisbach equation was selected
as the benchmark since it is commonly accepted as
providing a more accurate estimate of energy loss
due to fluid flow, hereafter referred to as friction
loss. This position is supported by the Sprinkler
Handbook as indicated in the introduction of this
article. The reason it is viewed as more accurate
is that it is based on scientific principles (the
continuity equation) and accounts for changes such
as water velocity and viscosity; whereas, the Hazen-Williams
equation, being empirically based from the observations
of many pipeline flows, treats the effect of these
variables as a constant.
One of the main problems with the
Darcy-Weisbach equation is that it is much more difficult to apply.
For instance, the friction factor is not a constant for the same pipe
material but changes with the water velocity as well as the pipe diameter.
The typical method for determining the friction factor is the use of
the Moody diagram (see Figure 1). As shown by the diagram, as the Reynolds
number increases (due to an increase in water velocity), the friction
factor decreases. A severe restriction on its use is that there is
little or no information readily available on the roughness of pipe
material except for new piping. Hazen-Williams uses the same friction
factor (the "C" value)
based solely on pipe material, regardless of pipe size or water velocity.
Though more difficult to apply, the Darcy-Williams equation can be used
for any fluid, at any temperature, whereas the Hazen-Williams equation
is applicable to only water and is most accurate when the water temperature
is near 60° F.
Results
The results from
the two equations were compared over a flow range
of 10 feet per second (ft/s) to 40 ft/s. Friction
factors were based on the relative roughness for
new pipe which is determined by dividing the specific
roughness by the pipe diameter. The assigned specific
roughness, as indicated by a pipe manufacturer,
was 0.00015 ft and the Hazen-Williams "C" factor
was 140. As indicated in the Sprinkler handbook new
steel pipe actually has a "C" value of
140 but 120 is used in standard hydraulic calculations
to account for the reduced "C"
value of aged pipe. In order to evaluate the impact
of the roughness of the material, both steel (schedule
40) and copper (type M) were evaluated.Even though
copper is called tubing, for simplicity it will hereafter
be referred to as pipe.The specific roughness assigned
to the copper was 0.000005 ft and the "C" factor
was 150. The pipe diameter was also varied from 1
inch to 4 inches. As final parameter needing identification
is that the water viscosity was determined based
on a temperature of 60° F.
See Figure 2 for the results of the
friction loss calculations.Within this figure the columns showing the
friction loss are identified as Loss/ft Ð HW
for Hazen-Williams and DW for Darcy-Weisbach. The indicated value for
Darcy-Weisbach was converted from ft to psi by multiplying it by 0.434.
Since the Darcy-Weisbach is being used as the benchmark, the percentage
that Hazen-Williams varies from it is identified as Difference HW/DW,
which is actually (HW-DW)/DW. The percentages indicated were calculated
using the whole number, not the rounded-off values shown in the figure.
A negative value indicates Hazen-Williams has estimated a lower friction
loss.
Discussion
In discussing the results, references to deviation generally
relates to the difference by percentage of the friction loss calculated
by the Hazen-Williams equation when compared to that by Darcy-Weisbach.
It is also worth mentioning that a seemingly large percentage deviation
is not always significant, particularly when evaluating small numbers.
For instance, 0.003 psi deviates from .0025 psi by 20% but there is only
a real difference of 0.0005 psi. Since the roughness of the material
completely changes the pattern of the results, steel and copper are addressed
separately.
In looking at the results for steel pipe, the only thing
that is clearly indicated is that an increase in velocity always increases
the deviation. Unfortunately, (from perspective of identifying a maximum
value), as the velocity increases, the amount of deviation generally
decreases. For example, in a 1 inch pipe there is a 2.45 increase in
percentage between 10 ft/s and 15 ft/s but between 35 ft/s and 40 ft/s
the increase is only 0.89. The most prominent issue is that at 10 ft/s
1-inch pipe deviates 11.2% and 4 inch deviates 2.7% while at 40 ft/s
1- inch deviates 21.9% and 4 inch 14.8%.These are large percentages but
presented as a real numbers, 40 ft/s in a 1-inch pipe cerates a deviation
of 0.6 psi but in the 4-inch pipe it is only 0.08 psi.
There are two items of interest that
I would also like to address. We've identified that, up to a point,
the smaller the pipe size, the greater the deviation.
It appears though, that for pipe sizes starting at 3 inches, there
is a consistent deviation. This eliminates, beyond this point,
pipe diameter as an influential parameter. It is also worth noting
that steel pipe has a negative deviation indicating a lower calculated
friction loss, which is less conservative.
As already mentioned the roughness
of the material is important since smoother material experiences less
deviation. For example, at 40 ft/s 1 inch steel deviates 21.9%, whereas
copper is only 4.8% and 4 inch steel is 14.8% with copper at 7.5%.
Allow me to revert to steel pipe momentarily as it relates to pipe
roughness. Even though this evaluation
did not review existing pipe with lower "C" values, it does
support the position that Hazen-Williams becomes less accurate for lower "C" values.The
SFPE Handbook of Fire Protection Engineering, Second Edition, indicates
that lower "C" values below 100 consists of a 5% reduction
when the velocity exceeds 3 ft/s, with an additional 5 % reduction each
time the velocity doubles.
Copper does not produce a significant deviation but it does
produce an interesting change of pattern in the results. The rougher
steel pipe had a negative deviation, whereas copper has a positive deviation
(except for 1-inch pipe at 10 ft/s which has a deviation of less than
1%). This indication of additional friction loss actually provides a
conservative estimate. Another shift is that the larger the diameter,
the greater the deviation. Again, one should greater the deviation in
percentage by the change in actual values. In this case, at 40 ft/s 4
inch copper deviates 7.5% but the actual difference in pressure is only
0.03 psi.
Conclusions
There is not a single velocity that
stands out as a logical maximum, when applied to the different pipe
sizes and materials, which would allow us to say "stop here, do not exceed."
I wouldn't even call the variances large, except for 1-inch steel pipe
where we see up to a 22% variance.Even though this is a large variance,
in real numbers it only represents a difference of 0.6 psi. Considering
the lack of accuracy of the water supply data, which is typically no
better than plus or minus 10 psi (anyone who has performed a water flow
test and seen how much the pressure gauge bounces, knows what I mean),
is 0.6 psi significant? I wouldn't think so, particularly when the high
deviation is typically accompanied by a high friction loss (discussed
further below). The important thing is the use of a consistent methodology
throughout the industry. I don't want to imply that allowing a design
to go over by 0.6 psi is acceptable, since this would interfere with
the consistency of the methodology. The theoretical, calculated difference
of 0.6 psi is what I'm referring to.
It doesn't seem appropriate to have
a single maximum velocity applied to all pipe size when only the smaller
diameter pipe is adversely affected. After all, what do we gain by
having a velocity restriction of 40 ft/s in a 4-inch steel pipe? Sure
the deviation is 14.8% (sounds big) but the actual pressure difference
is only 0.08 psi (yes, in the hundredths). Thus, we would be restricting
the velocity of all pipe sizes when only the smallest diameter pipes
are moderately affected. Or for the sake of theoretical purity, should
we restrict the deviation to a specified percentage? This also does
not seem appropriate since large percentages often do not reflect a
meaningful real value.
>Another method could be assigning a maximum real value allowed for deviation
such as 0.2 psi.
The problem with this approach is that there will be a different maximum
velocity for each pipe size as well as for each type pipe material. Would
there be sufficient improvement in sprinkler performance to warrant the
increase in confusion? I wouldn't think so.
There is a naturally occurring restriction which is called
friction loss. In small diameter pipe where a high velocity creates the
greatest variance, the friction loss is also the greatest. Few sprinkler
designs can tolerate a friction loss of 2.2 psi/ft (1 inch steel at 40
ft/s) which creates a 22 psi loss between sprinklers at a 10 ft. spacing
(and that's with no equivalent lengths for fittings). From the perspective
of accuracy of the friction loss equations, this seems the best method.
There is another drastic method for obtaining a perceived
increase in accuracy, and that is to start using the Darcy-Weisbach instead
of the Hazen-Williams equation. There are several items which indicated
this approach is undesirable particularly for standard sprinkler systems.
The first item is that even though the Darcy-Weisbach equation is more
accurate for specific conditions, many of these conditions for a sprinkler
system will change over time.This offsets the accuracy since an average
must now be assumed. A second item is the extensive amount of information
accrued on the Hazen-Williams equation, particularly for existing piping.
A final item is the successful performance of current systems, which
demonstrates an acceptable degree of accuracy. Thus, there is no reason
to change to a different equation for performing hydraulic calculations
for sprinkler systems.
In closing, there is not sufficient deviation to warrant
a maximum water velocity of 20 ft/s simply for the sake of a more conservative
design nor does a high velocity create sufficient inaccuracies to invalidate
hydraulic calculations based on the Hazen-Williams equation.
Figure 2 Result of the friction loss calculations.
4" Schedule 40 Steel
Pipe (4.026 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
397
|
140
|
0.035
|
0.018
|
0.036
|
-2.68%
|
15
|
595
|
140
|
0.075
|
0.175
|
0.079
|
-5.80%
|
20
|
794
|
140
|
0.127
|
0.174
|
0.14
|
-9.26%
|
25
|
992
|
140
|
0.192
|
0.173
|
0.217
|
-11.74%
|
30
|
1190
|
140
|
0.269
|
0.171
|
0.309
|
-13.12%
|
32
|
1270
|
140
|
0.303
|
0.171
|
0.352
|
-13.96%
|
35
|
1389
|
140
|
0.357
|
0.169
|
0.416
|
-14.10%
|
40
|
1587
|
140
|
0.457
|
0.167
|
0.537
|
-14.79%
|
|
|
|
|
|
|
|
3" Schedule 40 Steel
Pipe (3.068 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
230
|
140
|
0.048
|
0.0188
|
0.05
|
-2.41%
|
15
|
346
|
140
|
0.102
|
0.0185
|
0.11
|
-6.68%
|
20
|
461
|
140
|
0.174
|
0.0183
|
0.193
|
-9.65%
|
25
|
576
|
140
|
0.263
|
0.018
|
0.297
|
-11.16%
|
30
|
691
|
140
|
0.369
|
0.0178
|
0.422
|
-12.59%
|
32
|
737
|
140
|
0.416
|
0.0178
|
0.48
|
-13.43%
|
35
|
806
|
140
|
0.491
|
0.0177
|
0.572
|
-14.10%
|
40
|
922
|
140
|
0.628
|
0.0175
|
0.738
|
-14.85%
|
|
|
|
|
|
|
|
2" Schedule 40 Steel
Pipe (2.067 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
105
|
140
|
0.077
|
0.021
|
0.082
|
-6.57%
|
15
|
157
|
140
|
0.163
|
0.0202
|
0.178
|
-8.60%
|
20
|
209
|
140
|
0.277
|
0.0199
|
0.311
|
-11.14%
|
25
|
261
|
140
|
0.418
|
0.0196
|
0.479
|
-12.75%
|
30
|
314
|
140
|
0.586
|
0.0195
|
0.687
|
-14.67%
|
32
|
335
|
140
|
0.66
|
0.0194
|
0.777
|
-15.05%
|
35
|
366
|
140
|
0.779
|
0.0193
|
0.925
|
-15.75%
|
40
|
418
|
140
|
0.998
|
0.0192
|
1.202
|
-17.00%
|
|
|
|
|
|
|
|
1" Schedule 40 Steel
Pipe (1.049 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
27
|
140
|
0.17
|
0.0248
|
0.191
|
-11.22%
|
15
|
40
|
140
|
0.359
|
0.024
|
0.416
|
-13.67%
|
20
|
54
|
140
|
0.612
|
0.0238
|
0.734
|
-16.62%
|
25
|
67
|
140
|
0.925
|
0.0235
|
1.132
|
-18.34%
|
30
|
81
|
140
|
1.296
|
0.0232
|
1.61
|
-19.51%
|
32
|
86
|
140
|
1.46
|
0.0232
|
1.831
|
-20.29%
|
35
|
94
|
140
|
1.723
|
0.0231
|
2.181
|
-21.01%
|
40
|
108
|
140
|
2.206
|
0.0229
|
2.825
|
-21.90%
|
|
|
|
|
|
|
|
4" Type M Copper Tubing
(3.935 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
379
|
150
|
0.032
|
0.015
|
0.031
|
3.19%
|
15
|
569
|
150
|
0.067
|
0.014
|
0.065
|
4.04%
|
20
|
758
|
150
|
0.115
|
0.0134
|
0.11
|
4.11%
|
25
|
948
|
150
|
0.173
|
1.0128
|
0.164
|
5.40%
|
30
|
1137
|
150
|
0.243
|
0.0124
|
0.229
|
5.87%
|
32
|
1213
|
150
|
0.274
|
0.0123
|
0.259
|
5.70%
|
35
|
1327
|
150
|
0.323
|
0.0121
|
0.305
|
6.01%
|
40
|
1516
|
150
|
0.413
|
0.0117
|
0.385
|
7.46%
|
|
|
|
|
|
|
|
1" Type M Copper Tubing
(1.055 inch)
|
|
|
|
|
Velocity
|
Q
|
C
|
Loss/ft
|
Fric Factor
|
Loss/ft
|
Difference
|
(ft/s)
|
(gpm)
|
HW
|
HW
|
DW
|
DW
|
HW/DW
|
10
|
27
|
150
|
0.148
|
0.0195
|
0.149
|
-0.71%
|
15
|
41
|
150
|
0.314
|
0.018
|
0.31
|
1.22%
|
20
|
54
|
150
|
0.535
|
0.017
|
0.521
|
2.64%
|
25
|
68
|
150
|
0.808
|
0.0165
|
0.79
|
2.27%
|
30
|
82
|
150
|
1.133
|
0.0158
|
1.09
|
3.92%
|
32
|
87
|
150
|
1.276
|
0.0157
|
1.232
|
3.58%
|
35
|
95
|
150
|
1.507
|
0.0153
|
1.437
|
4.87%
|
40
|
109
|
150
|
1.929
|
0.015
|
1.84
|
4.84%
|
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