The pressure drop that occurs in a conduction is the loss of dynamic energy of the fluid due to the friction of the fluid particles against one another and against the walls of the conduit that contains them. The losses may be continuous, along the entire length of regular ducts, or accidental or localized, due to particular circumstances, such as a narrowing, a change of direction, the presence of a valve, etc.

We can therefore distinguish between two types of pressure drop: primary losses and secondary losses.

*Primary losses*

These occur when the fluid comes into contact with the surface of the duct. This makes some layers rub up against others – laminar flow – or friction among particles – turbulent flow.

*Secondary losses or losses in singularities*

These occur at transitions of the duct – where it narrows or expands – and in all kinds of accessories: valves, dampers, elbows, etc.

## Calculations in thermal fluid equipment

Calculating the head loss in a thermal fluid circuit is essential. If we limit ourselves to thermal fluid as the fluid to consider, this allows us to design the diameter of the pipes correctly so that each device that consumes fluid receives the required flow for its productive process. The calculation is also necessary to be able to make a suitable selection of the main recirculation pump of the installation or of the secondary circuit pumps.

Within this section, we must also consider the head losses that occur at the consumption points themselves – exchangers, reactors – or in the boiler.

Calculating the head loss is also necessary in the fumes circuit of a boiler to correctly determine which burner needs to be installed. This head loss is what we normally call the boiler’s overpressure, in this case being the fluid that circulates and the combustion gases.

Finally, and also looking at the combustion gases as the fluid in question, we must calculate the head losses that occur in the fumes evacuation duct – the stack – in order to determine the most satisfactory diameter for it and its height, as well as whether the head losses detected can be withstood by the stack’s own damper or must be partially dealt with by the burner ventilator.

An erroneous calculation of the pressure drop in any of these sections which make up a thermal fluid installation, will not only affect the production and its quality, but may pose serious problems in the operation of the boiler in the medium term, or even – especially in the cases of burner selection and stack dimensioning -, the impossibility of starting it up.

All of these calculations, given their specific nature, must be made by specialized technicians, whether these be the engineers responsible for the project, the manufacturer of the equipment or the installer.

Below, we show the main formulas used in these calculations, although some of them are not perhaps the most appropriate for thermal fluid installations; however, they are useful for auxiliary or secondary installations, using hot water or liquid/gaseous fuel.

We hope that they will serve on the one hand to understand the complexity of these calculations and their importance in the smooth operation of a fluid installation, and on the other hand as a reference list or reminder for those who are already familiar with this topic.

## Formulas

Thanks to information technology, the calculation of the pressure drop in installations is quite accessible these days, based on software and spreadsheets, in empirical formulas we see below and that are the most widely-known.

In all of them, we can see that there are two factors that are always necessary and important. Firstly, the type of piping used for the duct: the materials, finishes, and therefore its roughness. The speed of the fluid, whether expressed directly as such or via the Reynolds number, is the other determining factor in all of the expressions.

### Primary losses

For primary losses, the best-known and most widely used formulas are:

- Darcy-Weisbach
- Manning
- Hazen-Williams
- Scimeni
- Scobey

In all of these formulas, the pipe is assumed to be circular in section. However, they can be used for pipes with non-circular sections, by using what is known as the hydraulic diameter. By using this term, one can study the behavior of the flow in the same way as if it were a pipe circular in section.

The hydraulic diameter, *D** _{h}* ,

Where:

A area of the of the duct cross-section

P Perimeter moistened by the fluid

Section form | Cross-section area | Moistened perimeter | Hydraulic diameter |

${\pi}{\times}{{D}}^{{4}}$ | ${\pi}{\times}{D}$ | ${D}$ | |

${{a}}^{{2}}$ | ${4}{\times}{a}$ | ${a}$ | |

${a}{\times}{b}$ | ${2}\left({a}{+}{b}\right)$ | $\frac{{2}{\times}{a}{\times}{b}}{{a}{+}{b}}$ | |

$\frac{{\pi}{\times}\left({{D}}_{{1}}^{{2}}{\u2013}{{D}}_{{2}}^{{2}}\right)}{{4}}$ | ${\pi}{\times}\left({{D}}_{{1}}{+}{{D}}_{{2}}\right)$ | ${{D}}_{{1}}{\u2013}{{D}}_{{2}}$ |

Table 1. Hydraulic diameter of the most common sections

#### Darcy-Weisbach

Without a doubt, the most precise formula hydraulic calculations is the Darcy-Weisbach, and it is the most appropriate one for thermal fluid installations.

The original formula is:

Depending on the flow, the expression is as follows:

Where:

h – Pressure drop or loss of energy (m)

f – friction co-efficient (non-dimensional)

L – Length of pipe (m)

D – Internal diameter of pipe (m)

v – Average speed (m/sec)

g – acceleration due to gravity (m/s^{2})

Q – flow (m^{3}/s)

The f factor will depend on:

v: velocity (m/sec)

D: diameter (l)

ρ : density of the fluid (kg/m^{3})

μ : viscosity of the fluid (N·s/m^{2})

ε : absolute roughness (l)

εr, relative roughness of the pipe’s walls, non-dimensional, is εr = ε / D

As stated earlier, the characteristics of the material of the duct and its condition determine the resistance that the fluid will encounter and therefore the loss of pressure that will occur.

As well as this value, called roughness, the friction factor F is of vital importance in the Darcy-Weisbach expression, which has led to the conceiving of many different expressions and graphs to determine it.

Some of these are as follows:

#### Blasius

It proposes an expression in which the friction coefficient f is found according to the Reynolds number, and is valid for smooth pipes, in which the relative roughness εr does not affect the flow where the laminar sub-layer eliminates the irregularities.

It is valid up to Re < 100000. The expression is:

#### Prandtl and Von-Karman

They extend the range of validity of the Blasius formula for smooth pipes.

#### Nikuradse

Proposes a valid equation for rough pipes.

#### Colebrook-White

Colebrook-White groups together the Prandtl-Von Karman and Nikuradse expressions in a single expression, which is also valid for all kinds of flows and roughnesses.

It is the most exact and comprehensive formula, but it is complex and to obtain the friction coefficient f, one has to use iterative methods.

The standard UNE 149201:2008, which is the benchmark for water installations, indicates that the friction factor must be obtained based on the Colebrook-White equation.

The field of application of this formula is found in the transition zone from laminar flow to turbulent flow and in turbulent flow.

For very high Reynolds numbers, the second sum total situated within the parenthesis of the Colebrook-White equation is negligible. In this case, the viscosity does not influence in practice the determining of the friction coefficient; this depends solely on the relative roughness of the pipe. This is clear in the Moody diagram – see Fig.1 – in which the curve for high Reynolds values becomes a horizontal straight line.

#### Moody

Moody succeeded in representing Colebrook-White’s expression in an easy to use calculation table – Fig. 1. To calculate “f” according to the Reynolds number (Re) and with the relative roughness (εr) acting as a differentiating parameter of the curves.

Fig 1. Moody’s diagram

#### Churchill

#### Swamee-Jain

The calculation is direct, without iterations. It can be classified as an explicit equation for calculating the friction factor. The equation offers very similar results to the Colebrook-White one.

If the Reynolds number is very high, in completely turbulent flow, the second fractional from the bracket in the denominator can be simplified, giving the expression:

Table 2 shows some absolute roughness values for different materials.

Material | ε (mm) |

Plastic (PE, PVC) | 0,0015 |

Asphalt-dipped cast iron | 0,06- 0,18 |

Polyester reinforced with fiber glass | 0,01 |

Cast iron | 0,12-0,60 |

Drawn steel tubes | 0,0024 |

Commercial and welded steel | 0,03-0,09 |

Tin or copper tubes | 0,0015 |

Wrought iron | 0,03-0,09 |

Cement-coated cast iron | 0,0024 |

Galvanized steel | 0,06-0,24 |

Cast iron with bituminous coating | 0,0024 |

Wood | 0,18-0,90 |

Centrifuged cast iron | 0,003 |

Concrete | 0,3-3,0 |

Table 2. Absolute roughness of materials for the expressions of Colebrook-White, Swamee-Jain and Nikuradse

#### Manning

Manning’s equations are normally used to calculate pressure drops in channels. They are valid for pipes when the channel is circular and is partially or completely full, or when the pipe’s diameter is very large. One of the formula’s drawbacks is that it only takes into account a roughness coefficient (n) that has been obtained empirically, and not the variations in viscosity with temperature.

The expression is as follows:

where:

h: Pressure drop or loss of energy (m)

n: roughness co-efficient (non-dimensional)

D: Internal diameter of pipe (m)

Q: flow (m^{3}/s)

L: Length of pipe (m)

Calculating the roughness coefficient “n” is complex, as there is no exact method. For the case of pipes, one can look at the “n” values in tables – See Table 3.

Some of these values are summarized in the following table:

Material | n |

Plastic (PE, PVC) | 0,006-0,010 |

Cast iron | 0,012-0,015 |

Polyester reinforced with fiber glass | 0,009 |

Concrete | 0,012-0,017 |

Steel | 0,010-0,011 |

Galvanized steel | 0,015-0,017 |

Bituminous coating | 0,013-0,016 |

Table 3. Roughness coefficient n for Manning’s expression

#### Hazen-Williams

The Hazen-Williams method is one of the best-known and widely-used, as the formula it employs is simple as is its calculation thanks to the fact that the roughness coefficient “C” is not a function of the velocity nor of the pipe diameter.

However, it is only valid for cast iron and steel pipes, if the fluid circulating is water and only at temperatures of between 5 ºC and 25 ºC.

where:

h: Pressure drop or loss of energy (m)

Q: flow (m^{3}/s)

C: roughness co-efficient (non-dimensional)

D: Internal diameter of pipe (m)

L: Length of pipe (m)

Material | C |

Asbestos cement | 140 |

Galvanized steel | 120 |

Tin | 130-140 |

Glass | 140 |

Sanitation brick | 100 |

Lead | 130-140 |

Cast iron, new | 130 |

Plastic (PE, PVC) | 140-150 |

Cast iron, 10 years old | 107-113 |

Smooth new piping | 140 |

Cast iron, twenty years old | 89-100 |

New steel | 140-150 |

Cast iron, 30 years old | 75-90 |

Steel | 130 |

Cast iron, 40 years old | 64-83 |

Rolled steel | 110 |

Concrete | 120-140 |

Tin | 130 |

Copper | 130-140 |

Wood | 120 |

Ductile iron | 120 |

Concrete | 120-140 |

Table 4. Coefficient of absolute roughness of materials for the Hazen-Williams expression

#### Hagen-Poiseuille

It is a valid formula for calculating pressure drops of fluids at very low velocities – laminar flow – in cylindrical ducts. This is due to the fact that the profile of velocities in a pipe is similar in shape to a parabola, where the maximum speed is on the axis of the pipe and the velocity is zero on the wall of the pipe, and losses due to friction with the wall can be ignored, minimizing the roughness of the duct and therefore the characteristics of the material thereof.

In this way, the loss of energy – pressure drop – is proportional to the average velocity, and therefore to the Reynolds number – formula (15).

We recall that flow is deemed to be laminar when the Reynolds number – formula (14) is less than 2014. For higher Reynolds numbers, the flow is deemed to be turbulent.

However, the critical Reynolds number that delimits turbulent and laminar flow depends on the geometry of the system.

See in the Moody – Fig 1 – flow zones according to the Reynolds number and the roughness.

The Hagen-Poiseuille expression is:

where:

h: Pressure drop or loss of energy (m)

vmedia: the fluid’s average velocity along the z axis of the cylindrical system of coordinates (m/sec)

D: Internal diameter of pipe (m)

L: Length of pipe (m)

g: acceleration due to gravity (m/s^{2})

Re: Reynolds number, whose expression is:

If we compare the Hagen-Poiseuille expression (13), with the Darcy-Weisbach formula (2), we can see that they are identical if we consider the friction coefficient f as:

#### Scimeni

It is used exclusively for fiber cement pipes, the roughness coefficient being integrated in the expression, and is not the valid formula for other types of materials other than fiber cement.

The formula is as follows:

Where:

h: Pressure drop or loss of energy (m)

Q: flow (m^{3}/s)

D: Internal diameter of pipe (m)

L: Length of pipe (m)

#### Scobey

It is mainly used in aluminum pipes with flows in the turbulent transition zone – see in the Moody diagram, Fig 1, the different flow zones according to the Reynolds number.

As in Scimeni’s formula, the expression is only valid for pipes of the specified material.

The equation is:

Where:

h: Pressure drop or loss of energy (m)

K: Scobey roughness co-efficient (non-dimensional)

Q: flow (m^{3}/s)

D: Internal diameter of pipe (m)

L: Length of pipe (m)

Material | K |

Galvanized steel | 0,42 |

Steel | 0,36 |

Aluminum | 0,40 |

Fiber cement and plastics | 0,32 |

Table 5. Scobey K coefficient for different materials

## Pressure drops in singularities

In addition to the load losses due to friction, other types of losses occur that originate at singular points of pipes (changes of direction, elbows, joints …) that are due to turbulence phenomena. The sum of these accidental or localized load losses plus friction losses gives the total head losses.

Currently, most valve manufacturers provide information on the load losses of their products extensively, including for example, graphs of loss of load – Fig 2 -, depending on the opening of the valve.

These graphs facilitate what is called the flow coefficient – Kv in metric units, Cv in imperial units -, which by the expression (18) allows us to obtain the pressure drop (head loss) of the valve:

where:

h: head loss (bars)

Q: circulating fluid flow (m^{3}/h)

Kv: valve flow coefficient

Fig 2. Load loss graph of a manual valve provided by the manufacturer. This gives us the value Kv according to the turns made of its wheel – Handwheel rotations -, and therefore its degree of opening

If we don’t have this information, or for calculations of pipe singularities, such as elbows, reductions, etc., we can consider two methods of calculating the secondary load losses. By the K-factor method, and by means of the equivalent length method.

#### K Factor

Except in exceptional cases, localized pressure drops can only be determined experimentally, and since they are due to an energy dissipation caused by turbulence, they can be expressed as a function of the corrected kinetic height by means of an empirical coefficient, called the K factor.

This coefficient depends, therefore, on the type of singularity and its geometric shape, basically, and is similar to the flow coefficient provided by valve manufacturers, although much more general and approximate.

In Table 6, we can see different values of K for certain singularities.

The fundamental equation of secondary losses by means of the K factor, has the expression

Where:

h: Pressure drop or loss of energy (m)

K: Non-dimensional resistance coefficient, which depends on the element that causes the pressure drop.

v: Average velocity (m/sec) in the element

g: acceleration due to gravity (m/s^{2})

Singularity | K factor |

Spherical valve (fully open) | 10 |

Right-angle valve (fully open) | 5 |

Safety valve (fully open) | 2,5 |

Retention valve (fully open) | 2 |

Gate valve (fully open) | 0,2 |

Gate valve (3/4 open) | 1,15 |

Gate valve (1/2 open) | 5,6 |

Gate valve (1/4 open) | 24 |

T with side outlet | 1,80 |

90º elbow short-radius (with flanges) | 0,90 |

90º elbow normal-radius (with flanges) | 0,75 |

90º elbow large-radius (with flanges) | 0,60 |

45º elbow short-radius (with flanges) | 0,45 |

45º elbow normal-radius (with flanges) | 0,40 |

45º elbow large-radius (with flanges) | 0,35 |

#### Equivalent length

This method, undoubtedly the oldest one, consists in assigning a length of cylindrical piping which is supposed to produce a pressure drop in the system of similar readings. The assigning of the equivalent length depends on the type of singularity and on its diameter, and this is specified in tables. – see Table 7 -.

However, this does not take into account the state of the valve, however. If it is completely open, partially closed, etc., so it is not currently an overused method.

Table 7. Equivalent length of some singularities