## Pipe Friction Models

What Is Friction.

Friction can be defined as a force which opposes the relative motion between two elements in contact. Friction converts the kinetic energy of the moving bodies into heat energy causing useful work loss.

Friction in fluids occur due to a the friction between the layers of fluids moving relative to each other. The friction in fluids is dependent on a property called viscosity. This is the resistance offered to flow or deformation by a fluid.

Fluid Friction In Pipes

Pressure drops or head loss can be experienced in pipes carrying fluids due to the frictional forces between the boundary layer(outer layer) of the moving fluid and the internal surface of the pipe combined with the viscosity. The extent of head loss is dependent on some factors in the fluid, some of these factors are:

• Types Of Flow: : Flow in pipes can either be laminar, turbulent or transitional. Laminar flow is characterized by smooth orderly streamlines which do not cross while turbulent flow is characterized by disorderly streamlines and erratic, small, whirlpool-like circles called eddies.
• Reynolds Number :This is a dimensionless parameter used to help predict the type of flow in the fluid at different conditions. It is given as:

Re = Vavg × ρ × D / μ

• Where Vavg = Average Velocity, ρ = density, D = Characteristic length, μ= Kinematic viscosity. For laminar flow: Re < 2100, Turbulent flow: Re > 4000, Transitional flow: 2000< Re > 4000
• Relative Roughness: This is a measure of the roughness of the inner pipe surface. It is gotten by dividing the height of surface irregularities by the diameter of the pipe.

Other factors which influence the head loss from friction in a pipe are: viscosity, temperature, pipe diameter, flow velocity etc.

When designing piping systems, identifying the head loss due to friction is of paramount importance because it makes up the majority of losses in pipe flow. To that end, several models have been developed to try and approximate the operation of these pipes under various condition so that the losses from friction can be easily approximated and accounted for. Here are some of them:

Moody Friction Model

The Moody Diagram is a chart plotted by engineer Lewis F. Moody in the early 20th century. Through experiments, he found out that the type of flow in pipes could be described by four dimensionless parameters: Reynolds number, pressure loss coefficient of the pipe, diameter ratio and the relative roughness of the pipe). He then represented this on a graph by plotting the Reynold number, relative roughness and Darcy - Weisbach friction factor against each other.

Formula

• The surface roughness is calculated by dividing the height of surface irregularities (ϵ) by the pipe diameter (D).
• The Reynolds number (Re) is calculated using the formula given in eqn(1).
• The Darcy friction factor (fd) is gotten by consulting the chart with the values gotten for the Reynolds number and the relative roughness.

Applications

The Moody chart is used extensively in calculating head loss and pressure drops in pipes in tandem with the Darcy-Weisbach equation. The friction factor for different flow regimes (laminar, turbulent and transitional) can be gotten from the Moody diagram. which is then substituted into the Darcy-Weisbach equation. To find the head loss.

Example 1

Water at 35oC is flowing through a steel pipe of diameter 100mm at 3m/s. Calculate the Moody friction factor for:

• A new smooth pipe
• A corroded pipe.

Solution
From the values given, we have: temperature = 35C, pipe internal diameter = 0.1m, velocity = 3m/s, density = 994.1 kg/m 3 , kinematic viscosity = 0.000720 kg/m sec (From standard steam tables). With these values, we find the Reynolds number with (1):

Re = Vavg × ρ × D / μ = 3 * 994.1 * 0.1 / 0.000720 = 414208. approx 4.14× 105

We then find the absolute roughness by dividing the relative roughness(ϵ) by the pipe diameter(D). For a smooth steel pipe we have relative roughness = 0.000061m. For a corroded pipe we have relative roughness = 0.004m(The American Iron & Steel Institute Committee for Steel Pipe Producers)

Calculating the absolute roughness:
For the smooth pipe = 0.000061/0.1 = 0.00061
For the corroded pipe = 0.004/0.1 = 0.04

Consulting the Moody chart, we have Darcy friction factors.
For the smooth pipe: 0.018
For the corroded pipe: 0.029

• The Moody chart provides an easy to use graphical aid for calculating the Darcy friction factor.
• The Moody chart has a special area devoted to the Darcy frictional factor for transitional flow which cannot be gotten accurately with most conventional equations.

• At very high Reynolds numbers, the friction factor becomes independent of the Reynolds number.

Hazen - Williams Friction Model

The Hazen -Williams friction model was developed for use in water pipe systems, although in recent times, it has found use in refined petroleum pipelines. It was initially developed as an alternative to the Darcy - Weisbach equation because the Darcy friction factor was difficult to calculate for turbulent flows in the past.

Formula

The Hazen-Williams equation calculates the head loss or pressure drop in water pipelines when the pipe diameter and flow rate is given, taking into account a factor C which is dependent on the internal smoothness of the pipe. The formula used to calculate the

Where L= Length of pipe(m), Q = Flow rate(m3/s), D =Pipe inside diameter(m), C = Hazen - Williams C factor.
*The coefficient of smoothness C is a factor that increases with increasing smoothness of the pipe.

Applications

The Hazen -Williams equation is primarily used in the design of water distribution systems such as irrigation pipelines, water sprinklers and fire stations. It is also applied in refined petroleum pipelines.

Example 1

For the conditions given in example 1, find the head loss due to friction for both the smooth and corroded pipe using the Hazen - Williams equation.

Solution
In example I, we had velocity(v) = 3m/s, equivalent length = 500m, pipe internal diameter = 0.1m. Q = velocity *area of pipe = 0.024m3 /s

To find the head loss due to friction, we need the Coefficient of Smoothness C which is 140 for smooth steel pipe and 100 for the corroded steel pipe.(The American Iron & Steel Institute Committee for Steel Pipe Producers)

So, substituting the values into (2), we have:
Head loss in the smooth pipe:
hL = 10.67 * 500 * (0.024)1.87 ÷ (1401.852 * 0.14.87 ) = 40.54m
Head loss in the corroded steel pipe:
hL = 10.67 * 500 * (0.024)1.87 ÷ (1001.852 * 0.14.87 ) = 75.6m

• It is easier to solve than the Darcy - Weisbach equation.
• The roughness coefficient C is not dependent on the Reynolds number which makes it easier to use than other equations.

• It can only be used for water and select refined petroleum products .
• It does not take into account the temperature of the fluid and is valid for only a narrow range of temperature.
• It does not take into account the viscosity and density of the fluid.
• The model suffers significant accuracy loss as pipe diameter increases.
• The value of C varies over the lifetime of the pipe.

Darcy - Weisbach Friction Model

The Darcy-Weisbach equation is the most commonly used formula in determining the head loss in pipes. The Darcy - Weisbach equation is an empirical equation that relates the head loss or pressure loss due to friction in a pipe to the average velocity of the fluid.

Formula

The head loss formula for the Darcy Weisbach equation utilizes a special friction factor fd known as the Darcy Friction Factor, the equivalent length and pipe diameter in order to relate the head loss to friction to the velocity head. It is expressed as:

hL = fD × (L/D) × V2/2g……..(4)

Where fd =Darcy friction factor, L=Equivalent length of pipe, D = Diameter of pipe, V = Velocity of the fluid, g = acceleration due to gravity

The Darcy friction factor can be gotten by a number of methods. The easiest way is to read it from the Moody Chart. It can also be gotten by solving the respective equations.

For Laminar flow: fd = 64/Re

For turbulent flow, we can solve the Colebrook-White equation for fd

Where e = Absolute roughness(m), D = Pipe internal diameter, Re =Reynolds number.

*The Darcy Weisbach equation assumes the flow is unaffected by the pipes roughness when the flow is laminar.

Applications

The Darcy-Weisbach is widely viewed as the most accurate equation used in calculating the friction loss for incompressible Newtonian fluid flow in pipes. It is applied in designing piping systems for various fluids and even in some low speed gas flows over short distances provided there aren’t any major changes in density.

Other equations like the Hazen-Williams equation were used before in determining the head loss because of the difficulty encountered in obtaining the friction factor fd. But with the advent of new and faster computing methods the Darcy-Weisbach formula has emerged as the preferred equation for calculating the head loss in incompressible fluid flow in pipes.

Example 3

For the conditions given in example 1, calculate the head loss due to friction in the pipe using the Darcy - Weisbach equation.

Solution
We have velocity(v) = 3m/s, acceleration due to gravity(g) = 9.81m/s2 , equivalent
length = 500m, pipe ID = 0.1m, Darcy friction factor = 0.018 for smooth steel pipe,
0.04 for corroded pipe(example 1)

Substituting these values into (4) to find the head loss, we have:
For the smooth pipe, we have
hL = 0.018 * (500/0.1) * (32 /2 * 9.81) = 41.2m
For the corroded pipe, we have
hL = 0.04 * (500/0.1) * (32 /2 *9.81) = 91.73m

• It can be used for a wide range of Newtonian fluids.
• It takes into account the pipe roughness, fluid viscosity and density.
• It is the most accurate formula available for modelling losses in fluid flow in pipes.

• Without the Moody chart, the Friction factors for turbulent flows are hard to determine.

Shell-MIT Friction Model.

This model was developed by the petroleum company Shell in conjunction with MIT. It was primary developed for use in oil pipeline engineering. The pipelines were used for transporting heavy crude oils that undergo heating to reduce their viscosity and enhance pipe flow..

Formula

The Shell-MIT formula uses a modified Reynolds number in its calculations. This
modified Reynolds number is gotten by diving the Reynolds number by 772.
Rem = Re/ 772.……(5)
Where Rem =Modified Reynolds number, Re= Reynolds number

The friction factor is then calculated with the modified Reynold’s number:

f = 0.00207 + Rem ( For laminar flow)
f = 0.0018 +0.00662(1/Rem)0.355 (For turbulent flow)
*This is different from the Darcy - Weisbach friction factor.
The friction factor is then used to calculate the pressure drop in the pipeline using this
formula.

Pm = 6.2191× 1010 (f ×Sg×Q2 ) / D5 ……..(6)

Where Pm = Frictional pressure drop (kPa/km), f = friction factor, Sg =Specific
gravity of the liquid, Q = flow rate(m3/hr), D = pipe internal diameter(mm).

Application

The Shell-MIT equation is used specifically in the petroleum industry to calculate the pressure drop in heavy crude oil and heated pipelines.

• It is simpler to use and solve than some of the other oil pipeline equations like the Miller equation, Aude equation and the Colebrook-White equation.
• It considers the Reynold’s number and takes into account the roughness of the pipe.

• It is only suitable for heavy crude oil and high pour fuel oils.

Other equations like the Miller and Aude equations are used in the oil industry for calculating the pressure drop in crude oil pipelines. They have been replaced by the Shell-MIT equation due to certain restrictions and difficulties encountered when using
them.

Conclusion
Comparing the values from the Hazen -Williams and the Darcy Weisbach factor.

• Hazen- Williams Equation
• Darcy-Weisbach equation
• Smooth steel pipe
• 40.54m
• 41.2m
• Corroded steel pipe
• 75.6m
• 91.73m

From the results gotten, we can see that:

1. There is clear agreement between the two methods for a smooth (clean) pipe with
just a little variation in the values gotten.

2. As the pipe roughness increases, the variation between the two methods also
increases. This increase is due to the discrepancies between the various
experimental methods used to derive the relative toughness and the smoothness
coefficient C for the rough pipes.

In conclusion, the Darcy-Weisbach model is the best model to use by default in fluid
pipe flow.
It takes into consideration factors such as density and kinematic viscosity which have
a pronounced effect on the Reynolds number which can in turn impact the losses in
the pipe flow. It is also retains it’s validity over a wide range of temperatures for
Newtonian fluids.
With advances in computation, the Darcy friction factor has also become easier to
calculate which makes using the Darcy- Weisbach equation easier.

References

• Liu H., 2005, Pipeline Engineering, Lewis Publishers.
• Meno E.S., 2015, Transmission Pipeline and Simulations Manual, Elsevier Publishers.
• Silowash B., 2010, Piping Systems Manual, Mc-Graw Hill, inc..