He **moody diagram** It 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.

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 fluid’s viscosity and the friction between it. and the pipe walls.

[toc]

**Equations for the energy of a moving fluid**

Between two sections of a pipe, denoted as *1* and *2,* It is possible to establish the following balance, which is an expansion of Bernoulli’s equation: Where:

*–p1* and *p2* are the pressures at each point,

*–z1* and *z2* are the heights with respect to the reference point,

–v1 and *v2* are the respective velocities of the fluid,

*– ha* is the energy added by pumps, *hr* is the energy taken by some device such as a motor, and *hL* It covers the energy losses of the fluid due to friction between it and the walls of the pipes, as well as other minor losses.

The value of *hL* is calculated using the Darcy-Weisbach equation:

Where *L* is the length of the pipe, *D.* is its inner diameter, *v* is the velocity of the fluid and *g* is the value of the acceleration due to gravity. The dimensions of *hL* are of length, and usually the units in which it is represented are meters or feet.

**-Friction factor and Reynolds number**

To calculate *F* Empirical equations obtained from experimental data can be used. It is necessary to distinguish whether it is a fluid in a laminar regime or in a turbulent regime. For the laminar regime *F* is easily evaluated:

*f = 64/NR*

Where *NR* is the Reynolds number, whose value depends on the regime in which the fluid is found. The criteria is:

*Laminar flow: NR < 2000 the flow is laminar; Turbulent flow NR > 4000; Transition regime: 2000 < NR < 4000*

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 *no* of the fluid, whose value is obtained by means of tables:

*NR = vD /n*

**Colebrook equation**

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

In this equation the quotient *e/D* is the relative roughness of the pipe and *NR* is the Reynolds number. When observing it carefully, it is noticed that it is not easy to leave *F* to the left side of the equality, so it is not convenient for immediate calculations.

Colebrook himself suggested this approximation that is explicit, valid with some limitations:

**What is it for?**

The Moody diagram is useful for finding the friction factor *F* included in the Darcy equation, since in the Colebrook equation it is not easy to express *F* directly in terms of other values.

Its use simplifies obtaining the value of *F*as it contains the graphical representation of *F *in function of *NR* for different values of relative roughness on a logarithmic scale.

These curves have been created from experimental data with various materials commonly used in the manufacture of pipes. The use of a logarithmic scale for both *F* as for *NR* It is necessary, since they cover a very wide range of values. This makes it easier to plot values of different orders of magnitude.

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

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

**How is it made and how is it used?**

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

– Calculate the Reynolds number *NR* to determine if the flow is laminar or turbulent.

– Calculate the relative roughness using the equation *er = e/D*where *and* is the absolute roughness of the material and D is the internal diameter of the pipe. These values are obtained through tables.

– Now that you have *er *and *NR*project vertically until reaching the curve corresponding to the *er* obtained.

– Project horizontally and to the left to read the value of *F*.

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

**-Worked example 1**

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

**Solution**

Necessary data (found in the tables):

*Kinematic viscosity of water at 160ºF: 4.38 x 10-6 ft2/s*

*Absolute roughness of uncoated wrought iron: **1.5 x 10 -4 feet*

**First step**

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

*1 inch = 0.0833 feet*

*NR = (22 x 0.0833)/ 4.38 x 10-6= 4.18 x 10 5*

According to the criterion shown before, it is a turbulent flow, so 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:

*er = 1.5 x 10 -4 / 0.0833 = 0.0018*

**Third step**

In the provided Moody diagram it is necessary to locate to the extreme right and look for the closest relative roughness to the obtained value. There is none that corresponds exactly to 0.0018 but there is one that is quite close, that of 0.002 (red oval in the figure).

Simultaneously, the corresponding Reynolds number is found 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 it is a logarithmic scale on both the horizontal and vertical axis.

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

There are online calculators 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, provided the length of the pipe, the difference in height between the two points to be considered, the velocity and the internal diameter of the pipe.

– Determination of the flow, given the length and diameter of the pipe, plus the specific pressure drop.

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

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

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

**-Worked example 2**

Water at 160°F is flowing steadily through a 1-in.-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 horizontal piece of pipe L = 200 ft long.

**Solution**

Data Needed: Acceleration due to gravity is 32 ft/s2; the specific weight of water at 160ºF is γ = 61.0 lb-force/ft3

This is the pipe from worked example 1, therefore the friction factor is already known* F*, which has been estimated at 0.0024. This value is taken to Darcy’s equation to evaluate friction losses:

The required pumping power is:

*W = v. A. (p1 – p2)*

Where A is the cross sectional area of the tube: A = p. (D2/4) = e.g. (0.08332/4) ft2 = 0.00545 ft2

* **W = 22 ft/s. 2659.6 lb-force/ft2. 0.00545 ft2= 318.9 lb-force. feet*

* *Power is best expressed in Watts, for which the conversion factor is required:

*1 Watt = 0.737 lb-force. feet*

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

**References**

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