**Introduction**

With the
widespread adoption of engineering software and computerized
design processes, it is commonly thought that engineering calculations
are very accurate. However, while engineering software can eliminate
mathematical errors and perform calculations with greater speed
than hand calculations, there are many factors that must be
acknowledged which can result in the calculated values being
substantially different from actual measurements. This article
will discuss some of the factors that can make calculations
vary from measured results, including factors related to system
construction as well as to system modeling. These factors can
then be considered during system design to ensure that the design
will meet all necessary requirements. In addition, consideration
of these factors is also prudent when troubleshooting existing
systems to avoid drawing unwarranted or incorrect conclusions.

**Construction
Issues **

When
analyzing fluid flow problems, the designer typically uses nominal
dimensions for pipe and fittings. A nominal dimension is similar
to an average dimension. No specific pipe or fitting is guaranteed
to match the nominal dimensions exactly. Consider a nominal
NPS 1, schedule 40 carbon steel pipe dimensioned using ANSI
B36.10, Welded and Seamless Wrought Steel Pipe and constructed
to ASTM A 106 - 91, Standard Specification for Seamless Carbon
Steel Pipe for High-Temperature Service. These are common specifications
for carbon steel pipe. For this nominal pipe, the nominal inside
diameter is 1.049 inches. The material specification imposes
limits on variation in dimensions, including limits on the pipe
outside diameter, limits on the pipe wall thickness, and limits
on the pipe weight. According to these limits, this pipe could
have an inside diameter as large as 1.079 inches or as small
as 0.978 inches (localized measurements could even be outside
this range). These as-built dimensions could result in a calculated
head loss from 13.7 percent under to as much as 44.6 percent
over the calculated head loss for the nominal pipe (both calculated
with the same volumetric flow rate and with fully turbulent
flow). Similarly for a large pipe such as a NPS 24, schedule
20 to the same specifications, the calculated head loss could
range from 3.4 percent under to 2.5 percent over the nominal
head loss. In general, this potential error is larger for smaller
diameter pipe, larger nominal wall thickness, and higher flow
rates.

Similarly, for items other than pipe, most analyses
use resistance values based on generic items. For example, many
sources provide a resistance value for an inline globe valve
equal to 340 diameters of pipe. (Some sources are slightly different,
but this doesn’t change the conclusion). This resistance
value is combined with a turbulent friction factor for the appropriate
size to calculate the resistance of the valve. When you compare
this resistance for a generic valve to actual manufacturer data,
there can be considerable differences. For example, comparing
the resistance of a generic, 2 inch inline globe valve to a
small sample of commercially available valves shows that the
resistance of the generic valve can be as much as 38 percent
under to 25 percent over that of the commercial valve which
would result in an identical variation in head loss at a constant
flow. This difference is caused largely because different valves
are designed for different characteristics, such as good throttling
or low resistance, and are not truly represented by a single
generic value. In general, where data is available for a specific
valve or fitting, this data should be used instead of the generic
information.

Further, the resistance of a fitting is generally
determined in isolation, meaning that a single fitting is placed
between two long sections of pipe and the flow and head loss
are measured. If fittings in a system are not separated sufficiently,
then the head loss produced by each fitting can differ from
that of an isolated component (the normal resistance for that
component). While there is no formula for calculating the effect
and this difference has not been quantified for all arbitrary
combination of fittings and spacing, it has been measured for
simple combinations such as two 90 degree bends separated by
small lengths of straight pipe. With two bends, the use of normal
resistance values determined in isolation can overstate the
actual resistance by as much as 50 percent or understate the
resistance by 300 percent. The actual resistance of two closely
spaced bends depends on the specific type of bends (including
factors such as the bend type, angle, and radius), the orientation
of the bends, and the spacing between the bends. This is even
more complex for other combinations of fittings, and currently
no general method exists to calculate modified resistance values.
In general, however, the use of an isolated resistance value
generally (but not always) overstates the actual resistance
of the item.

**Modeling
Issues**

A common way of expressing resistance of an item is with a
dimensionless factor denoted ‘K’. If a fitting has
the same geometry for different sizes, the K factor for the
fitting will be the same for each size. However, the geometry
of fittings is generally different for each different size (in
other words, it does not change proportionally with size). Consequently,
common practice is to represent the resistance of a fitting
as the number of diameters of straight pipe that would cause
the same pressure drop (generally denoted ‘L/D’).
This number is generally the same for different sizes of a given
fittings. This value is then combined with the turbulent friction
factor for the size of the fitting to obtain a resistance value.

This assumption to use a turbulent friction factor is based
on extensive testing and has been shown to be valid down to
small flow rates. In the case of a simple bend, the assumption
of a turbulent friction factor is valid down to a Reynolds number
of less than 500. For other fittings, with more extreme changes
in flow direction or flow area, it would be expected that this
assumption would hold at even lower Reynolds numbers. However,
beyond this value, the resistance of the item can increase substantially
(in the case of a simple bend by a factor of over 100 compared
to the turbulent value at a Reynolds number of 10).

Pipes on the other hand are always analyzed with a friction
factor determined based on the actual flow rate and internal
surface condition. In most cases, the internal surface of the
pipe will become rougher with age, resulting in increased head
loss compared to new pipe. Analysis for a large water distribution
system showed increases in head loss through cast iron pipe
of up to 800% over 40 years. Other studies with steel pipe have
shown smaller but still significant increases, with increases
in head loss ranging from 40% to 200% over 20 years.

Beyond the issue of what roughness to use, and what friction
factors are appropriate, there is still the issue of the accuracy
of the head loss equations themselves. There are several to
choose from, including empirical relationships such as the Hazen-Williams
formula for liquids, and the Panhandle and Weymouth formulas
for gases. However, because of its range of applicability and
overall accuracy, for liquids the Darcy-Weisbach equation is
generally used. First proposed in this form in 1845 by Julies
Weisbach, this equation relates head loss through pipe to the
square of the average fluid velocity in the pipe. The current
form of this equation is routinely called the Darcy equation,
after Henry Darcy, who made significant achievements in understanding
and documenting the friction factor and its relationship to
the pipe inside surface condition. This equation is also commonly
used for valves and fittings in addition to pipe. However, experiments
have shown that while the Darcy equation is a practical solution
method for valves and fittings, actual flow measurements have
shown that the head loss for any given valve or fitting design
is proportional to velocity raised to an exponent of between
1.8 to 2.1 instead of the exponent of 2 assumed in the Darcy
equation. Nonetheless, common practice is to use the Darcy equation
unmodified for valves and fittings.

**Conclusion**

The factors discussed here are generally applicable to all
fluid flow problems. However, for an individual problem these
factors may or may not be significant in the context of the
whole problem. When performing a fluid flow calculations, an
engineer should review these factors and determine what, if
any, actions should be taken to accommodate the uncertainty
they introduce. In general, actions taken by an engineer may
include performing additional analysis (such as calculations
with different pipe diameters or with different pipe roughness),
adding additional margin to a design, or performing actual tests
on as-built systems.