estimation of pressure drop in pipe systems

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Process Engineering Guide: GBHE-PEG-FLO-300

Estimation of Pressure Drop in Pipe Systems Information contained in this publication or as otherwise supplied to Users is believed to be accurate and correct at time of going to press, and is given in good faith, but it is for the User to satisfy itself of the suitability of the information for its own particular purpose. GBHE gives no warranty as to the fitness of this information for any particular purpose and any implied warranty or condition (statutory or otherwise) is excluded except to the extent that exclusion is prevented by law. GBHE accepts no liability resulting from reliance on this information. Freedom under Patent, Copyright and Designs cannot be assumed.



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 7

5.2 Static and Stagnation Pressures 8 5.3 Sonic Flow 10

6 INCOMPRESSIBLE FLOW IN PIPES OF CONSTANT

CROSS-SECTION 11

6.1 Straight Circular Pipes 11 6.2 Ducts of Non-circular Cross-section 16 6.3 Coils 19 6.4 General Equation for Incompressible Flow

in Pipes of Constant Cross-section 20



7 COMPRESSIBLE FLOW IN PIPES OF CONSTANT CROSS-SECTION 21

7.1 Isothermal Flow 21 7.2 Adiabatic Flow 22 7.3 Estimation of Pressure Drop for Adiabatic

Flow in Pipes of Constant Cross-section 29 7.4 Ratio of Isothermal to Adiabatic Pressure Drop 32

8 FLOW IN PIPE FITTINGS 33

8.1 Incompressible Flow 33 8.2 Compressible Flow 35

9 FLOW IN BENDS 38

9.1 Incompressible Flow in Bends 38 9.2 Compressible Flow in Bends 47

10 CHANGES IN CROSS-SECTIONAL AREA 48


11 ORIFICES, NOZZLES AND VENTURIS 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 64 12.2 Incompressible Flow in Valves 65 12.2 Compressible Flow in Valves 76



13 COMBINING AND DIVIDING FLOW 78


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 7 APPROXIMATE BASE LOSS COEFFICIENTS K*b FOR 90°

MITRE BENDS 42 8 INTERACTION CORRECTION FACTOR FOR COMBINATIONS

OF TWO 90° BENDS 45 9 LOSS COEFFICIENTS Kb FOR 90° BENDS –

COMPRESSIBLE FLOW 47

10 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

16 VALVE TYPE CONSTANTS C1 77



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 10 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 CORRECTION FACTORS FOR 90° COMBINED BENDS 46 16 CORRECTION FACTORS FOR 180° COMBINED BENDS 47 17 TYPES OF CONTRACTION 49



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 25 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



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 non-Newtonian 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 GBHE-PEG-FLO-301. It does not cover unsteady fluid flow. An introduction to unsteady state flow (pressure surge) in incompressible systems is given in GBHE-PEG-FLO-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 Engineering Sciences Data Unit HTFS 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.

3.1 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 - m Diameter - m Time - s Viscosity - 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) Force - N (1 N = 1 kg.m.s-2) Pressure - N.m-2 (1 bar = 105 N.m-2) Mass flow - kg.s-1 Volume flow - m3.s-1 Temperature - 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 = acceleration due to gravity (m.s-2) K = number of velocity heads lost due to friction,

or frictional pressure loss coefficient (-) P = static pressure (N.m-2) v = velocity of the fluid (m.s-1) z = elevation of pipe above datum (m) α = kinetic energy factor (-) ρ = 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 = acceleration due to gravity m.s-2 Hs = height of liquid column m Pg = static pressure above atmospheric pressure

(gauge pressure) N.m-2 ρ = 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 = the isentropic exponent for the fluid in question M = 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 = the isentropic exponent for the fluid in question R = universal gas constant (= 8314 J. kmole-1.K-1) T = static temperature (K) v = fluid velocity (m.s-1) W = molecular weight of the gas (kg.kmole-1) Z = 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 = Friction pressure drop N.m-2 d = pipe diameter m τ w = wall shear stress N.m-2 dL = 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 = a friction factor n = 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 = pipe cross-sectional area (m2) F = 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 SHAPE FACTOR K2 FOR NON-CIRCULAR DUCTS

6.3 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 = pipe internal diameter (m) D = diameter of helix (m) Rec = 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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