# Head Loss – Pressure Loss

## Article Summary & FAQs

### What is head loss or pressure loss?

In fluid flow, head loss or pressure loss reduces the total head (sum of the potential headvelocity head, and pressure head) of a fluid caused by the friction present in the fluid’s motion.

### Key Facts

• The Darcy friction factor is a dimensionless quantity used in the Darcy–Weisbach equation to describe frictional losses in pipe or duct and open-channel flow. This is also called the Darcy–Weisbach friction factorresistance coefficient, or simply friction factor.
• 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 easily integrated into the Darcy-Weisbach equation through resistance coefficient K.
How to calculate pressure loss of hydraulic system?
How to calculate the pressure loss of a hydraulic system?

Sometimes, engineers use the pressure loss coefficientPLC. It is noted K or ξ  (pronounced “xi”). This coefficient characterizes pressure loss of a certain hydraulic system or 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.

How to calculate the pressure drop for two-phase fluid flow?
How to calculate the pressure drop for two-phase fluid flow?

In contrast, to single-phase pressure drops, calculation and prediction of two-phase pressure drops is a much more sophisticated problem, and leading methods differ significantly. Experimental data indicates that the frictional pressure drop in the two-phase flow (e.g.,, in a boiling channel) is substantially higher than for a single-phase flow with the same length and mass flow rate. A homogenous flow model or some friction multiplicators can be used for this purpose.

See more: Two-phase Fluid Flow

## Head Loss – Pressure Loss

In the practical analysis of piping systems, the quantity of most importance is the pressure loss due to viscous effects along the length of the system, as well as additional pressure losses arising from other technological equipment like valves, elbows, piping entrances, fittings, and tees.
At first, an extended Bernoulli’s equation must be introduced. This equation permits account of viscosity to be included empirically and quantify this with a physical parameter known as head loss.

Extended Bernoulli’s Equation
There are two main assumptions that were applied on the derivation of the simplified Bernoulli’s equation.

• The first restriction on Bernoulli’s equation is that no work is allowed to be done on or by the fluid. This is a significant limitation because most hydraulic systems (especially in nuclear engineering) include pumps. This restriction prevents two points in a fluid stream from being analyzed if a pump exists between the two points.
• The second restriction on simplified Bernoulli’s equation is that no fluid friction can solve hydraulic problems. In reality, friction plays a crucial role. The total head possessed by the fluid cannot be transferred completely and is lossless from one point to another. In reality, one purpose of pumps incorporated in a hydraulic system is to overcome the losses in pressure due to friction.

Due to these restrictions, most practical applications of the simplified Bernoulli equation to real hydraulic systems are very limited. The simplified Bernoulli equation must be modified to deal with both head losses and pump work.

The Bernoulli equation can be modified to take into account gains and losses of the head. The resulting equation referred to as the extended Bernoulli’s equation is very useful in solving most fluid flow problems. The following equation is one form of the extended Bernoulli equation.

where:
h = height above reference level (m)
v = average velocity of fluid (m/s)
p = pressure of fluid (Pa)
Hfriction = head loss due to fluid friction (m)
g = acceleration due to gravity (m/s2)

The head loss (or the pressure loss) due to fluid friction (Hfriction) represents the energy used in overcoming friction caused by the pipe walls. The head loss that occurs in pipes is dependent on the flow velocity, pipe diameter, and length, and a friction factor based on the roughness of the pipe and the Reynolds number of the flow. A piping system containing many pipe fittings and joints, tube convergence, divergence, turns, surface roughness, and other physical properties will also increase the head loss of a hydraulic system.

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 is conserved 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 fluid’s internal energy (temperature increases).

Most methods for evaluating head loss due to friction are based almost exclusively on experimental evidence. This will be discussed in the following sections.

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.

In fluid dynamics, the 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 Bernoulli’s equation can also be measured in distance units. 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 the static head and represents the maximum height (pressure) it can deliver. Therefore the characteristics of all pumps can usually be read from their 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.ρw: 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. The height in feet is that a flowing fluid would rise in a column if all of its kinetic energy were converted to potential energy.

The sum of a fluid’s elevation head, kinetic head, and pressure head 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. Suppose this pipe undergoes a gradual expansion in diameter. In that case, the continuity equation tells us that as the pipe diameter increases, the flow velocity must decrease 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 the elevation head (the pipe lies horizontally), the decrease in the kinetic head must be compensated for by an increase in pressure head.

The head loss (or the pressure loss) represents the reduction in the total head or pressure (sum of elevation head, velocity head, and pressure head) of the fluid as it flows through a hydraulic system. The head loss also represents the energy used in overcoming friction caused by the pipe walls and other technological equipment. Head loss is unavoidable in real moving fluids. It is present because of the friction between adjacent fluid particles moving relative to one another (especially in turbulent flow).

The head loss that occurs in pipes is dependent on the flow velocity, pipe diameter, and length, and a friction factor based on the roughness of the pipe and 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 is conserved 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 fluid’s internal energy (temperature increases).

Most methods for evaluating head loss due to friction are based almost exclusively on experimental evidence. This will be discussed in the following sections.

Water at 20°C is pumped through a smooth 12-cm-diameter pipe 10 km long, at a flow rate of 75 m3/h. The inlet is fed by a pump at an absolute pressure of 2.4 MPa.
The exit is at standard atmospheric pressure (101 kPa) and is 200 m higher.

Calculate the frictional head loss Hf, and compare it to the velocity head of the flow v2/(2g).

Solution:

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

vout = Q/A = 75 [m3/h] * 3600 [s/h] / 0.0113 [m2] = 1.84 m/s

Velocity head = vout2/(2g) = 1.842 / 2*9.81 = 0.173 m

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

2 400 000 [Pa] / 1000 [kg/m3] * 9.81 [m/s2]  + 0.173 [m] + 0 [m] = 101 000 [Pa] / 1000 [kg/m3] * 9.81 [m/s2] + 0.173 [m]+ 200 [m] + Hf

H = 244.6 – 10.3 – 200 = 34.3 m

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 the 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.

The head loss can be then expressed as:

hloss = Σ hmajor_losses + Σ hminor_losses

## Why is head loss very important?

As can be seen from the picture, the head loss is formed 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 determine the flow rate through the system.

## Major Head Loss – Frictional Loss

Major losses, which are associated with frictional energy loss per length of the 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 is conserved 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 fluid’s internal energy (temperature increases).

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 (head loss form).

where:

• Δh = the head loss due to friction (m)
• fD = the Darcy friction factor (unitless)
• L = the pipe length (m)
• D = the hydraulic diameter of the pipe D (m)
• g = the gravitational constant (m/s2)
• V = the mean flow velocity V (m/s)
Pressure loss form
The Darcy–Weisbach equation in the pressure loss form can be written as:

where:

• Δp = the pressure loss due to friction (Pa)
• fD = the Darcy friction factor (unitless)
• L = the pipe length (m)
• D = the hydraulic diameter of the pipe D (m)
• g = the gravitational constant (m/s2)
• V = the mean flow velocity V (m/s)

___________

Evaluating the Darcy-Weisbach equation provides insight into factors affecting 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). Thus, reducing the pipe diameter by half increases the head loss by a factor of 16. This is a 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 fluid’s viscosity is reduced by half.

Except for 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 using various empirical relations or read from published charts (e.g.,, Moody chart).

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 the following equations:

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:

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 the Blasius formula:

f = (100.Re)

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.

## Minor Head Loss – Local Pressure Loss

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 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.

Generally, most methods that are used in the industry define a coefficient K as a value for a certain technological component.

Like pipe friction, the minor losses are roughly proportional to the square of the flow rate, and therefore they can be easily 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.

The following methods are of practical importance in local pressure loss calculations:

• Equivalent Length Method
• K-Method – Resistance Coefficient Method
• 2K-Method
• 3K-Method

Example: The head loss for one loop of primary piping
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: ReD = 17 [m/s] x 0.7 [m] / 0.12×10-6 [m2/s] = 99 000 000
• The Darcy friction factor is equal to fD = 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:

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

Example: Change in head loss due to a decrease in viscosity.
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, the resulting head loss becomes proportional to viscosity. Therefore, the head loss is reduced by half when the fluid’s viscosity is reduced by half when the flow rate and thus the average velocity are held constant.

Pressure Drop - Fuel Assembly
In general, total fuel assembly pressure drop is formed by fuel bundle frictional drop (dependent on relative roughness of fuel rods, Reynolds number, hydraulic diameter, etc.) and other pressure drops of structural elements (top and bottom nozzle, spacing grids or mixing grids).

In general, it is not so simple to calculate pressure drops in fuel assemblies (especially the spacing grids), and it belongs to the key know-how of certain fuel manufacturers. Mostly, pressure drops are measured in experimental hydraulic loops rather than calculated.

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 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.

Using the data of the below-mentioned example, the pressure loss coefficient (only frictional from a straight pipe) is equal to ξ = fDL/DH = 4.9. But the overall pressure loss coefficient (including spacing grids, top, and bottom nozzles, etc.) is usually about three times higher. This PLC (ξ = 4.9) causes that the pressure drop is of the order of (using the previous inputs) Δpfriction =  4.9 x 714 x 52/ 2 = 43.7 kPa (without spacing grids, top, and bottom nozzles). About three times higher real PLC means about three times higher Δpfuel will be.

The overall reactor pressure loss, Δpreactor, must include:

• downcomer and reactor bottom
• lower support plate
• fuel assembly including spacing grids, top and bottom nozzles, and other structural components – Δpfuel
• upper guide structure assembly

As a result, the overall reactor pressure loss – Δpreactor is usually of the order of hundreds kPa (let say 300 – 400 kPa) for design parameters.

Pressure Loss Coefficient - PLC
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 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.

## Head Loss of Two-phase Fluid Flow

In contrast, to single-phase pressure drops, calculation and prediction of two-phase pressure drops is a much more sophisticated problem, and leading methods differ significantly. Experimental data indicates that the frictional pressure drop in the two-phase flow (e.g.,, in a boiling channel) is substantially higher than for a single-phase flow with the same length and mass flow rate. Explanations include an increased surface roughness due to bubble formation on the heated surface and increased flow velocities.

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:

Bernoulli’s Principle