Moody Diagram

Moody Diagram

The Moody diagram (also known as the Moody chart) is a graph in non-dimensional form that relates the Darcy friction factor, Reynolds number, and the relative roughness for fully developed flow in a circular pipe.

Moody Diagram
Source: Donebythesecondlaw at the English language Wikipedia, CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php?curid=4681366
 
Darcy Friction Factor
The Darcy friction factor is a dimensionless quantity used in the Darcy–Weisbach equation, for the description of frictional losses in pipe or duct as well as for open-channel flow. This is also called the Darcy–Weisbach friction factor, resistance coefficient, or simply friction factor.

The friction factor has been determined to depend on the Reynolds number for the flow and the degree of roughness of the pipe’s inner surface (especially for turbulent flow). The friction factor of laminar flow is independent of roughness of the pipe’s inner surface.
darcy friction factor
The pipe cross-section is also important, as deviations from circular cross-section will cause secondary flows that increase the head loss. Non-circular pipes and ducts are generally treated by using the hydraulic diameter.

Relative Roughness
The quantity used to measure the roughness of the pipe’s inner surface is called the relative roughness, and it is equal to the average height of surface irregularities (ε) divided by the pipe diameter (D).

relative roughness - equation

,where both the average height surface irregularities and the pipe diameter are in millimeters.

If we know the relative roughness of the pipe’s inner surface, then we can obtain the value of the friction factor from the Moody Chart.

The Moody chart (also known as the Moody diagram) is a graph in non-dimensional form that relates the Darcy friction factor, Reynolds number, and the relative roughness for fully developed flow in a circular pipe.

relative roughness - absolute roughness

Reynolds Number
The Reynolds number is the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. It can be interpreted that when the viscous forces are dominant (slow flow, low Re) they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. Even very low Re indicates viscous creeping motion, where inertia effects are negligible. When the inertial forces dominate over the viscous forces (when the fluid is flowing faster and Re is larger) then the flow is turbulent.

reynolds number

It is a dimensionless number comprised of the physical characteristics of the flow. An increasing Reynolds number indicates an increasing turbulence of flow.

It is defined as:
Reynolds number

where:
V is the flow velocity,
D is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.)
ρ fluid density (kg/m3),
μ dynamic viscosity (Pa.s),
ν kinematic viscosity (m2/s);  ν = μ / ρ.

Example: Moody Diagram

Determine the friction factor (fD) for fluid flow in a pipe of 700mm in diameter that has the Reynolds number of 50 000 000 and an absolute roughness of 0.035 mm.

Solution:

The relative roughness is equal to ε = 0.035 / 700 = 5 x 10-5. Using the Moody Chart, a Reynolds number of 50 000 000 intersects the curve corresponding to a relative roughness of 5 x 10-5 at a friction factor of 0.011.

Moody chart, moody diagram
Example: Moody chart.
Source: Donebythesecondlaw at the English language Wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4681366

Darcy Friction Factor for various flow regime

The most common classification of flow regimes is according to the Reynolds number. The Reynolds number is a dimensionless number comprised of the physical characteristics of the flow and it determines whether the flow is laminar or turbulent. An increasing Reynolds number indicates an increasing turbulence of flow. As can be seen from the Moody chart, also Darcy friction factor is highly dependent on the flow regime (i.e. on the Reynolds number).

 
Darcy Friction Factor for Laminar Flow
For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Red,crit = 2300. For laminar flow, the head loss is proportional to velocity rather than velocity squared, thus the friction factor is inversely proportional to velocity.

The Darcy friction factor for laminar (slow) flows is a consequence of Poiseuille’s law that and it is given by following equations:

darcy friction factor for laminar flow

Darcy Friction Factor for Transitional Flow
At Reynolds numbers between about 2000 and 4000 the flow is unstable as a result of the onset of turbulence. These flows are sometimes referred to as transitional flows. The Darcy friction factor contains large uncertainties in this flow regime and is not well understood.
Darcy Friction Factor for Turbulent Flow
If the Reynolds number is greater than 3500, the flow is turbulent. Most fluid systems in nuclear facilities operate with turbulent flow. In this flow regime the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. The Darcy friction factor depends strongly on the relative roughness of the pipe’s inner surface.

The most common method to determine a friction factor for turbulent flow is to use the Moody chart. The Moody chart (also known as the Moody diagram) is a log-log plot of the Colebrook correlation that relates the Darcy friction factor, Reynolds number, and the relative roughness for fully developed flow in a circular pipe. The Colebrook–White equation:

darcy friction factor for turbulent flow

which is also known as the Colebrook equation, expresses the Darcy friction factor f as a function of pipe relative roughness ε / Dh and Reynolds number.

In 1939, Colebrook found an implicit correlation for the friction factor in round pipes by fitting the data of experimental studies of turbulent flow in smooth and rough pipes.

For hydraulically smooth pipe and the turbulent flow (Re < 105) the friction factor can be approximated by Blasius formula:

 f = (100.Re)

darcy friction factor - relative roughnessIt must be noted, at very large Reynolds numbers, the friction factor is independent of the Reynolds number. This is because the thickness of laminar sublayer (viscous sublayer) decreases with increasing Reynolds number. For very large Reynolds numbers the thickness of laminar sublayer is comparable to the surface roughness and it directly influences the flow. The laminar sublayer becomes so thin that the surface roughness protrudes into the flow. The frictional losses in this case are produced in the main flow primarily by the protruding roughness elements, and the contribution of the laminar sublayer is negligible.

 
References:
Reactor Physics and Thermal Hydraulics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
  6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
  7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
  8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
  9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.
  10. White Frank M., Fluid Mechanics, McGraw-Hill Education, 7th edition, February, 2010, ISBN: 978-0077422417

See above:

Major Loss