**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.

**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):

The pressure head represents the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid.**Pressure potential – Pressure head:****ρ**_{w}: 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.**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.

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.

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:

** h_{loss} = Σ h_{major_losses} + Σ h_{minor_losses}**

## Major Head Loss – Frictional Loss

**Major losses**, which are associated with

**frictional energy loss**per length of pipe depends on the

**flow velocity, pipe length, pipe diameter, and a friction factor**based on the roughness of the pipe, and whether the flow is laminar or turbulent (i.e. the Reynolds number of the flow).

Although the **head loss represents a loss of energy**, it **does not represent a loss of total energy** of the fluid. The total energy of the fluid conserves as a consequence of the **law of conservation of energy**. In reality, the head loss due to friction results in an equivalent **increase in the internal energy** (increase in temperature) of the fluid.

By observation, the **major head loss is roughly proportional to the square of the flow rate** in most engineering flows (fully developed, turbulent pipe flow).

The most common equation used to calculate major head losses in a tube or duct is the **Darcy–Weisbach equation** .

## Darcy-Weisbach Equation

**the Darcy–Weisbach equation**is a phenomenological equation, which relates the

**major head loss**, or pressure loss, due to

**fluid friction**along a given length of pipe to the average velocity. This equation is valid for

**fully developed, steady, incompressible single-phase flow**.

The Darcy–Weisbach equation can be written in two forms (**pressure loss form** or **head loss form**). In the head loss form can be written as:

where:

- Δh = the head loss due to friction (m)
*f*= the Darcy friction factor (unitless)_{D}- L = the pipe length (m)
- D = the hydraulic diameter of the pipe D (m)
- g = the gravitational constant (m/s
^{2}) - V = the mean flow velocity V (m/s)

**pressure loss form**can be written as:

where:

- Δp = the pressure loss due to friction (Pa)
*f*= the Darcy friction factor (unitless)_{D}- L = the pipe length (m)
- D = the hydraulic diameter of the pipe D (m)
- g = the gravitational constant (m/s
^{2}) - V = the mean flow velocity V (m/s)

___________

### 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…

**Darcy’s equation**can be used to calculate**major losses**.- The
**friction factor**for fluid flow can be determined using a**Moody chart**. **The friction factor**for laminar flow is**independent of roughness**of the pipe’s inner surface.**f = 64/Re****The friction factor**for turbulent flow depends strongly on the**relative roughness.**It is determined by the Colebrook equation. It must be noted,**at very large Reynolds numbers**, the friction factor is independent of the Reynolds number.

### 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.

**Darcy-Weisbach equation**provides insight into factors affecting the head loss in a pipeline.

- Consider that the
**length of the pipe**or channel is**doubled**, the resulting**frictional head loss will double**. - At constant flow rate and pipe length, the
**head loss is inversely proportional to the 4th power of diameter**(for laminar flow), and thus reducing the pipe diameter by half increases the head loss by a factor of 16. This is a very significant increase in head loss, and shows why larger diameter pipes lead to much smaller pumping power requirements. - Since the head loss is roughly proportional to the square of the flow rate, then if the
**flow rate is doubled**, the**head loss increases by a factor of four**. - The
**head loss is reduced by half**(for laminar flow) when the**viscosity of the fluid is reduced by half**.

With the exception of the **Darcy friction factor**, each of these terms (the flow velocity, the hydraulic diameter, the length of a pipe) can be easily measured. The Darcy friction factor takes the fluid properties of density and viscosity into account, along with the **pipe roughness**. This factor may be evaluated by the use of various empirical relations, or it may be read from published charts (e.g. **Moody chart**).

**10 km**long, at a flow rate of

**75 m**. The inlet is fed by a pump at an absolute pressure of

^{3}/h**2.4 MPa**.

The exit is at standard

**atmospheric pressure**(101 kPa) and is

**200 m higher**.

Calculate the **frictional head loss H _{f}**, and compare it to the

**velocity head**of the flow v

^{2}/(2g).

Solution:

Since the pipe diameter is constant, the average velocity and velocity head is the same everywhere:

**v _{out} **=

**Q/A**= 75 [m

^{3}/h] * 3600 [s/h] / 0.0113 [m

^{2}] =

**1.84 m/s**

Velocity head:

**Velocity head** = v_{out}^{2}/(2g) = 1.84^{2} / 2*9.81 = **0.173 m**

In order to find the frictional head loss, we have to use extended Bernoulli’s equation:

Head loss:

2 400 000 [Pa] / 1000 [kg/m^{3}] * 9.81 [m/s^{2}] + 0.173 [m] + 0 [m] = 101 000 [Pa] / 1000 [kg/m^{3}] * 9.81 [m/s^{2}] + 0.173 [m]+ 200 [m] + H_{f}

**H _{f }** = 244.6 – 10.3 – 200 =

**34.3 m**

## Darcy Friction Factor

**the Darcy and the Fanning friction factors**.

f

_{D}= 4.f

_{F}

**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.

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

**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).

,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.

**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**.

## Darcy Friction Factor for various flow regime

**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).

**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 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.

**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 /**and Reynolds number.

*D*_{h}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 < 10^{5}) the friction factor can be approximated by **Blasius formula:**

** f = (100.Re) ^{-¼}**

It 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.

## Examples

**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 constant temperature of
**290°C**(**⍴ ~ 720 kg/m**).^{3} - The
**kinematic viscosity**of the water at 290°C is equal to**0.12 x 10**.^{-6}m^{2}/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}[m^{2}/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 17^{2} / 0.7 = **4.2 m**

**Pressure loss form:**

**Δp** = 0.01 x ½ x 720 x 20 x 17^{2} / 0.7 = 29 725 Pa ≈ **0.03 MPa**

**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:**

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