In general, the hydraulic head, or total head, is a measure of the potential of fluid at the measurement point. It can be used to determine a hydraulic gradient between two or more points.
Bernoulli Theorem - Equation

In fluid dynamics, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. The units for all the different forms of energy in the Bernoulli’s equation can be measured also in units of distance, and therefore these terms are sometimes referred to as “heads” (pressure head, velocity head, and elevation head). Head is also defined for pumps. This head is usually referred to as the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can be usually read from its Q-H curve (flow rate – height).

There are four types of potential (head):

  • Pressure potential – Pressure head: The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.Pressure Headρw: density of water assumed to be independent of pressure
  • Elevation potential – Elevation head: The elevation head represents the potential energy of a fluid due to its elevation above a reference level.Elevation Head
  • Kinetic potential – Kinetic head: The kinetic head represents the kinetic energy of the fluid. It is the height in feet that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.Kinetic Head

The sum of the elevation head, kinetic head, and pressure head of a fluid is called the total head. Thus, Bernoulli’s equation states that the total head of the fluid is constant.

Total Hydraulic Head

Consider a pipe containing an ideal fluid. If this pipe undergoes a gradual expansion in diameter, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease in order to maintain the same mass flow rate. Since the outlet velocity is less than the inlet velocity, the kinetic head of the flow must decrease from the inlet to the outlet. If there is no change in elevation head (the pipe lies horizontal), the decrease in kinetic head must be compensated for by an increase in pressure head.

The head loss of a pipe, tube or duct system, is the same as that produced in a straight pipe or duct whose length is equal to the pipes of the original systems plus the sum of the equivalent lengths of all the components in the system.

As can be seen, the head loss of piping system is divided into two main categories, “major losses” associated with energy loss per length of pipe, and “minor losses” associated with bends, fittings, valves, etc.

  • Major Head Loss – due to friction in pipes and ducts.
  • Minor Head Loss – due to components as valves, fittings, bends and tees.

The head loss can be then expressed as:

hloss = Σ hmajor_losses + Σ hminor_losses

Minor Head Loss – Local Losses

In industry any pipe system contains different technological elements as bends, fittings, valves or heated channels. These additional components add to the overall head loss of the system. Such losses are generally termed minor losses, although they often account for a major portion of the head loss. For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses (especially with a partially closed valve that can cause a greater pressure loss than a long pipe, in fact when a valve is closed or nearly closed, the minor loss is infinite).

The minor losses are commonly measured experimentally. The data, especially for valves, are somewhat dependent upon the particular manufacturer’s design.

Like pipe friction, 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. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss.minor head loss - equation

There are several methods how to calculate head loss from fittings, bends and elbows. In the following section these methods are summarized in the order from the simplest to the most sophisticated.

Sometimes, engineers use the pressure loss coefficient, PLC. It is noted K or ξ  (pronounced “xi”). This coefficient characterizes pressure loss of a certain hydraulic system or of a part of a hydraulic system. It can be easily measured in hydraulic loops. The pressure loss coefficient can be defined or measured for both straight pipes and especially for local (minor) losses.

PLC - Pressure loss coefficient - equations

Example - pressure loss coefficient

Equivalent Length Method

The equivalent length method (The Le/D method) allows the user to describe the pressure loss through an elbow or a fitting as a length of straight pipe.

This method is based on the observation that the major losses are also proportional to the velocity head (v2/2g).

equivalent length method

The Le/D method simply increases the multiplying factor in the Darcy-Weisbach equation (i.e. ƒ.L/D) by a length of straight pipe (i.e. Le) which would give rise to a pressure loss equivalent to the losses in the fittings, hence the name “equivalent length”. The multiplying factor therefore becomes ƒ(L+Le)/D and the equation for calculation of pressure loss of the system is therefore:

equivalent length

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

table of equivalent lengths - valves, elbows, bendsAll fittings, elbows, tees,  can be summed up to make one total length, and the pressure loss calculated from this length. It has been found experimentally that if the equivalent lengths for a range of sizes of a given type of fitting are divided by the diameters of the fittings then an almost constant ratio (i.e. Le/D) is obtained. The advantage of the equivalent length method is that a single data value is sufficient to cover all sizes of that fitting and therefore the tabulation of equivalent length data is relatively easy. Some typical equivalent lengths are shown in the table.

See also: Pipe Sizing and Flow Calculation Software

Resistance Coefficient Method – K Method – Excess head

table of K-values - valves, elbows, bendsThe resistance coefficient method (or K-method, or Excess head method) allows the user to describe the pressure loss through an elbow or a fitting by a dimensionless number – K. This dimensionless number (K) can be incorporated into the Darcy-Weisbach equation in a very similar way to the equivalent length method. Instead of of equivalent length data in this case the dimensionless number (K) is used to characterise the fitting without linking it to the properties of the pipe.

The K-value represents the multiple of velocity heads that will be lost by fluid passing through the fitting. The equation for calculation of pressure loss of the hydraulic element is therefore:
K-value methodTherefore the equation for calculation of pressure loss of entire hydraulic system is:
K-value - head loss
The K-value can be characterised for various flow regime (i.e. according to the Reynolds number) and this causes it is more accurate than the equivalent length method.

There are several other methods for calculating pressure loss for fittings, these methods are more sophisticated and also more accurate:

  • 2K-Method. The 2K method is a technique developed by Hooper B.W. to predict the head loss in an elbow, valve or tee. The 2K method improves on the excess head method by characterising the change in pressure loss due to varying Reynolds number. The 2-K method is advantageous over other method especially in the laminar flow region.2K method
  • 3K-Method. The 3K method (by Ron Darby in 1999) further improves the accuracy of the pressure loss calculation by also characterising the change in geometric proportions of a fitting as its size changes. This makes the 3K method particularly accurate for a system with large fittings.3K method

Flow through Elbow – Minor Loss

flow through elbow - minor lossThe 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.

Fluid Acceleration

Chart - density - water - temperature

Density of liquid (compressed) water as a function of temperature of water

It is known that when the fluid is heated (e.g. in a fuel channel), the fluid expands (change in the fluid density) and increases its flow velocity as a result of the continuity equation (the channel cross-section remains the same). For a control volume that has a single inlet and a single outlet, this equation states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.

fluid acceleration - pressure drop
Mass entering per unit time = Mass leaving per unit time

See also: Subcooled Water Properties

Another very important principle states (Bernoulli’s principle) that the increase in flow velocity in the heated channel causes the lowering of fluid pressure. This pressure loss can be also considered as a local pressure loss and can be calculated from the following equation:
fluid acceleration - equation

Flow rate through a reactor core – coolant acceleration

It is an illustrative example, following data do not correspond to any reactor design.

Continuity Equation - Flow Rates through Reactor

Example of flow rates in a reactor. It is an illustrative example, data do not represent any reactor design.

Pressurized water reactors are cooled and moderated by high-pressure liquid water (e.g. 16MPa). At this pressure water boils at approximately 350°C (662°F).  Inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) as the water flows through the core.

The primary circuit of typical PWRs is divided into 4 independent loops (piping diameter ~ 700mm), each loop comprises a steam generator and one main coolant pump. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the downcomer). From the bottom of the pressure vessel, the flow is reversed up through the core, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.

Calculate:

  • Pressure loss due to the coolant acceleration in an isolated fuel channel

 when

  • channel inlet flow velocity is equal to  5.17 m/s
  • channel outlet flow velocity is equal to  5.69 m/s

Solution:

The pressure loss due to the coolant acceleration in an isolated fuel channel is then:

coolant acceleration - example

This fact has important consequences. Due to the different relative power of fuel assemblies in a core, these fuel assemblies have different hydraulic resistance and this may induce local lateral flow of primary coolant and it must be considered in thermal-hydraulic calculations.

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: