MOODY DIAGRAM: EQUATIONS, WHAT IT IS FOR, APPLICATIONS - PHYSICAL - 2025

The Moody diagram consists of a series of curves drawn on logarithmic paper, which are used to calculate the friction factor present in the flow of a turbulent fluid through a circular duct.

With the friction factor f, the loss of energy due to friction is evaluated, an important value to determine the adequate performance of the pumps that distribute fluids such as water, gasoline, crude oil and others.

Pipes at industrial level. Source: Pixabay.

To know the energy in the flow of a fluid, it is necessary to know the gains and losses due to factors such as speed, height, the presence of devices (pumps and motors), the effects of the viscosity of the fluid and the friction between it. and the pipe walls.

Equations for the energy of a moving fluid

Where N _R is the Reynolds number, whose value depends on the regime in which the fluid is. The criteria is:

The Reynolds number (dimensionless) in turn depends on the velocity of the fluid v, the internal diameter of the pipe D and the kinematic viscosity n of the fluid, whose value is obtained by means of tables:

Colebrook equation

For a turbulent flow the most accepted equation in copper and glass pipes is that of Cyril Colebrook (1910-1997), but it has the disadvantage that f is not explicit:

In this equation the ratio e / D is the relative roughness of the pipe and N _R is the Reynolds number. A careful observation shows that it is not easy to leave f to the left side of the equality, so it is not suitable for immediate calculations.

Colebrook himself suggested this approach, which is explicit, valid with some limitations:

What is it for?

The Moody diagram is useful to find the friction factor f included in Darcy's equation, since it is not easy to express f directly in terms of other values ​​in the Colebrook equation.

Its use simplifies obtaining the value of f, by containing the graphical representation of f as a function of N _R for different values ​​of relative roughness on a logarithmic scale.

Moody diagram. Source:

These curves have been created from experimental data with various materials commonly used in pipe fabrication. The use of a logarithmic scale for both f and N _R is necessary, since they cover a very wide range of values. In this way the graphing of values ​​of different orders of magnitude is facilitated.

The first graph of the Colebrook equation was obtained by engineer Hunter Rouse (1906-1996) and shortly thereafter modified by Lewis F. Moody (1880-1953) in the form in which it is used today.

It is used for both circular and non-circular pipes, simply substituting the hydraulic diameter for these.

How is it made and how is it used?

As explained above, the Moody diagram is made from numerous experimental data, presented graphically. Here are the steps to use it:

- Calculate the Reynolds number N _R to determine if the flow is laminar or turbulent.

- Calculate the relative roughness using the equation e _r = e / D, where e is the absolute roughness of the material and D is the internal diameter of the pipe. These values ​​are obtained through tables.

- Now that e _r and N _{R are available}, project vertically until reaching the curve corresponding to the e _r obtained.

- Project horizontally and to the left to read the value of f.

An example will help to easily visualize how the diagram is used.

-Resolved example 1

Determine the friction factor for water at 160º F flowing at a rate of 22 ft / s in a duct made of uncoated wrought iron with an internal diameter of 1 inch.

Solution

Required data (found in tables):

First step

The Reynolds number is calculated, but not before passing the internal diameter from 1 inch to feet:

According to the criteria shown before, it is a turbulent flow, then the Moody diagram allows obtaining the corresponding friction factor, without having to use the Colebrook equation.

Second step

You have to find the relative roughness:

Third step

In the supplied Moody diagram, it is necessary to go to the extreme right and find the closest relative roughness to the value obtained. There is no one that corresponds exactly to 0.0018, but there is one that is quite close, 0.002 (red oval in the figure).

Simultaneously, the corresponding Reynolds number is searched on the horizontal axis. The closest value to 4.18 x 10 ⁵ is 4 x 10 ⁵ (green arrow in the figure). The intersection of both is the fuchsia point.

Fourth step

Project to the left following the blue dotted line and reach the orange point. Now estimate the value of f, taking into account that the divisions do not have the same size as they are a logarithmic scale on both the horizontal and vertical axes.

The Moody diagram provided in the figure does not have fine horizontal divisions, so the value of f is estimated at 0.024 (it is between 0.02 and 0.03 but it is not half but a little less).

There are calculators online that use the Colebrook equation. One of them (see References) supplied the value 0.023664639 for the friction factor.

Applications

The Moody diagram can be applied to solve three types of problems, provided the fluid and the absolute roughness of the pipe are known:

- Calculation of the pressure drop or the pressure difference between two points, given the length of the pipe, the difference in height between the two points to be considered, the speed and the internal diameter of the pipe.

- Determination of the flow, knowing the length and diameter of the pipe, plus the specific pressure drop.

- Evaluation of the diameter of the pipe when the length, flow and pressure drop between the points to be considered are known.

Problems of the first type are solved directly through the use of the diagram, while those of the second and third types require the use of a computer package. For example, in the third type, if the diameter of the pipe is not known, the Reynolds number cannot be evaluated directly, nor the relative roughness.

One way to solve them is to assume an initial internal diameter and from there successively adjust the values ​​to obtain the pressure drop specified in the problem.

-Resolved example 2

You have water at 160 ° F flowing steadily through a 1-inch diameter uncoated wrought iron pipe at a rate of 22 ft / s. Determine the pressure difference caused by friction and the pumping power required to maintain flow in a length of horizontal pipe L = 200 feet long.

Solution

Data needed: the acceleration of gravity is 32 ft / s ²; the specific gravity of water at 160ºF is γ = 61.0 lb-force / ft ³

This is the pipe from solved example 1, therefore the friction factor f is already known, which has been estimated at 0.0024. This value is taken into Darcy's equation to evaluate friction losses:

The required pumping power is:

Where A is the cross-sectional area of ​​the tube: A = p. (D ² /4) = p. (.0833 ² /4) foot ² = 0.00545 foot ²

Therefore the power required to maintain the flow is W = 432.7 W

References

1. Cimbala, C. 2006. Fluid Mechanics, Fundamentals and Applications. Mc. Graw Hill. 335- 342.
2. Franzini, J. 1999. Fluid Mechanics with Application is in Engineering. Mc. Graw Hill. 176-177.
3. LMNO Engineering. Moody Friction Factor Calculator. Recovered from: lmnoeng.com.
4. Mott, R. 2006. Fluid Mechanics. 4th. Edition. Pearson Education. 240-242.
5. The Engineering Toolbox. Moody Diagram. Recovered from: engineeringtoolbox.com
6. Wikipedia. Moody Chart. Recovered from: en.wikipedia.org