Fanning Friction Factor

Determine the friction factor (f_D) 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.

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 Re_d,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 the following equations:

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.

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:

which is also known as the Colebrook equation, expresses the Darcy friction factor f as a function of pipe relative roughness ε / D_h 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 the Blasius formula:

 f = (100.Re)^-¼

It must be noted that the friction factor is independent of the Reynolds number at very large Reynolds numbers. This is because the thickness of the laminar sublayer (viscous sublayer) decreases with increasing Reynolds number. For very large Reynolds numbers, the thickness of the 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.

The primary circuit of typical PWRs is divided into 4 independent loops (piping diameter of about 700mm), each loop comprises a steam generator and one main coolant pump.

Assume that (this data do not represent any certain reactor design):

• Inside the primary piping flows water at a constant temperature of 290°C (⍴ ~ 720 kg/m³).
• The kinematic viscosity of the water at 290°C is equal to 0.12 x 10^-6 m²/s.
• The primary piping flow velocity may be about 17 m/s.
• The primary piping of one loop is about 20m long.
• The Reynolds number inside the primary piping is equal to: Re_D = 17 [m/s] x 0.7 [m] / 0.12×10^-6 [m²/s] = 99 000 000
• The Darcy friction factor is equal to f_D = 0.01

Calculate the head loss for one loop of primary piping (without fitting, elbows, pumps, etc.).

Solution:

Since we know all inputs of the Darcy-Weisbach equation, we can calculate the head loss directly:

Head loss form:

Δh = 0.01 x ½ x 1/9.81 x 20 x 17² / 0.7 = 4.2 m

Pressure loss form:

Δp = 0.01 x ½ x 720 x 20 x 17² / 0.7 = 29 725 Pa ≈ 0.03 MPa

In fully developed laminar flow in a circular pipe, the head loss is given by:
where:

Since the Reynolds number is inverse proportional to viscosity, then the resulting head loss becomes proportional to viscosity. Therefore, the head loss is reduced by half when the viscosity of the fluid is reduced by half, when the flow rate and thus the average velocity are held constant.

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