Darcy-Weisbach equation
Encyclopedia
In fluid dynamics
, the Darcy–Weisbach equation is a phenomenological
equation, which relates the head loss — or pressure
loss — due to friction
along a given length of pipe to the average velocity of the fluid flow. The equation is named after Henry Darcy
and Julius Weisbach
.
The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also called the Darcy–Weisbach friction factor or Moody friction factor. The Darcy friction factor is four times the Fanning friction factor, with which it should not be confused.
where
loss Δp as the height of a column of fluid,
where ρ is the density of the fluid, the Darcy–Weisbach equation can also be written in terms of pressure loss:
where the pressure loss due to friction Δp (units: Pa or kg/ms2) is a function of:
Since the pressure loss equation can be derived from the head loss equation by multiplying each side by ρ and g.
The friction factor f or flow coefficient λ is not a constant and depends on the parameters of the pipe and the velocity of the fluid flow, but it is known to high accuracy within certain flow regimes. It may be evaluated for given conditions by the use of various empirical or theoretical relations, or it may be obtained from published charts. These charts are often referred to as Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes called the Moody friction factor. It is also sometimes called the Blasius
friction factor, after the approximate formula he proposed.
For laminar (slow) flows, it is a consequence of Poiseuille's law that λ = 64/Re, where Re is the Reynolds number calculated substituting for the characteristic length the hydraulic diameter of the pipe, which equals the inside diameter for circular pipe geometries.
For turbulent flow, methods for finding the friction factor f include using a diagram such as the Moody chart
; or solving equations such as the Colebrook–White equation, or the Swamee–Jain equation. While the diagram and Colebrook–White equation are iterative methods, the Swamee–Jain equation allows f to be found directly for full flow in a circular pipe.
Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is f = 16/Re, it's the Fanning factor, and if the formula for laminar flow is f = 64/Re, it's the Darcy–Weisbach factor.
Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:
The procedure above is similar for any available Reynolds number that is an integral power of ten. It is not necessary to remember the value 1000 for this procedure – only that an integral power of ten is of interest for this purpose.
; this variant was developed by Henry Darcy
of France, and further refined into the form used today by Julius Weisbach
of Saxony
in 1845. Initially, data on the variation of f with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of empirical equations valid only for certain flow regimes, notably the Hazen–Williams equation or the Manning equation, most of which were significantly easier to use in calculations. However, since the advent of the calculator
, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.
.
Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length, Δp/L, and the volumetric flow rate. The flow rate can be converted to an average velocity V by dividing by the wetted area of the flow (which equals the cross-sectional
area
of the pipe if the pipe is full of fluid).
Pressure has dimensions of energy per unit volume. Therefore, the pressure drop between two points must be proportional to (1/2)ρV2, which has the same dimensions as it resembles (see below) the expression for the kinetic energy per unit volume. We also know that pressure must be proportional to the length of the pipe between the two points L as the pressure drop per unit length is a constant. To turn the relationship into a proportionality coefficient of dimensionless quantity we can divide by the hydraulic diameter of the pipe, D, which is also constant along the pipe. Therefore,
The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as π, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter).
Note that (1/2)ρV2 is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of laminar flow
, where all the flow lines are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the mean-square velocity, which always exceeds the square of the mean velocity. In the case of turbulent flow, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.
applications, it is often desirable to express the head loss in terms of volumetric flow rate in the pipe. For this, it is necessary to substitute the following into the original head loss form of the Darcy–Weisbach equation
where
For the general case of an arbitrarily-full pipe, the value of Aw will not be immediately known, being an implicit function of pipe slope, cross-sectional shape, flow rate and other variables. If, however, the pipe is assumed to be full flowing and of circular cross-section, as is common in practical scenarios, then
where D is the diameter
of the pipe
Substituting these results into the original formulation yields the final equation for head loss in terms of volumetric flow rate
in a full-flowing circular pipe
where all symbols are defined as above.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the Darcy–Weisbach equation is a phenomenological
Phenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
equation, which relates the head loss — or pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
loss — due to friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
along a given length of pipe to the average velocity of the fluid flow. The equation is named after Henry Darcy
Henry Darcy
Henry Philibert Gaspard Darcy was a French engineer who made several important contributions to hydraulics.-Biography:...
and Julius Weisbach
Julius Weisbach
Julius Ludwig Weisbach was a German mathematician and engineer.-Life and work:...
.
The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also called the Darcy–Weisbach friction factor or Moody friction factor. The Darcy friction factor is four times the Fanning friction factor, with which it should not be confused.
Head loss form
Head loss can be calculated withwhere
- hf is the head loss due to friction (SI units: m);
- L is the length of the pipe (m);
- D is the hydraulic diameter of the pipe (for a pipe of circular section, this equals the internal diameter of the pipe) (m);
- V is the average velocity of the fluid flow, equal to the volumetric flow rateVolumetric flow rateThe volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...
per unit cross-sectional wetted area (m/s); - g is the local acceleration due to gravity (m/s2);
- f is a dimensionless coefficient called the Darcy friction factorDarcy friction factor formulaeIn fluid dynamics, the Darcy friction factor formulae are equations — based on experimental data and theory — for the Darcy friction factor. The Darcy friction factor is a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open...
. It can be found from a Moody diagram or more precisely by solving the Colebrook equation.
Pressure loss form
Given that the head loss hf expresses the pressurePressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
loss Δp as the height of a column of fluid,
where ρ is the density of the fluid, the Darcy–Weisbach equation can also be written in terms of pressure loss:
where the pressure loss due to friction Δp (units: Pa or kg/ms2) is a function of:
- the ratio of the length to diameter of the pipe, L/D;
- the density of the fluid, ρ (kg/m3);
- the mean velocity of the flow, V (m/s), as defined above;
- a (dimensionless) coefficient of laminar, or turbulent flow, f.
Since the pressure loss equation can be derived from the head loss equation by multiplying each side by ρ and g.
Darcy friction factor
- See also Darcy friction factor formulaeDarcy friction factor formulaeIn fluid dynamics, the Darcy friction factor formulae are equations — based on experimental data and theory — for the Darcy friction factor. The Darcy friction factor is a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open...
The friction factor f or flow coefficient λ is not a constant and depends on the parameters of the pipe and the velocity of the fluid flow, but it is known to high accuracy within certain flow regimes. It may be evaluated for given conditions by the use of various empirical or theoretical relations, or it may be obtained from published charts. These charts are often referred to as Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes called the Moody friction factor. It is also sometimes called the Blasius
Paul Richard Heinrich Blasius
Paul Richard Heinrich Blasius was a German fluid dynamics engineer.He was one of the first students of Prandtl who provided a mathematical basis for boundary-layer drag but also showed as early as 1911 that the resistance to flow through smooth pipes could be expressed in terms of the Reynolds...
friction factor, after the approximate formula he proposed.
For laminar (slow) flows, it is a consequence of Poiseuille's law that λ = 64/Re, where Re is the Reynolds number calculated substituting for the characteristic length the hydraulic diameter of the pipe, which equals the inside diameter for circular pipe geometries.
For turbulent flow, methods for finding the friction factor f include using a diagram such as the Moody chart
Moody chart
The Moody chart or Moody diagram is a graph in non-dimensional form that relates the Darcy friction factor, Reynolds number and relative roughness for fully developed flow in a circular pipe...
; or solving equations such as the Colebrook–White equation, or the Swamee–Jain equation. While the diagram and Colebrook–White equation are iterative methods, the Swamee–Jain equation allows f to be found directly for full flow in a circular pipe.
Confusion with the Fanning friction factor
The Darcy–Weisbach friction factor is 4 times larger than the Fanning friction factor, so attention must be paid to note which one of these is meant in any "friction factor" chart or equation being used. Of the two, the Darcy–Weisbach factor is more commonly used by civil and mechanical engineers, and the Fanning factor by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula.Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is f = 16/Re, it's the Fanning factor, and if the formula for laminar flow is f = 64/Re, it's the Darcy–Weisbach factor.
Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:
- Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.
- If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor: f = 64/Re.
- If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor: f = 16/Re.
The procedure above is similar for any available Reynolds number that is an integral power of ten. It is not necessary to remember the value 1000 for this procedure – only that an integral power of ten is of interest for this purpose.
History
Historically this equation arose as a variant on the Prony equationProny equation
The Prony equation is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe...
; this variant was developed by Henry Darcy
Henry Darcy
Henry Philibert Gaspard Darcy was a French engineer who made several important contributions to hydraulics.-Biography:...
of France, and further refined into the form used today by Julius Weisbach
Julius Weisbach
Julius Ludwig Weisbach was a German mathematician and engineer.-Life and work:...
of Saxony
Saxony
The Free State of Saxony is a landlocked state of Germany, contingent with Brandenburg, Saxony Anhalt, Thuringia, Bavaria, the Czech Republic and Poland. It is the tenth-largest German state in area, with of Germany's sixteen states....
in 1845. Initially, data on the variation of f with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of empirical equations valid only for certain flow regimes, notably the Hazen–Williams equation or the Manning equation, most of which were significantly easier to use in calculations. However, since the advent of the calculator
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...
, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.
Derivation
The Darcy–Weisbach equation is a phenomenological formula obtainable by dimensional analysisDimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
.
Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length, Δp/L, and the volumetric flow rate. The flow rate can be converted to an average velocity V by dividing by the wetted area of the flow (which equals the cross-sectional
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of the pipe if the pipe is full of fluid).
Pressure has dimensions of energy per unit volume. Therefore, the pressure drop between two points must be proportional to (1/2)ρV2, which has the same dimensions as it resembles (see below) the expression for the kinetic energy per unit volume. We also know that pressure must be proportional to the length of the pipe between the two points L as the pressure drop per unit length is a constant. To turn the relationship into a proportionality coefficient of dimensionless quantity we can divide by the hydraulic diameter of the pipe, D, which is also constant along the pipe. Therefore,
The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as π, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter).
Note that (1/2)ρV2 is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of laminar flow
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...
, where all the flow lines are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the mean-square velocity, which always exceeds the square of the mean velocity. In the case of turbulent flow, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.
Practical applications
In hydraulic engineeringHydraulic engineering
This article is about civil engineering. For the mechanical engineering discipline see Hydraulic machineryHydraulic engineering as a sub-discipline of civil engineering is concerned with the flow and conveyance of fluids, principally water and sewage. One feature of these systems is the extensive...
applications, it is often desirable to express the head loss in terms of volumetric flow rate in the pipe. For this, it is necessary to substitute the following into the original head loss form of the Darcy–Weisbach equation
where
- V is, as above, the average velocity of the fluid flow, equal to the volumetric flow rateVolumetric flow rateThe volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...
per unit cross-sectional wetted area; - Q is the volumetric flow rateVolumetric flow rateThe volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...
; - Aw is the cross-sectional wetted area;
For the general case of an arbitrarily-full pipe, the value of Aw will not be immediately known, being an implicit function of pipe slope, cross-sectional shape, flow rate and other variables. If, however, the pipe is assumed to be full flowing and of circular cross-section, as is common in practical scenarios, then
where D is the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of the pipe
Substituting these results into the original formulation yields the final equation for head loss in terms of volumetric flow rate
Volumetric flow rate
The volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...
in a full-flowing circular pipe
where all symbols are defined as above.