Estimation of Pressure Drop in Pipe Systems

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-FLO-300

Estimation of Pressure Drop in
Pipe Systems

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Process Engineering Guide:

Estimation of Pressure Drop in
Pipe Systems

CONTENTS
0

INTRODUCTION/PURPOSE

5

1

SCOPE

5

2

FIELD OF APPLICATION

5

3

DEFINITIONS

6

3.1 units

6

4

SOURCES OF DATA

6

5

BASIC CONCEPTS

7

5.1 Equation for Pressure Change in a Flowing
Fluid
5.2 Static and Stagnation Pressures
5.3 Sonic Flow

6

7
8
10

INCOMPRESSIBLE FLOW IN PIPES OF CONSTANT
CROSS-SECTION

11

6.1 Straight Circular Pipes
6.2 Ducts of Non-circular Cross-section
6.3 Coils
6.4 General Equation for Incompressible Flow
in Pipes of Constant Cross-section

11
16
19
20


7

COMPRESSIBLE FLOW IN PIPES OF CONSTANT
CROSS-SECTION

7.1 Isothermal Flow
7.2 Adiabatic Flow
7.3 Estimation of Pressure Drop for Adiabatic
Flow in Pipes of Constant Cross-section
7.4 Ratio of Isothermal to Adiabatic Pressure Drop

8

FLOW IN PIPE FITTINGS
8.1 Incompressible Flow
8.2 Compressible Flow

9

FLOW IN BENDS

CHANGES IN CROSS-SECTIONAL AREA

11

21
22
29
32

33
33
35
38

9.1 Incompressible Flow in Bends
9.2 Compressible Flow in Bends
10

21

ORIFICES, NOZZLES AND VENTURIS

38
47
48
48
51

55

11.1 Incompressible Flow through an Orifice
56
11.2 Compressible Flow through an Orifice or Nozzle 57
11.3 Venturi Choke Tubes
61

12

VALVES

64

12.1 General
12.2 Incompressible Flow in Valves
12.2 Compressible Flow in Valves

64
65
76


13

COMBINING AND DIVIDING FLOW

78


78
84

14

COMPUTER PROGRAMS FOR FLUID FLOW

86

15

NOMENCLATURE

86

16

REFERENCES

90

APPENDICES

A

BASIC THERMODYNAMICS

92

B

COMPRESSIBLE FLOW THROUGH ORIFICES

102

C

THE ‘TWO-K’ METHOD FOR FITTING PRESSURE LOSS

104


TABLES
1

DEFINITIONS OF FRICTION FACTOR

12

2

TYPICAL ROUGHNESS VALUES

13

3

HYDRAULIC MEAN DIAMETER FOR DIFFERENT CHANNELS

17

4

RATIO OF ISOTHERMAL TO ADIABATIC PRESSURE GRADIENT 32

5

LOSS COEFFICIENTS Kb FOR 90° BENDS

38

6

PRESSURE LOSS COEFFICIENTS FOR SINGLE MITRE
BENDS

40

APPROXIMATE BASE LOSS COEFFICIENTS K*b FOR 90°
MITRE BENDS

42

INTERACTION CORRECTION FACTOR FOR COMBINATIONS
OF TWO 90° BENDS

45

LOSS COEFFICIENTS Kb FOR 90° BENDS –
COMPRESSIBLE FLOW

47

PRESSURE DROP PARAMETERS FOR DISCS AND
ROUND WIRE GAUZES

56

11

CRITICAL PRESSURE RATIOS

58

12

FLOW AND LOSS COEFFICIENTS FOR DURCO PLUG VALVES 69

13

LOSS COEFFICIENTS FOR DURCO BUTTERFLY VALVES

72

14

LOSS COEFFICIENTS FOR BATLEY BUTTERFLY VALVES

72

15

LOSS COEFFICIENTS FOR SAUNDERS TYPE ’KB’
DIAPHRAGM VALVES

74

VALVE TYPE CONSTANTS C1

77

7

8

9

10

16


FIGURES
1

MOODY CHART - FRICTION FACTOR vs REYNOLDS NUMBER 14

2

SHAPE FACTOR K2 FOR NON-CIRCULAR DUCTS

18

3

SECONDARY FLOW PATTERNS IN COILS

19

4

FACTOR FOR GAS FLOW IN A PIPE

22

5

RELATIONSHIP BETWEEN PRESSURE AND FRICTION FACTOR
REQUIRED TO PRODUCE SONIC FLOW AT PIPE OUTLET
25

6

RELATIONSHIP BETWEEN STATIC PRESSURE AND
MACH NUMBER

26

7

STAGNATION PRESSURE

27

8

PRESSURE PLUS MOMENTUM FLUX

28

9

COMPARISON OF ‘OVERALL’ AND ‘TRUE’
PRESSURE DROP IN FITTINGS

37

APPROXIMATE LOSS COEFFICIENT FOR 90° CIRCULAR
BENDS OF r/d BETWEEN 1.5 AND 2.0 AT LOW REYNOLDS
NUMBERS

39

11

OUTLET TANGENT CORRECTION

40

12

MITRE BEND LOSS COEFFICIENTS

41

13

NOTATION FOR COMBINATIONS OF TWO 90° BENDS

44

14

CORRECTION FACTORS FOR 0° COMBINED BENDS

46

15


46

16


47

17

TYPES OF CONTRACTION

49

10


18

CONTRACTION

51

19

ENLARGEMEN

52

20

CHANGE IN MACH NUMBER FOR COMPRESSIBLE
FLOW THROUGH A SUDDEN ENLARGEMENT

54

21

SCHEMATIC OF FLOW THROUGH AN ORIFICE

55

22

ORIFICE FLOW vs PRESSURE RATIO

60

23

TYPICAL VENTURI CHOKE TUBE

61

24

VENTURI STATIC PRESSURE DROP AT CRITICAL
CONDITIONS

63

VARIATION OF VENTURI PRESSURE DROP WITH
FLOWRATE

64

26

VALVES WITH ALMOST LINEAR CHARACTERISTICS

66

27

CHARACTERISTICS OF EQUAL PERCENTAGE VALVES

67

28

BALL VALVE LOSS COEFFICIENTS

68

29

VARIATION OF Cv AND Kv WITH POSITION

69

30

APPROXIMATE EFFECT OF REYNOLDS NUMBER, GATE,
PLUG AND BUTTERFLY VALVES

70

31

LOSS COEFFICIENTS FOR BUTTERFLY VALVES

71

32

DIAPHRAGM VALVE LOSS COEFFICIENTS

73

33

APPROXIMATE EFFECT OF REYNOLDS NUMBER,
POPPET TYPE AND DIAPHRAGM VALVES

73

34

FULLY OPEN GATE VALVE LOSS COEFFICIENTS

75

35

GATE AND SLUICE VALVE LOSS COEFFICIENTS

75

36

APPROXIMATE LOSS COEFFICIENTS FOR FULLY OPEN
GLOBE VALVES

76

25


37

TEE JUNCTIONS WITH RIGHT-ANGLED BRANCHES

78

38

TEE JUNCTIONS WITH A DEAD LEG

82

39

TYPES OF 3-LEG JUNCTION

84

40

MODEL FOR COMPRESSIBLE FLOW IN A JUNCTION

85

41

SCHEMATIC OF FLOW THROUGH AN ORIFICE

102

DOCUMENTS REFERRED TO IN THIS PROCESS
ENGINEERING GUIDE

105


0

INTRODUCTION / PURPOSE
This Process Engineering Guide is one of a series on fluid flow produced
by GBH Enterprises.
1 SCOPE
This guide recommends methods for the estimation of pressure drop for
the steady state flow of single phase Newtonian liquids and compressible
fluids (gases and vapors) in pipe systems. It covers most of the
geometries likely to be encountered in normal design cases. Further
information on more unusual geometries may be found in Miller [1990].
It does not cover liquids that are non-Newtonian in their behavior, that is,
liquids for which, when sheared in laminar flow, the shear stress is not
directly proportional to the shear rate. For general information on nonNewtonian behavior see GBHE-PEG-FLO-302. For estimation of the
pressure drop for pipeline flow of such fluids see GBHE-PEG-FLO-303
and GBHE-PEG-FLO-304.
It does not cover slurries. Slurries with low solids contents may be
Newtonian in behavior, but in general most slurries are non-Newtonian.
It does not cover pressure drop for two phase gas-liquid flow.
It does not cover the gravity driven flow of liquids in partially filled pipes
and open channels. Information on this subject can be found in GBHEPEG-FLO-301.
It does not cover unsteady fluid flow. An introduction to unsteady state
flow (pressure surge) in incompressible systems is given in GBHE-PEGFLO-305.
GBHE has access to proprietary computer programs, that may be used to
model unsteady state compressible and incompressible flows.
The data and methods given in this PEG are suitable for hand
calculations, or for incorporation into computer codes.


2 FIELD OF APPLICATION
This guide applies to the process engineering community in GBH
Enterprises worldwide. GBHE-PEG-FLO-300
3

DEFINITIONS
For the purposes of this guide, the following definitions apply:
ESDU
HTFS

3.1

Engineering Sciences Data Unit
Heat Transfer and Fluid Flow Service. A co-operative
research organization, with headquarters at Harwell, UK,
involved in research into the fundamentals of heat transfer
and two phase flow and the production of design guides and
computer programs for the design of industrial heat
exchange equipment.

Units
All equations in this guide are in coherent units unless stated otherwise.
Normally, base units for the SI system are used. For example:
Length
Diameter
Time
Viscosity

-

Force
Pressure
Mass flow
Volume flow Temperature -

m
m
s
kg.m-1 s-1 (1 kg.m-1 s-1 = 1 N.s.m-2,
1 cP = 10-3 kg.m-1.s-1)
N (1 N = 1 kg.m.s-2)
N.m-2 (1 bar = 105 N.m-2)
kg.s-1
m3.s-1
K

Usually, symbols are defined after the equation in which they first occur,
and base SI units are given. However, the equations are, in general,
equally valid if the individual terms are expressed in any other coherent
set of units. Any constants in the equations will then be dimensionless.
There are a few equations that include dimensional constants. For these,
individual terms must be expressed in the appropriate units, which are
indicated with the equations.

A full list of symbols, with the appropriate base SI units, is given in clause
15.
4

SOURCES OF DATA
The pressure losses through pipe fittings are dependent on the exact
geometry of the fitting. Whereas the geometry for straight pipes and bends
is readily definable, this is not the case for proprietary items such as
valves, where nominally similar items from different manufacturers may
have markedly different characteristics. If such components form a
significant portion of the total pressure drop, manufacturers’ data should if
possible be used in preference to those contained in this guide.

5

BASIC CONCEPTS
5.1 Equation for Pressure Change in a Flowing Fluid
The change in static pressure during the flow of a fluid in a pipe is the sum
of three factors:
(a)

Pressure change due to change in elevation.

(b)

Pressure change due to change in kinetic energy.

(c)

Pressure change due to frictional effects.

The first two of these factors can result in either a pressure loss or a
pressure gain, and are known as reversible changes; the pressure change
can be reversed by reversing the process. Friction, on the other hand,
always results in a fall in pressure in the direction of flow, and is an
irreversible change. It is the process whereby mechanical energy is
degraded into heat.
The basic equation for pressure change for any flowing fluid is:


Where:
g
K

=
=

P
v
z
α
ρ

=
=
=
=
=

acceleration due to gravity (m.s-2)
number of velocity heads lost due to friction,
or frictional pressure loss coefficient (-)
static pressure (N.m-2)
velocity of the fluid (m.s-1)
elevation of pipe above datum (m)
kinetic energy factor (-)
density of the fluid (kg.m-3)

This equation can be deduced from basic thermodynamics for any fluid.
See Appendix A for details of the derivation.
The first of the terms on the right hand side of equation 5.1 is the change
in pressure due to changes in elevation, the second is the change due to
changes in kinetic energy and the third is the change due to frictional
losses.
The kinetic energy factor, α, allows for the effects of a non-uniform velocity
profile, as explained in Appendix A. This is important for laminar flow,
where the value of α for established flow in a straight pipe is equal to 2.
However, for established turbulent flow α has a value of only about 1.05.
In common with most texts on the subject, α will in general be omitted
from equations in the main body of this document. For most cases where
the effects of compressibility are important, the flow is likely to be
turbulent. For liquid flows, laminar flow is more of a possibility, and in
these cases the methods given should be modified. However, there are
few reliable data for laminar flow through fittings.
Equation 5.1 applies to all fluids. For an incompressible fluid, the density is
constant and the first two terms of equation 5.1 can be integrated directly:

For a pipe of constant diameter with an incompressible fluid, v is constant,
the second term disappears and the third can be integrated to give:


For a fitting where the cross sectional area is changing, the third term
cannot in general be integrated analytically. It is normal practice in this
case to express the loss coefficient for the fitting in terms of the velocity at
either the inlet or outlet position.

where v r refers to the reference conditions, either the inlet or outlet.
Note:
In general, the loss coefficient K is not constant, but a function of the
Reynolds number.
For a compressible fluid, the density is a function of both temperature and
pressure, and equation 5.1 cannot be so readily integrated. It is necessary
to introduce the equation of state for the fluid to relate the density to
temperature and pressure. See Appendix A3. However, the elevation term
for gas flow can generally be neglected, so equation 5.1 simplifies to:

5.2

Static and Stagnation Pressures
The term designated by P in the above equations is the static pressure.
For a fluid at rest, it is simply the pressure that would be measured by any
pressure-measuring device. For a fluid in motion, the static pressure is the
pressure measured in a direction normal to the local direction of flow.
GBHE-PEG-FLO-3009
If a flowing fluid with a velocity v is brought to rest in a reversible manner,
the pressure will rise to the "stagnation pressure" P0. (The term "total
pressure" is sometimes used instead of stagnation pressure.


5.2.1 Incompressible Flow
For an incompressible fluid the stagnation pressure is given by:

This can be readily seen by applying equation 5.2, omitting the elevation and
frictional terms and putting v2 to zero. For an incompressible fluid, the term ½ρv2
is known as the "dynamic pressure" or the "velocity pressure", Pv

Some references work in terms of fluid head rather than pressure, particularly
when considering the flow of liquids. A vertical column of liquid exerts a pressure
at its base given by:

where:
g
Hs
Pg

=
=
=

ρ

=

acceleration due to gravity m.s-2
height of liquid column m
static pressure above atmospheric pressure
(gauge pressure) N.m-2
liquid density Kg.m-3

If this column of liquid is connected at its base to a fluid system, then provided
that the fluid is at rest, or the base of the column is parallel to the local direction
of flow, the height Hs is a measure of the (gauge) static pressure at the point of
attachment, and is termed the "static head".
If working in terms of head rather than pressure, the quantity corresponding to
the dynamic pressure is the "velocity head" Hv and that corresponding to the
stagnation pressure is the "total head" H0.


Note:
The dynamic pressure Pv is sometimes loosely referred to as the velocity head.
5.2.2 Compressible Flow
For a compressible fluid the relation between static and stagnation pressures is
more complex, as the density of the fluid changes as it is brought to rest. The
stagnation pressure is related to the static pressure by the equation:

where:
k
M

=
=

the isentropic exponent for the fluid in question
the Mach number (see 5.3 below)

This equation is derived in Appendix A.
For a compressible fluid, the dynamic or velocity pressure may be defined either
by equation 5.7 above, or as the difference between the stagnation and static
pressures.

These two definitions are not equivalent, and do not give the same values.


5.3

Sonic Flow

If an incompressible fluid is flowing in a pipe of constant cross section, the
pressure gradient down the pipe is constant. For a compressible fluid, on the
other hand, as the pressure falls along the pipe so does the density, and the
pressure gradient increases with distance down the pipe. If the pressure in a
receiver at the downstream end of a pipe is lowered, a point will be reached
where the exit velocity from the pipe has reached the sonic velocity. Further
reductions in downstream pressure will not result in an increase in mass flowrate
down the pipe. Instead, there will be a pressure discontinuity at the exit of the
pipe, and the emerging jet will expand down to the pressure in the receiver
through a complex series of shocks. The pressure discontinuity is known as a
choke. Sonic velocity represents the maximum practical flowrate in normal pipe
systems.
Sonic velocity will usually only occur at locations where there is an increase in
flow area. Thus it can occur at the ends of pipes, at expansions within pipes and
at valves, orifices and nozzles in pipes. Following an expansion with a choke,
after the shock system the fluid in the larger diameter section will be flowing at a
sub-sonic velocity. As it flows along this second pipe the pressure will continue to
fall and the fluid will accelerate back towards sonic conditions. It is possible to
have more than one choke in a pipe if there are successive increases in flow
area, but the first position at which choking occurs dictates the maximum flowrate
possible.
It is convenient to express the flowrate for high speed compressible flow in terms
of the Mach number M, which is the ratio of the fluid velocity to the velocity of
sound under the local conditions of temperature and pressure:

where:
k
R
T
v
W
Z

=
=
=
=
=
=

the isentropic exponent for the fluid in question
universal gas constant (= 8314 J. kmole-1.K-1)
static temperature (K)
fluid velocity (m.s-1)
molecular weight of the gas (kg.kmole-1)
is the compressibility (correction factor for
deviation from ideal gas laws)


6

INCOMPRESSIBLE FLOW IN PIPES OF CONSTANT CROSS-SECTION

6.1

Straight Circular Pipes
The pressure drop due to friction for steady state flow in a straight pipe of
constant diameter can be related to the wall shear stress by the force
balance:

where:
dP f
d
τw
dL

=
=
=
=

Friction pressure drop N.m-2
pipe diameter m
wall shear stress N.m-2
increment of pipe length m

Hence, re-arranging and expressing in terms of the dynamic pressure, as defined
by equation 5.7,

This equation may also be written in two alternative ways:

where K is the frictional pressure loss coefficient. This form corresponds to the
last term of equation 5.4.


where:
C
n

=
=

a friction factor
a constant, whose value depends on the form taken
by the friction factor.

Unfortunately, there is no general agreement on the form that the friction factor
should take. Table 1 shows the forms quoted in several sources commonly used
in GBH Enterprises. The 4th column of Table 1 shows the relationship between
the value of the friction factor in laminar flow, C lam and the Reynolds number Re.
The Reynolds number for flow in a circular pipe is defined as:

where:

The mass flux G is defined as the mass flow per unit cross sectional area.
TABLE 1

DEFINITIONS OF FRICTION FACTOR


It is obviously essential that, when a friction factor is quoted, the form used
should be clearly understood. An easy way to determine the form used is to look
at the variation of friction factor with Reynolds number for laminar flow, as given
in the last column of Table 1. This guide, to maintain continuity with V/CP/17 and
P/CP/4, uses the Moody form of the friction factor. Thus the frictional pressure
drop for straight pipe flow is given by:

The friction factor is a function of the Reynolds number, Re, and the relative
roughness, ε/d, where ε is the absolute roughness of the inside surface of the
pipe and d is the pipe inside diameter. Suggested values of ε for some typical
pipe materials are given in Table 2.
Note:
Different sources may quote different values for the same material.


TABLE 2

TYPICAL ROUGHNESS VALUES

A useful way of presenting friction factors is the Moody Chart, which shows the
friction factor as a function of the Reynolds number for a range of values of
relative roughness (Figure 1). Figure 1 is based on the Moody definition of friction
factor. Different sources may be based on different definitions of friction factor:
they will have different scales for the vertical axis, but may still refer to the chart
as a "Moody Chart".


FIGURE 1

MOODY CHART - FRICTION FACTOR vs REYNOLDS NUMBER


Note:
For laminar flow, the friction factor is independent of roughness, and is given by:

Equation 6.5 applies in the laminar flow region, up to a Reynolds number of
about 2100. For turbulent flow, the Moody chart is a plot of the Colebrook-White
equation:

This equation was derived from a systematic analysis of experimental data taken
on a range of smooth and artificially roughened pipes.
The Colebrook-White equation has the disadvantage that it cannot be solved
directly, as the friction factor appears on both sides of the equation. Several
equations that give the friction factor explicitly have been developed. V/CP/17
and Miller [1990] suggest the following equation, which agrees with the
Colebrook-White equation to within 2%:

Equations 6.6 and 6.7 apply for Reynolds numbers above about 4000. Between
2100 and 4000, the transition region, the friction factor is uncertain, and depends
amongst other things on the upstream flow conditions and also the past history of
the flow; hysteresis effects may occur. This region should be avoided for design if
possible. If calculations are required for this region, the recommendation is to
interpolate between the laminar value at Re = 2100 and the turbulent value at Re
= 4000.
The pressure drop in a pipe is sensitive to the pipe diameter. Equation 6.4 may
be written as:


where:
A
F

=
=

pipe cross-sectional area (m2)
mass flowrate (kg.s-1)

For a rough pipe at high Reynolds number, the friction factor is independent of
Reynolds number, and thus the pressure drop varies inversely with the fifth
power of the pipe diameter.
For a smooth pipe, the friction factor is reasonably represented by the Blasius
equation:

Hence, as the Reynolds number for a given mass flow is inversely proportional to
the diameter, for a smooth pipe the pressure drop varies inversely with the
diameter to the power 4.75.
For laminar flow, the friction factor varies as the inverse of the Reynolds number,
and thus the pressure drop is inversely proportional to the diameter raised to the
power 4. It is given by Poiseuille’s equation:

Poiseuille’s equation can be derived theoretically; details are given in standard
text books. See, for example, Coulson & Richardson. It can also be deduced
from equations 6.4 and 6.5, thus demonstrating a theoretical basis for equation
6.5.
Hence, it is essential that the actual internal diameter of the pipe be used when
calculating pressure drops. A 10% reduction in pipe diameter results in a 52 to
70% increase in pressure drop, even assuming no change in the relative
roughness. If the reduction is due to fouling, the relative roughness could
increase significantly, and the overall effect on pressure drop be even higher.
Note:
The thickness of a heavy duty plastic lining in a pipe may be 4 mm, reducing the
bore by 8 mm, which is approximately 10% for a 3" nominal bore pipe.


The main difficulty in calculating friction losses in straight pipes is the uncertainty
in selecting a value for the pipe roughness. Consider, for example, water at 20°C
flowing at 2 m/s in a mild steel pipe of diameter 0.1 m. The Reynolds number in
this case is 2x10 5. Table 6.2 suggests three different values for the roughness of
carbon steel, corresponding to the conditions ’slightly corroded’, ’moderate rust
layer’ and ’badly corroded’. The values are 0.05, 0.25 and 1.0 mm respectively.
Using these values, the friction factors at Re = 2x105 are 0.19. 0.275 and 0.38
respectively. There is thus a factor of two between the extremes of calculated
pressure drop.
6.2

Ducts of Non-circular Cross-section

The friction factors given in Figure 1 or equations 6.5 to 6.7 may be used for
calculating pressure drops in non-circular straight ducts with the following
adjustments.
(a) The Reynolds number should be calculated based on the true mass flux
through the duct, i.e. the actual mass flowrate divided by the cross
sectional area for flow, but the characteristic dimension used in the
Reynolds number should be the hydraulic mean diameter de:

(b) The friction factor based on the above Reynolds number should be
corrected by a factor K2 to allow for geometric effects (see below)

(c) The pressure drop is then given by equation 6.4, with the diameter d
replaced by the hydraulic mean diameter de.
Table 3 shows the definition of hydraulic mean diameter for some common
non-circular channels. Note that for a circular cross-section, the hydraulic
mean diameter equates to the actual diameter.


TABLE 3 HYDRAULIC MEAN DIAMETER FOR DIFFERENT CHANNELS

Figures 2a to 2d show the values of K2 for the cross-sections given in Table 3.
Note the following points:
(1)

K2 is independent of Reynolds number for laminar flow, and nearly
independent for turbulent flow, and is taken as independent for all
Reynolds numbers.

(2)

In general, the effects of K2 are greater for laminar flow than for turbulent
flow. For the concentric annulus and the rectangle, K2 is unity for turbulent
flow.

(3)

The values of K2 for turbulent flow apply to hydraulically smooth surfaces.
No information is available on the effects of roughness in these
geometries. In the absence of any data, assume that the value of K2 will
apply equally to rough tubes, where the relative roughness is the ratio of
the absolute roughness to the equivalent diameter.

(4)

For other ducts having shapes which depart significantly from circular or
rectangular cross sections, consult section 8.3.4 of Miller [1990].


FIGURE 2

6.3

SHAPE FACTOR K2 FOR NON-CIRCULAR DUCTS

Coils

Single phase flow in helically coiled tubes differs from that in straight tubes
because of the effects of centrifugal force on the fluid. This is greatest on the
fastest moving fluid at the centre of the tube; as a result, a secondary flow
pattern is set up, with two recirculation systems (see Figure 3).


FIGURE 3

SECONDARY FLOW PATTERNS IN COILS

This secondary flow system tends to stabilize laminar flow, increasing the
transition Reynolds number between laminar and turbulent flow. The
recommended transition Reynolds number is given in HTFS Handbook [HTFS
1976].

where:
d
D
Rec

=
=
=

pipe internal diameter (m)
diameter of helix (m)
transition Reynolds number


The secondary flow, by increasing the dissipation of turbulent energy, increases
the friction factor above that for straight pipe flow. The following equations for
friction factor in coiled pipes, taken from HTFS 1976, are recommended.
Note:
These equations have been changed from their original form to allow for the
different definition of friction factor used in this guide from that in HTFS
Handbook.
For laminar flow, the friction factor is a function of the Dean number, DN, defined
as:


Estimation of Pressure Drop in Pipe Systems

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