Flow through Elbow – Minor Loss

Flow through Elbow – Minor Loss

The flow through elbows is quite complicated. In fact, any curved pipe always induces a larger loss than the simple straight pipe. This is due to the fact in a curved pipe the flow separates on the curved walls. For very small radius of curvature the incoming flow is even unable to make the turn at the bend, therefore the flow separates and in part stagnates against the opposite side of the pipe. In this part of the bend the pressure raises (as a result of the Bernoulli’s principle) and the velocity decreases.

An interesting feature of the K-values for elbows is their non-monotone behavior as R/D ratio increases. The K-values include both the local losses and frictional losses of the pipe. The local losses, caused by flow separation and secondary flow, decrease with R/D, while the frictional losses increase because the bend length increases. Therefore there is a minimum in the K-value near the normalized radius of curvature of 3.

flow through elbow - minor loss

Summary:

  • Head loss of hydraulic system is divided into two main categories:
    • Major Head Loss – due to friction in straight pipes
    • Minor Head Loss – due to components as valves, bends…
  • A special form of Darcy’s equation can be used to calculate minor losses.
  • The minor losses are roughly proportional to the square of the flow rate and therefore they can be easy integrated into the Darcy-Weisbach equation through resistance coefficient K.
  • As a local pressure loss fluid acceleration in a heated channel can be also considered.

There are following methods:

  • Equivalent length method
  • K-method (resistance coeff. method)
  • 2K-method
  • 3K-method

Why the head loss is very important?

As can be seen from the picture, the head loss is forms key characteristic of any hydraulic system. In systems, in which some certain flowrate must be maintained (e.g. to provide sufficient cooling or heat transfer from a reactor core), the equilibrium of the head loss and the head added by a pump determines the flowrate through the system.

Q-H characteristic diagram of centrifugal pump and of pipeline
Q-H characteristic diagram of centrifugal pump and of pipeline
 
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:

Minor Loss