Introduction
The following notes should enable a mechanical engineer to establish basic flow conditions and head losses along pipe routes in which fluids are flowing. The equations are most relevant to liquids although approximate sizing for gases can be carried out if appropriate correction factors are used,where necessary, and low gas velocities are considered.
Symbols
|
A = Pipe Cross Section Area (m2) a = Velocity of sound ( m /s) c p = Specific Heat Capacity at Constant pressure (kJ/(kg K)) c v = Specific Heat Capacity at Constant Volume (kJ/(kg K)) ε = Pipe roughness (m) ε mm = Pipe roughness (mm) D = diameter (m) f = friction factor h = Specific Enthalpy (kJ/kg ) k = Thermal Conductivity (W/(m K)) K = f (L/D ) L = Pipe Length (m) p = Absolute Pressure N / m2 Pr = Prantl Number =c p. mu / k (Dimensionless) |
Q = Volume flow Rate (m3 /s ) q = Heat Input per unit mass ( kJ /kg ) R = Gas Constant = R o / M (kJ /(kg.K) Re = Reynolds Number = v.ρD/μ t = Temperature (C ) T = Absolute Temperature (K) u = Specific Internal Energy (kJ/kg) v =Fluid Velocity (m/s) w = Work Output per unit mass (kJ/kg) ρ = Density ( kg /m3 ) μ =Fluid Viscosity = (Ns/m2 = Pa s) z = Elevation (m ) g = gravitational acceleration ( 9.81 m /s2) |
Fluid Flow
Fluid flowing in pipes has two primary flow patterns. It can be either laminar when all of the fluid particles flow in parallel lines at even velocities and it can be turbulent when the fluid particles have a random motion interposed on an average flow in the general direction of flow. There is also a critical zone when the flow can be either laminar or turbulent or a mixture. It has been proved experimentally by Osborne Reynolds that the nature of flow depends on the mean flow velocity (v), the pipe diameter (D), the density (ρ) and the fluid viscosity Fluid Viscosity( μ). A dimensionless variable for the called the Reynolds number which is simply a ratio of the fluid dynamic forces and the fluid viscous forces , is used to determine what flow pattern will occur. The equation for the Reynold Number is
For normal engineering calculations , the flow in pipes is considered laminar if the
relevant Reynolds number is less than 2000, and it is turbulent if the Reynolds number is greater than
4000. Between these two values there is the critical zone in which the flow can be either laminar or turbulent
or the flow can change between the patterns...
It is important to know the type of flow in the pipe when assessing friction losses
when determining the relevant friction factors
Steady Flow Equation....
Reference :
The steady flow equation steady flow equation (energy per unit mass ) for a system is identified below...
Reference... Steady Flow
If q = w = 0 and the fluid is incompressible and frictionless
and if the variables are converted to measured heads of the fluid , that is the units are per
unit weight (ρg) - then the Bernoulli's equation results ..
Reference .. Bernoulli's Equation ideal fluids
..
In real flow systems there are losses due to internal and wall friction which result in increase in the internal energy of the fluid. (q > 0). Reference Bernoulli's Equation real Fluids . The bernoulli equation is modified to reflect these losses by adding a term h f = Head loss due to friction.= (u2 -u1 - q) The modified bernoullis equation is therefore ..
The object of most pipe flow head loss calculations is to determine the friction head loss and allow estimation of the pump /compressor power required to pump the fluid along the piping. In most fluid transfer cases the fluid is a incompressible (a liquid) and the flow rate (Q) is constant along the pipe run and therefore the velocity at any point can easily be calculated. The head (z) can also be easily obtained from the pipeline geometry. The system pressure and head loss are therefore the variables generally subject to the detailed pipeline calculations....
Pipe Flow Calculations
In determining the head loss (pressure drop) along a pipe as
a result of friction losses it is first necessary to determine the following:
Diameter (m), Length (m), Fluid Viscosity( μ),
Fluid density (ρ) and the fluid velocity (v). It is then necessary to obtain the
relevant Reynolds number..
The equation for the Reynold Number
Consistent units to be used i.e Typically ρ = kg/m3,
v = m/s, D= m, μ = Ns/m2
( 1 Ns/m2 = 103cP)
The value for the Reynold number is to be used to evaluate if
the flow is laminar or turbulent and can be used to obtain the friction factor " f "
from a moody chart. The moody chart plots the friction factor (f) against the Reynold number
with a number of different plotted lines for different values of absolute roughness/Diameter .
The head loss along the pipe can now be calculated using the Darcy-Weisbach equation
The result of the calculation is in units of head of the fluid. . It is based on the pipe being all one dia and the fluid is incompressible
For a single pipe line with a number of fittings the total head loss is calculated as
Kp = f (L/D) for the length of pipe. ( this may be made up of ∑ f(L/D). for a number of different pipe lengths of different diameters )
K1..n = f (L/D) equivalent for each fitting
A moody chart and tables for roughness values and (L/D) factors for various fittings are provided below
Moody Chart
Various typical values of hydraulic roughness (ε)
Note:
In the moody chart above (ε /D ) is identified with both
numerator and denominator in metres (for consistency with all other equations on this page.
It is probably more convenient to express both in (mm).i.e a 50mm cast iron pipe
(ε mm; /Dmm ) would simply be (0,203 /50 ).
| Type of Pipe | ε .103..( = εmm ) |
| Cast Iron | 0,203 |
| Galvanised Steel | 0,152 |
| Steel/Wrought Iron | 0,051 |
| Rivetted Steel | 0,91 - 9,1 |
| Asphalted Cast Iron | 0,12 |
| Wood-Stave | 0,18 - 0,91 |
| Concrete | 3,0 |
| Spun Concrete | 0,203 |
| Drawn Copper, Brass Steel,Glass | Smooth |
Typical Values of L/D for Fittings
The losses through fittings are generally evaluated by obtaining K = f (L/D)
| Fitting | L/D |
| Globe Valve | 340 |
| Gate Valve | 8 |
| Lift Check Valve | 600 |
| Swing Check Valve | 50 |
| Ball Valve | 6 |
| Butterfly Valve | 35 |
| Flush Pipe Entrance Sharp Corner | K = 0.5 |
| Flush Pipe Entrance radius >0,15 | K = 0.04 |
| Pipe Exit | K = 1 |
| Tee Through | 20 |
| Tee- Branch flow | 60 |
| Elbow-90 | 30 |
| Elbow -45 | 16 |
| Bend r /D=1 | 20 |
| Bend r /D=2 | 12 |
| Bend r /D=3 | 12 |
| Bend r / D=6 | 17 |
| Bend r / D=8 | 24 |
| Bend r / D=10 | 30 |
| Bend r / D=12 | 34 |
| Bend r / D=14 | 38 |
| Bend r / D=16 | 42 |
| Bend r / D=16 | 46 |
| Bend r / D=20 | 50 |
K values for Sudden Expansion-Contraction & Orifice
The losses through these fitting are generally evaluated by first obtaining β = d2 / d1
| β | K e | K c | K o | β | K e | K c | K o | |
| 0.15 | 1887.42 | 965.43 | 2852.85 | 0.6 | 3.16 | 2.47 | 5.63 | |
| 0.2 | 576 | 300 | 876 | 0.65 | 1.87 | 1.62 | 3.49 | |
| 0.25 | 225 | 120 | 345 | 0.7 | 1.08 | 1.06 | 2.14 | |
| 0.3 | 102.23 | 56.17 | 158.4 | 0.75 | 0.6 | 0.69 | 1.29 | |
| 0.35 | 51.31 | 29.24 | 80.55 | 0.8 | 0.32 | 0.44 | 0.76 | |
| 0.4 | 27.56 | 16.41 | 43.97 | 0.85 | 0.15 | 0.27 | 0.42 | |
| 0.45 | 15.51 | 9.72 | 25.23 | 0.9 | 0.06 | 0.14 | 0.2 | |
| 0.5 | 9 | 6 | 15 | 0.95 | 0.01 | 0.06 | 0.07 | |
| 0.55 | 5.32 | 3.81 | 9.13 | 1 | 0 | 0 | 0 |
Reasonable Velocities of fluid in Pipes
| Medium | Pressure (bar) | Service | Velocity (m/s) | Notes |
| Steam (sat) | 0 - 1.7 | Heating | 20 to 30 | + 100mm dia |
| Steam (sat) | over 1.7 | Process | 30 to 50 | +150mm dia |
| Steam (sup) | over 14 | Process | 30 to 100 | +150mm dia |
| Air | Forced Air Flow | 5 to 8 | e.g. AC Reheat | |
| Water | - | General | 1 to 3 | |
| Water | Concrete Pipe | 4,7 | ||
| Water | Pump Suction | 1,2 | ||
| Water | Horizontal Sewer | 0,75 | Minimum | |
| Water | Pump discharge | 1,2 to 2,5 | Minimum | |
| Water | Boiler Feed | 2,4 to 4,6 | Minimum | |
| Oil | Hydraulic Systems | 2,1 to 4,6 | Minimum | |
| Ammonia | Compressor Suction | 25 | Max. Permissable | |
| Ammonia | Compressor Discharge | 30 | Max. Permissible |