# Energy Equation - Pressure Loss vs. Head Loss

The total energy per mass unit in a given point in a fluid flow consists of elevation (potential) energy, velocity (kinetic) energy and pressure energy.

The Energy Equation states that energy can not disappear - the energy upstream in the fluid flow will always be equal to the energy downstream in the flow and the energy loss between the two points.

*E _{1} = E_{2 }+ E_{loss }(1)*

*where*

*E _{1} = energy upstream (J/kg, Btu/lb)*

*E _{2 }= energy downstream (J/kg, Btu/lb)*

*E _{loss}* = energy loss

*(J/kg, Btu/lb)*The energy in a specific point in the flow

* E _{flow} = E_{pressure} + E_{kinetic} + E_{potential} (2)*

*where *

*E _{pressure} = p / ρ = pressure energy (J/kg, Btu/lb)*

*E _{kinetic} = v^{2} / 2 = velocity (kinetic) energy (J/kg, Btu/lb)*

*E _{potential} = g h = elevation (potential) energy (J/kg, Btu/lb) *

*E*_{loss}* = Δp_{loss} / ρ = major and minor energy loss in the fluid flow (J/kg, Btu/lb) *

*p** = pressure in fluid (Pa (N/m ^{2}), psi (lb/in^{2}))*

*Δ*p_{loss}* = major and minor pressure loss in the fluid flow (Pa (N/m ^{2}), psi (lb/in^{2}))*

*ρ** = density of fluid (kg/m ^{3}, slugs/ft^{3})*

*v** = flow velocity (m/s, ft/s)*

*g** = acceleration of gravity (m/s ^{2}, ft/s^{2})*

*h** = elevation (m, ft)*

Eq. 1 and 2 can be combined to express the equal energies in two different points in a stream line as

*p _{1} / ρ + v_{1}^{2} / 2 +*

*g h*

_{1}= p_{2}/*ρ*+ v_{2}^{2}/ 2 + g h_{2}+ Δp_{loss}/*ρ**(3)*

or alternatively* *

*p _{1} + ρ v_{1}^{2} / 2 +*

*ρ*

*g h*

_{1}= p_{2}+ ρ v_{2}^{2}/ 2 + ρ g h_{2}+ Δp_{loss}*(3b)*

For a horizontal steady state flow *v _{1} = v_{2}* and

*h*, - and

_{1}= h_{2}*(3b)*can be simplified to:

Δp_{loss}=p_{1}- p_{2}(3c)

The pressure loss is divided in

**major loss**due to friction and**minor loss**due to change of velocity in bends, valves and similar

The major friction loss in a pipe or tube depends on the flow velocity, pipe or duct length, pipe or duct diameter, and a friction factor based on the roughness of the pipe or duct, and whether the flow us turbulent or laminar - the Reynolds Number of the flow. The pressure loss in a tube or duct due to friction, major loss, can be expressed as:

Δp_{major_loss}= λ (l / d_{h}) (ρ v^{2}/ 2)(4)

where

major friction pressure loss (Pa, (N/mΔp_{major_loss}=^{2}), lb/ft^{2})

λ= friction coefficient

l= length of duct or pipe (m, ft)

d_{h}= hydraulic diameter (m, ft)

Eq. *(3)* is also called the D'Arcy-Weisbach Equation. *(3)* is valid for fully developed, steady, incompressible flow.

The minor or dynamic loss depends flow velocity, density and a coefficient for the actual component.

*Δ*p_{minor_loss} = *ξ * ρ v^{2}* / 2 (5)*

*where *

*Δ*p_{minor_loss} = minor pressure loss (Pa (N/m^{2}), lb/ft^{2}*)*

### Head and Head Loss

The Energy equation can be expressed in terms of head and head loss by dividing each term by the specific weight of the fluid. The total head in a fluid flow in a tube or a duct can be expressed as the sum of elevation head, velocity head and pressure head.

**Note**! The heads in the equations below are based on the fluid itself as a reference fluid. Read more about head here.

p_{1}/ γ + v_{1}^{2}/ 2 g + h_{1}= p_{2}/ γ + v_{2}^{2}/ 2 g + h_{2}+ Δh_{loss}(6)

where

Δh_{loss}= head loss (m "fluid", ft "fluid")

γ=ρ g =specific weight of fluid (N/m^{3}, lb/ft^{3})

For horizontal steady state flow *v _{1} = v_{2}* and

*p*

*, - (4) can be transformed to:*

_{1}= p_{2}

h_{loss}=h_{1}- h_{2}(6a)

where

Δh=p / γ =head (m "fluid", ft "fluid")

The major friction head loss in a tube or duct due to friction can be expressed as:

Δh_{major_loss}=λ (l / d_{h}) (v^{2}/ 2 g)(7)

where

Δh_{loss}= head loss (m, ft)

The minor or dynamic head loss depends flow velocity, density and a coefficient for the actual component.

*Δ*p_{minor_loss} = *ξ *v^{2}* / (2 g) (8)*

### Friction Coefficient - *λ*

The friction coefficient depends on the flow - if it is

- laminar,
- transient or
- turbulent

and the roughness of the tube or duct.

To determine the friction coefficient we first have to determine if the flow is laminar, transient or turbulent - then use the proper formula or diagram.

#### Friction Coefficient for Laminar Flow

For fully developed laminar flow the roughness of the duct or pipe can be neglected. The friction coefficient depends only the Reynolds Number - *Re* - and can be expressed as:

λ= 64 / Re(9)

where

The flow is

- laminar when
*Re < 2300* - transient when
*2300 < Re < 4000* - turbulent when
*Re > 4000*

#### Friction Coefficient for Transient Flow

If the flow is transient - 2300 < Re < 4000 - the flow varies between laminar and turbulent flow and the friction coefficient is not possible to determine.

#### Friction Coefficient for Turbulent Flow

For turbulent flow the friction coefficient depends on the Reynolds Number and the roughness of the duct or pipe wall. On functional form this can be expressed as:

λ = f( Re, k / d_{h})(10)

where

k=absolute roughnessof tube or duct wall (mm, ft)

k / d_{h}=the relative roughness - or roughness ratio

Roughness for materials are determined by experiments. Absolute roughness for some common materials are indicated in the table below

Surface | Absolute Roughness - k | |
---|---|---|

(10^{-3 }m) | (feet) | |

Copper, Lead, Brass, Aluminum (new) | 0.001 - 0.002 | 3.3 - 6.7 10^{-6} |

PVC and Plastic Pipes | 0.0015 - 0.007 | 0.5 - 2.33 10^{-5} |

Epoxy, Vinyl Ester and Isophthalic pipe | 0.005 | 1.7 10^{-5} |

Stainless steel, bead blasted | 0.001 - 0.006 | (0.00328 - 0.0197) 10^{-3} |

Stainless steel, turned | 0.0004 - 0.006 | (0.00131 - 0.0197) 10^{-3} |

Stainless steel, electropolished | 0.0001 - 0.0008 | (0.000328 - 0.00262) 10^{-3} |

Steel commercial pipe | 0.045 - 0.09 | 1.5 - 3 10^{-4} |

Stretched steel | 0.015 | 5 10^{-5} |

Weld steel | 0.045 | 1.5 10^{-4} |

Galvanized steel | 0.15 | 5 10^{-4} |

Rusted steel (corrosion) | 0.15 - 4 | 5 - 133 10^{-4} |

New cast iron | 0.25 - 0.8 | 8 - 27 10^{-4} |

Worn cast iron | 0.8 - 1.5 | 2.7 - 5 10^{-3} |

Rusty cast iron | 1.5 - 2.5 | 5 - 8.3 10^{-3} |

Sheet or asphalted cast iron | 0.01 - 0.015 | 3.33 - 5 10^{-5} |

Smoothed cement | 0.3 | 1 10^{-3} |

Ordinary concrete | 0.3 - 1 | 1 - 3.33 10^{-3} |

Coarse concrete | 0.3 - 5 | 1 - 16.7 10^{-3} |

Well planed wood | 0.18 - 0.9 | 6 - 30 10^{-4} |

Ordinary wood | 5 | 16.7 10^{-3} |

The friction coefficient - *λ* - can be calculated by the ** Colebrooke Equation**:

1 / λ^{1/2}= -2,0 log_{10}[ (2,51 / (Re λ^{1/2})) + (k / d_{h}) / 3,72 ](11)

Since the friction coefficient - *λ* - is on both sides of the equation, it must be solved by iteration. If we know the Reynolds number and the roughness - the friction coefficient - *λ* - in the particular flow can be calculated.

A graphical representation of the Colebrooke Equation is the **Moody Diagram**:

- The Moody Diagram - The Moody diagram in a printable format.

With the Moody diagram we can find the friction coefficient if we know the **Reynolds Number** - *Re* - and the

Relative Roughness Ratio-k / d_{h}

In the diagram we can see how the friction coefficient depends on the Reynolds number for laminar flow - how the friction coefficient is undefined for transient flow - and how the friction coefficient depends on the roughness ratio for turbulent flow.

For hydraulic smooth pipes - the roughness ratio limits zero - and the friction coefficient depends more or less on the Reynolds number only.

For a fully developed turbulent flow the friction coefficient depends on the roughness ratio only.

### Example - Pressure Loss in Air Ducts

Air at *0 ^{o}C* is flows in a

*10 m*galvanized duct -

*315 mm*diameter - with velocity

*15 m/s*.

Reynolds number can be calculated:

Re = d_{h}v ρ / μ(12)

where

Re= Reynolds number

v= velocity (m/s)

ρ= density of air(kg/m^{3})

μ= dynamic or absolute viscosity(Ns/m^{2})

Reynolds number calculated:

Re = (15 m/s) (315 mm) (10^{-3}m/mm ) (1.23 kg/m^{3}) / (1.79 10^{-5}Ns/m^{2})

= 324679 (kgm/s^{2})/N

= 324679 ~ Turbulent flow

Turbulent flow indicates that Colebrooks equation *(9)* must be used to determine the friction coefficient - *λ* -.

With roughness - *ε* *-* for galvanized steel *0.15 mm*, the roughness ratio can be calculated:

Roughness Ratio= ε / d_{h}

=(0.15 mm) / (315 mm)

= 4.76 10^{-4}

Using the graphical representation of the Colebrooks equation - the Moody Diagram - the friction coefficient - *λ* - can be determined to:

λ= 0.017

The major loss for the *10 m* duct can be calculated with the Darcy-Weisbach Equation *(3)* or *(6):*

Δp_{loss}= λ ( l / d_{h}) ( ρ v^{2}/ 2 )

=0.017 ((10 m) / (0.315 m)) ( (1.23 kg/m^{3}) (15 m/s)^{2}/ 2 )

= 74 Pa (N/m^{2})

## Related Topics

### • Fluid Flow and Pressure Loss

Pipe lines - fluid flow and pressure loss - water, sewer, steel pipes, pvc pipes, copper tubes and more.

### • Fluid Mechanics

The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.

### • Ventilation

Systems for ventilation and air handling - air change rates, ducts and pressure drops, charts and diagrams and more.

## Related Documents

### Air Ducts - Friction Loss Diagram

A major friction loss diagram for air ducts - Imperial units ranging *10 - 100 000 cfm*.

### Air Ducts - Friction Loss Diagram

A major friction loss diagram for air ducts - in imperial units ranging *10 000 - 400 000 cfm*.

### Air Ducts - Friction Loss Diagram

A major friction loss diagram for air ducts - SI units.

### Air Ducts - Minor Loss Coefficient Diagrams

Minor loss coefficient diagrams for air ductwork, bends, expansions, inlets and outletsĀ - SI units.

### Colebrook Equation

Friction loss coefficients in pipes, tubes and ducts.

### Copper Tubes - Pressure Loss vs. Water Flow

Water flow and pressure loss *(psi/ft)* due to friction in copper tubes ASTM B88 Types K, L and M.

### Darcy-Weisbach Equation - Major Pressure and Head Loss due to Friction

The Darcy-Weisbach equation can be used to calculate the major pressure and head loss due to friction in ducts, pipes or tubes.

### Density vs. Specific Weight and Specific Gravity

An introduction to density, specific weight and specific gravity.

### Ducts - Sheet Metal Gauges

Thickness of sheet metal used in ductwork.

### Energy and Hydraulic Grade Line

The hydraulic grade line and the energy line are graphical presentations of the Bernoulli equation.

### Energy Equation - Pressure Loss vs. Head Loss

Calculate pressure loss - or head loss - in ducts, pipes or tubes.

### Energy Equation - Pressure Loss vs. Head Loss

Calculate pressure loss - or head loss - in ducts, pipes or tubes.

### Fluid Flow - Hydraulic Diameter

Calculate hydraulic diameter for pipes and ducts.

### Hot Water Systems - Equivalent Length vs. Fittings Resistance

Equivalent length of fittings like bends, returns, tees and valves in hot water heating systems - equivalent length in feet and meter.

### Lined Pipes - Pressure Loss vs. Water Flow

Pressure drop diagrams for PTFE, PP, PFA and PVDF lined pipes.

### Mechanical Energy Equation vs. Bernoulli Equation

The Mechanical Energy Equation compared to the Extended Bernoulli Equation.

### Moody Diagram

Calculate fluid flow friction coefficients from a Moody diagram.

### Pipe and Tube System Components - Minor (Dynamic) Loss Coefficients

Minor loss coefficients for components used in pipe and tube systems.

### Pressure Gradient Diagrams

Static pressure graphical presentation throughout a fluid flow system.

### Reynolds Number

Introduction and definition of the dimensionless Reynolds Number - online calculators.

### Roughness & Surface Coefficients

Surface coefficients that can be used to calculate friction and major pressure loss for fluid flow with surfaces like concrete, galvanized steel, corroded steel and more.

### Steel Pipes - Maximum Water Flow Capacities vs. Size

Maximum water flow capacities in steel pipes - pipe dimensions ranging *2 - 24 inches*.

### Steel Pipes Schedule 40 - Pressure Loss

Water flow and pressure loss in schedule 40 steel pipes - Imperial and SI units - gallons per minute, liters per second and cubic meters per hour.

### System Curve and Pump Performance Curve

Utilize the system curve and the pump performance curve to select the proper pump for a particular application.

### Velocity Classification of Ventilation Ducts

Recommended air velocities in ventilation ducts

### Ventilation Ducts - Minor Loss Resistance

Minor pressure or head loss in ventilation ducts vs. air velocity - minor loss coefficient diagram.

### Viscosity - Absolute (Dynamic) vs. Kinematic

Vicosity is a fluid's resistance to flow and can be valued as dynamic (absolute) or kinematic.