Pipeline Calculation Formulae

Laminar Flow and Turbulent Flow of Fluids Resistance to flow in a pipe When a fluid flows through a pipe the internal roughness (e) of the pipe wall can create local eddy currents within the fluid adding a resistance to flow of the fluid. Pipes with smooth walls such as glass, copper, brass and polyethylene have only a small effect on the frictional resistance. Pipes with less smooth walls such as concrete, cast iron and steel will create larger eddy currents which will sometimes have a significant effect on the frictional resistance. The velocity profile in a pipe will show that the fluid at the centre of the stream will move more quickly than the fluid towards the edge of the stream. Therefore friction will occur between layers within the fluid. Fluids with a high viscosity will flow more slowly and will generally not support eddy currents and therefore the internal roughness of the pipe will have no effect on the frictional resistance. This condition is known as laminar flow. Reynolds Number The Reynolds number (Re) of a flowing fluid is obtained by dividing the kinematic viscosity (viscous force per unit length) into the inertia force of the fluid (velocity x diameter) Kinematic viscosity = dynamic viscosity / fluid density Reynolds number = (Fluid velocity x Internal pipe diameter) / Kinematic viscosity Note: Information on Viscosity and Density Units and formula are included at the end of this article. Laminar Flow Where the Reynolds number is less than 2300 laminar flow will occur and the resistance to flow will be independent of the pipe wall roughness. The friction factor for laminar flow can be calculated from 64 / Re. Turbulent Flow Turbulent flow occurs when the Reynolds number exceeds 4000.

Eddy currents are present within the flow and the ratio of the internal roughness of the pipe to the internal diameter of the pipe needs to be considered to be able to determine the friction factor. In large diameter pipes the overall effect of the eddy currents is less significant. In small diameter pipes the internal roughness can have a major influence on the friction factor. The ‘relative roughness’ of the pipe and the Reynolds number can be used to plot the friction factor on a friction factor chart. The friction factor can be used with the Darcy-Weisbach formula to calculate the frictional resistance in the pipe. (See separate article on the Darcy-Weisbach Formula). Between the Laminar and Turbulent flow conditions (Re 2300 to Re 4000) the flow condition is known as critical. The flow is neither wholly laminar nor wholly turbulent. It may be considered as a combination of the two flow conditions. The friction factor for turbulent flow can be calculated from the Colebrook-White equation:

Internal roughness (e) of common pipe materials. Cast iron (Asphalt dipped) Cast iron Concrete Copper PVC Steel Steel (Galvanised)

0.1220 mm 0.4000 mm 0.3000 mm 0.0015 mm 0.0050 mm 0.0450 mm 0.1500 mm

0.004800” 0.001575” 0.011811” 0.000059” 0.000197” 0.001811” 0.005906”

Darcy-Weisbach Formula Flow of fluid through a pipe The flow of liquid through a pipe is resisted by viscous shear stresses within the liquid and the turbulence that occurs along the internal walls of the pipe, created by the roughness of the pipe material. This resistance is usually known as pipe friction and is measured is feet or metres head of the fluid, thus the term head loss is also used to express the resistance to flow. Many factors affect the head loss in pipes, the viscosity of the fluid being handled, the size of the pipes, the roughness of the internal surface of the pipes, the changes in elevations within the system and the length of travel of the fluid. The resistance through various valves and fittings will also contribute to the overall head loss. A method to model the resistances for valves and fittings is described elsewhere. In a well designed system the resistance through valves and fittings will be of minor significance to the overall head loss, many designers choose to ignore the head loss for valves and fittings at least in the initial stages of a design. Much research has been carried out over many years and various formulae to calculate head loss have been developed based on experimental data. Among these is the Chézy formula which dealt with water flow in open channels. Using the concept of ‘wetted perimeter’ and the internal diameter of a pipe the Chézy formula could be adapted to estimate the head loss in a pipe, although the constant ‘C’ had to be determined experimentally. The Darcy-Weisbach equation Weisbach first proposed the equation we now know as the Darcy-Weisbach formula or DarcyWeisbach equation: hf = f (L/D) x (v2/2g) where: hf = head loss (m) f = friction factor

L = length of pipe work (m) d = inner diameter of pipe work (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s²) or: hf = head loss (ft) f = friction factor L = length of pipe work (ft) d = inner diameter of pipe work (ft) v = velocity of fluid (ft/s) g = acceleration due to gravity (ft/s²) However the establishment of the friction factors was still an unresolved issue which needed further work. Friction Factors Fanning did much experimentation to provide data for friction factors, however the head loss calculation using the Fanning Friction factors has to be applied using the hydraulic radius equation (not the pipe diameter). The hydraulic radius calculation involves dividing the cross sectional area of flow by the wetted perimeter. For a round pipe with full flow the hydraulic radius is equal to ¼ of the pipe diameter, so the head loss equation becomes: hf = f f(L/Rh) x (v2/2g) where Rh = hydraulic radius, f f = Fanning friction factor Darcy introduced the concept of relative roughness, where the ratio of the internal roughness of a pipe to the internal diameter of a pipe, will affect the friction factor for turbulent flow. In a relatively smoother pipe the turbulence along the pipe walls has less overall effect, hence a lower friction factor is applied. The work of many others including Poiseuille, Hagen, Reynolds, Prandtl, Colebrook and White have contributed to the development of formulae for calculation of friction factors and head loss due to friction. The Darcy Friction factor (which is 4 times greater than the Fanning Friction factor) used with Weisbach equation has now become the standard head loss equation for calculating head loss in pipes where the flow is turbulent. Initially the Darcy-Weisbach equation was difficult apply, since no electronic calculators were available and many calculations had to be carried out by hand. The Colebrook-White equation which provides a mathematical method for calculation of the friction factor (for pipes that are neither totally smooth nor wholly rough) has the friction factor term f on both sides of the formula and is difficult to solve without trial and error (i.e. mathematical iteration is normally required to find f).

where: f = friction factor e = internal roughness of the pipe D = inner diameter of pipe work Due to the difficulty of solving the Colebrook-White equation to find f, the use of the empirical ‘Hazen-Williams’ formulae for flow of water at 60º F (15.5º C) has persisted for many years. To use the Hazen-Williams formula a head loss coefficient must be used. Unfortunately the value of the head loss coefficient can vary from around 80 up to 130 and beyond and this can make the ‘Hazen-Williams’ formulae unsuitable for accurate prediction of head loss. The Moody Chart In 1944 LF Moody plotted the data from the Colebrook equation and this chart which is now known as ‘The Moody Chart’ or sometimes the Friction Factor Chart, enables a user to plot the Reynolds number and the Relative Roughness of the pipe and to establish a reasonably accurate value of the friction factor for turbulent flow conditions. The Moody Chart encouraged the use of the Darcy-Weisbach friction factor and this quickly became the method of choice for hydraulic engineers. Many forms of head loss calculator were developed to assist with the calculations, amongst these a round slide rule offered calculations for flow in pipes on one side and flow in open channels on the reverse side. The development of the personnel computer from the 1980’s onwards reduced the time needed to perform the friction factor and head loss calculations, which in turn has widened the use of the Darcy-Weisbach formula to the point that all other formula are now largely unused. Hazen-Williams Formula Empirical formulae are occasionally still used to calculate the approximate head loss in a pipe when water is flowing and the flow is turbulent. Prior to the availability of personal computers the Hazen-Williams formula was very popular with engineers because of the relatively simple calculations required. Unfortunately the results depend upon the value of the friction factor C hw which must be used with the formula and this can vary from around 80 up to 130 and higher, depending on the pipe type, pipe size and the water velocity. The imperial form of the Hazen-Williams formula is: hf = 0.002083 L (100/C)1.85 x (gpm1.85/d4.8655) where: hf = head loss in feet of water L = length of pipe in feet C = friction coefficient gpm = gallons per minute (USA gallons not imperial gallons)

d = inside diameter of the pipe in inches The empirical nature of the friction factor C hw makes the ‘Hazen-Williams’ formula unsuitable for accurate prediction of head loss. The results are only valid for fluids which have a kinematic viscosity of 1.13 centistokes, where the fluid velocity is less than 10 feet per sec and the pipe size is greater than 2” diameter. Water at 60º F (15.5º C) has a kinematic viscosity of 1.13 centistokes. Common Friction Factor Values of C hw used for design purposes are: Asbestos Cement 140 Brass tube 130 Cast-Iron tube 100 Concrete tube110 Copper tube130 Corrugated steel tube 60 Galvanized tubing 120 Glass tube130 Lead piping130 Plastic pipe140 PVC pipe 150 General smooth pipes 140 Steel pipe 120 Steel riveted pipes 100 Tar coated cast iron tube 100 Tin tubing130 Wood Stave 110 These factors include some allowance to provide for the effects of changes to the internal pipe surface due to the build up of deposits or pitting of the pipe wall during long periods of use. Fanning Friction Factor The frictional head loss in pipes with full flow may be calculated by using the following formula and an appropriate Fanning friction factor. hf = f f (L/Rh) x (v2/2g) where: hf = head loss (m) f f = Fanning friction factor L = length of pipe work (m) Rh = hydraulic radius of pipe work (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s²) or: hf = head loss (ft)

f f = Fanning friction factor L = length of pipe work (ft) Rh = hydraulic radius of pipe work (ft) v = velocity of fluid (ft/s) g = acceleration due to gravity (ft/s²) The Fanning friction factor is not the same as the Darcy Friction factor (which is 4 times greater than the Fanning Friction factor) The above formula is very similar to the Darcy-Weisbach formula but the Hydraulic Radius of the pipe work must used, not the pipe diameter. The hydraulic radius calculation involves dividing the cross sectional area of flow by the wetted perimeter. For a round pipe with full flow the hydraulic radius is equal to ¼ of the pipe diameter. i.e. Cross sectional area of flow / Wetted perimeter = (π x d2 / 4) / (π x d) = d/4 Published tables of Fanning friction factors are usually only applicable to the turbulent flow of water at 60º F (15.5º C). The development of ‘The Moody Chart’ which enables engineers to plot the Darcy Friction factor and the use of the personnel computer to calculate the Darcy Friction factor has led to a large reduction in the use of Fanning friction factors. Non-Circular Pipe Friction The frictional head loss in circular pipes is usually calculated by using the Darcy-Weisbach formula with a Darcy Friction factor. For circular pipes the inner pipe diameter is used is used to calculate the Reynolds number and to calculate the relative roughness of the pipe, which are both used to calculate the Darcy Friction factor. To calculate the frictional head loss non-circular pipes the method must be adapted to use the Hydraulic Diameter instead of the internal dimensions of the pipe. Hydraulic Diameter = 4 x cross sectional area of flow / wetted perimeter For a round pipe the Dh = 4 x (π x d2 / 4) / (π x d) = d For a rectangular duct the Dh = 4 x (w x h) / 2 x (w + h)

where w = width, h = height

For an elliptical duct the Dh = 4 x (π x a x b) / π x √ [(2 x (a2 + b2)) – ((a - b)2/2)] where a = major diameter / 2, b = minor diameter /2 , Note: the formula uses an approximation for the circumference of an elliptical duct. For an annulus formed by placing a smaller diameter pipe inside a larger diameter pipe the cross sectional area of flow will be the cross sectional area of the larger pipe calculated using the inner pipe diameter minus the cross sectional area of the smaller pipe calculated using the outer

pipe diameter. The wetted perimeter will be the inner circumference of the larger pipe plus the outer circumference of the smaller pipe. Dh = 4 x (π x (d12 – d22) / 4) / (π x d1 + d2) where d1 = inner diameter of larger pipe, d2 = outer diameter of smaller pipe Example calculation of pipe friction factors: 1. Round pipe: A round steel pipe 0.4 m internal diameter x 10.0 m long carries a water flow rate of 349.1 litres/sec (20.946 m3/min). The temperature of the water is 10o C (50o F). Dh = Internal diameter of pipe = 0.4 m Pipe cross sectional area = π x 0.4002/4 = 0.1256 m2 Flow velocity = 20.94/0.1256/60 = 2.778 m/s Relative roughness = 0.000046/0.4 = 0.000115 Re = v x Dh / (kinematic viscosity in m2/s) = 2.778 x 0.4 / 0.000001307 = 850191 Friction factor = 0.014 (plotted from Moody chart) hf = f (L / Dh) x (v2 / 2g) = 0.014 x (10 / 0.4) x (2.7782 / (2 x 9.81)) = 0.138 m head where: hf = frictional head loss (m) f = friction factor L = length of pipe work (m) Dh = Hydraulic diameter (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s ²) 2. Rectangular duct: A rectangular steel duct 0.6 m wide x 0.3 m high x 10.0 m long carries a water flow rate of 500 litres/sec (30 m3/min). The temperature of the water is 10o C (50o F). Dh = 4 x (0.6 x 0.3) / 2 x (0.6 + 0.3) = 0.4 m Duct cross sectional area = 0.6 x 0.3 = 0.18 m2 Flow velocity = 30.00/0.18/60 = 2.778 m/s Relative roughness = 0.000046/0.4 = 0.000115 Re = v x Dh / (kinematic viscosity in m2/s) = 2.778 x 0.4 / 0.000001307 = 850191 Friction factor = 0.014 (plotted from Moody chart) hf = f (L / Dh) x (v2 / 2g) = 0.014 x (10 / 0.4) x (2.7782 / (2 x 9.81)) = 0.1377 m head where: hf = frictional head loss (m) f = friction factor L = length of pipe work (m) Dh = Hydraulic diameter (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s ²)

Pseudo check calculation: A steel pipe with an internal diameter of 0.400 m x 10 m long carrying a water flow rate of 349.1 litres/sec (20.946 m3/min) will have the same flow velocity as the rectangular duct. If the water temperature is 10o C (50o F) the calculated frictional pressure drop through the steel pipe is 0.138 m head. 3. Elliptical duct: An elliptical duct made from aluminium has internal dimensions of 0.8 m at its widest point and 0.3 m at is highest point. The duct is 10.0 m long and carries a water flow rate of 400 litres/sec (24 m3/min). The temperature of the water is 10o C (50o F). a = major diameter / 2 = 0.800 / 2 = 0.400 b = minor diameter / 2 = 0.300 / 2 = 0.150 Duct cross sectional area = π x a x b = π x 0.400 x 0.150 = 0.1885 m2 Duct circumference = π x √ [(2 x (a2 + b2)) – ((a - b)2/2)] = π x √ [(2 x (0.42 + 0.152)) – ((0.4 – 0.15)2/2)] = π x √[0.365 – 0.03125] = 1.8149 m Dh = 4 x 0.1885 / 1.8149 = 0.415 m Flow velocity = 24.00 / 0.1885 / 60 = 2.1220 m/s Relative roughness = 0.0000015 / 0.415= 0.000003615 Re = v x Dh / (kinematic viscosity in m2/s) = 2.1220 x 0.415 / 0.000001307 = 673780 Friction factor = 0.0123 (plotted from Moody chart) hf = f (L / Dh) x (v2 / 2g) = 0.0123 x (10 / 0.415) x (2.12202 / (2 x 9.81)) = 0.068 m head where: hf = frictional head loss (m) f = friction factor L = length of pipe work (m) Dh = Hydraulic diameter (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s ²) Pseudo check calculation: An aluminium pipe with an internal diameter of 0.415 m x 10 m long carrying a water flow rate of 287.1 litres/sec (17.226 m3/min) will have the same flow velocity as the elliptical duct. If the water temperature is 10o C (50o F) the calculated frictional pressure drop is 0.069 m head. 4. Annulus: An annulus section is formed by placing a stainless steel pipe with an outer diameter of 350 mm inside a stainless steel pipe with an inner diameter of 600. The annulus section is 10 m long and carries a water flow rate of 600 litres/sec (36.00 m3/min). The water temperature is 20o C (68o F). Inner cross sectional area of the larger pipe = π x 0.6002 / 4 = 0.2827 m2 Outer cross sectional area of the smaller pipe = π x 0.3502 / 4 = 0.0962 m2 Cross sectional area of the annulus = 0.2827 - 0.0962 = 0.1865 m2 Inner circumference of the larger pipe = π x 0.600 = 1.8850 m

Outer circumference of the smaller pipe = π x 0.350 = 1.0995 m Wetted perimeter = 1.8850 + 1.0995 = 2.9845 m Dh = 4 x 0.1865 / 2.9845 = 0.250 m Flow velocity = 36.00 / 0.1865 / 60 = 3.217 m/s Relative roughness = 0.000045 / 0.250 = 0.000180 Re = v x Dh / (kinematic viscosity in m2/s) = 3.217 x 0.250 / 0.000001004 = 801045 Friction factor = 0.0146 (plotted from Moody chart) hf = f (L / Dh) x (v2 / 2g) = 0.0146 x (10 / 0.250) x (3.2172 / (2 x 9.81)) = 0.307 m head where: hf = frictional head loss (m) f = friction factor L = length of pipe work (m) Dh = Hydraulic diameter (m) v = velocity of fluid (m/s) g = acceleration due to gravity (m/s ²) Pseudo check calculation: A stainless steel pipe with an internal diameter of 0.250 m x 10 m long carrying a water flow rate of 157.917 litres/sec (9.475 m3/min) will have the same flow velocity as the annulus. If the water temperature is 20o C (68o F) the calculated frictional pressure drop through the steel pipe is 0.307 m head. Viscosity and Density (Metric SI Units) In the SI system of units the kilogram (kg) is the standard unit of mass, a cubic meter is the standard unit of volume and the second is the standard unit of time. Density p The density of a fluid is obtained by dividing the mass of the fluid by the volume of the fluid. Density is normally expressed as kg per cubic meter. p = kg/m3 Water at a temperature of 20°C has a density of 998 kg/m3 Sometimes the term ‘Relative Density’ is used to describe the density of a fluid. Relative density is the fluid density divide by 1000 kg/m3 Water at a temperature of 20°C has a Relative density of 0.998 Dynamic Viscosity μ Viscosity describes a fluids resistance to flow. Dynamic viscosity (sometimes referred to as Absolute viscosity) is obtained by dividing the Shear stress by the rate of shear strain. The units of dynamic viscosity are: Force / area x time The Pascal unit (Pa) is used to describe pressure or stress = force per area This unit can be combined with time (sec) to define dynamic viscosity. μ = Pa•s

1.00 Pa•s = 10 Poise = 1000 Centipoise Centipoise (cP) is commonly used to describe dynamic viscosity because water at a temperature of 20°C has a viscosity of 1.002 Centipoise. This value must be converted back to 1.002 x 10-3 Pa•s for use in calculations. Kinematic Viscosity v Sometimes viscosity is measured by timing the flow of a known volume of fluid from a viscosity measuring cup. The timings can be used along with a formula to estimate the kinematic viscosity value of the fluid in Centistokes (cSt). The motive force driving the fluid out of the cup is the head of fluid. This fluid head is also part of the equation that makes up the volume of the fluid. Rationalizing the equations the fluid head term is eliminated leaving the units of Kinematic viscosity as area / time v = m2/s 1.0

m2/s = 10000 Stokes = 1000000 Centistokes

Water at a temperature of 20°C has a viscosity of 1.004 x 10-6 m2/s This evaluates to 1.004000 Centistokes. This value must be converted back to 1.004 x 10-6 m2/s for use in calculations. The kinematic viscosity can also be determined by dividing the dynamic viscosity by the fluid density. Kinematic Viscosity and Dynamic Viscosity Relationship Kinematic Viscosity = Dynamic Viscosity / Density v=μ/p Centistokes = Centipoise / Density To understand the metric units involved in this relationship it will be necessary to use an example: Dynamic viscosity μ = Pa•s Substitute for Pa = N/m2 and N = kg• m/s2 Therefore μ = Pa•s = kg/(m•s) Density p = kg/m3 Kinematic Viscosity = v = μ/p = (kg/(m•s) x 10-3) / (kg/m3) = m2/s x 10-6 Viscosity and Density (Imperial Units)

In the Imperial system of units the pound (lb) is the standard unit of weight, a cubic foot is the standard unit of volume and the second is the standard unit of time. The standard unit of mass is the slug. This is the mass that will accelerate by 1 ft/s when a force of one pound (lbf) is applied to the mass. The acceleration due to gravity (g) is 32.174 ft per second per second. To obtain the mass of a fluid the weight (lb) must be divided by 32.174. Density p Density is normally expressed as mass (slugs) per cubic foot. The weight of a fluid can be expressed as pounds per cubic foot. p = slugs/ft 3 Water at a temperature of 70°F has a density of 1.936 slugs/ft3 (62.286 lbs/ft3) Dynamic Viscosity μ The units of dynamic viscosity are: Force / area x time μ = lb•s/ft2 Water at a temperature of 70°F has a viscosity of 2.04 x 10-5 lb•s/ft2 1.0 lb•s/ft2 = 47880.26 Centipoise Kinematic Viscosity v The units of Kinematic viscosity are area / time v = ft2/s 1.00 ft 2/s = 929.034116 Stokes = 92903.4116 Centistokes Water at a temperature of 70°F has a viscosity of 10.5900 x 10-6 ft2/s (0.98384713 Centistokes) Kinematic Viscosity and Dynamic Viscosity Relationship Kinematic Viscosity = Dynamic Viscosity / Density v=μ/p The imperial units of kinematic viscosity are ft2/s To understand the imperial units involved in this relationship it will be necessary to use an example: Dynamic viscosity μ = lb•s/ft2 Density p = slugs/ft3 Substitute for slug = lb/32.174 ft•s2

Density p = (lb/32.174 ft•s2)/ft3= (lb/32.174•s2)/ft4 Note: slugs/ft3 can be expressed in terms of lb•s2/ft 4 Kinematic Viscosity v = (lb•s/ft2)/(slugs/ft3) Substitute lb•s2/ft 4 for slugs/ft3 Kinematic Viscosity v = (lb•s/ft2 )/(lb•s2/ft4) = ft2/s