Borda–Carnot equation
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In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
the Borda–Carnot equation is an
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
description of the
mechanical energy In Outline of physical science, physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, t ...
losses of the
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
due to a (sudden)
flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psych ...
expansion. It describes how the
total head In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
reduces due to the losses. This is in contrast with
Bernoulli's principle In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
for
dissipation In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a ...
less flow (without irreversible losses), where the total head is a constant along a
streamline Streamline may refer to: Business * Streamline Air, American regional airline * Adobe Streamline, a discontinued line tracing program made by Adobe Systems * Streamline Cars, the company responsible for making the Burney car Engineering ...
. The equation is named after
Jean-Charles de Borda Jean-Charles, chevalier de Borda (4 May 1733 – 19 February 1799) was a French mathematician, physicist, and Navy officer. Biography Borda was born in the city of Dax to Jean‐Antoine de Borda and Jeanne‐Marie Thérèse de Lacroix. In 175 ...
(1733–1799) and
Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician. He was known as the "Organizer of Victory" in the French Revolutionary Wars and Napoleonic Wars. Education and early ...
(1753–1823). This equation is used both for
open channel flow In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but diff ...
as well as in
pipe flow In fluid mechanics, pipe flow is a type of liquid flow within a closed conduit, such as a pipe or tube. The other type of flow within a conduit is open channel flow. These two types of flow are similar in many ways, but differ in one important ...
s. In parts of the flow where the irreversible energy losses are negligible, Bernoulli's principle can be used.


Formulation

The Borda–Carnot equation is:Chanson (2004), p. 231.Massey & Ward-Smith (1998), pp. 274–280. :\Delta E\, =\, \xi\, \, \rho\, \left( v_1\, -\, v_2 \right)^2, where *''ΔE'' is the fluid's mechanical energy loss, *''ξ'' is an empirical loss coefficient, which is
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
and has a value between zero and one, 0 ≤ ''ξ'' ≤ 1, *''ρ'' is the fluid
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, *''v''1 and ''v''2 are the mean flow velocities before and after the expansion. In case of an abrupt and wide expansion the loss coefficient is equal to one. In other instances, the loss coefficient has to be determined by other means, most often from
empirical formula In chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms present in a compound. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, would simply be SO, as is th ...
e (based on data obtained by
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
s). The Borda–Carnot loss equation is only valid for decreasing velocity, ''v''1 > ''v''2, otherwise the loss ''ΔE'' is zero – without
mechanical work In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stren ...
by additional external
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s there cannot be a gain in mechanical energy of the fluid. The loss coefficient ''ξ'' can be influenced by streamlining. For example, in case of a pipe expansion, the use of a gradual expanding
diffuser Diffuser may refer to: Aerodynamics * Diffuser (automotive), a shaped section of a car's underbody which improves the car's aerodynamic properties * Part of a jet engine air intake, especially when operated at supersonic speeds * The channel betw ...
can reduce the mechanical energy losses.


Relation to the total head and Bernoulli's principle

The Borda–Carnot equation gives the decrease in the constant of the
Bernoulli equation In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
. For an incompressible flow the result is – for two locations labelled 1 and 2, with location 2 downstream to 1 – along a
streamline Streamline may refer to: Business * Streamline Air, American regional airline * Adobe Streamline, a discontinued line tracing program made by Adobe Systems * Streamline Cars, the company responsible for making the Burney car Engineering ...
: : p_1\, +\, \,\rho\,v_1^2\, +\, \rho\,g\,z_1\, =\, p_2\, +\, \,\rho\,v_2^2\, +\, \rho\,g\,z_2\, +\, \Delta E, with *''p''1 and ''p''2 the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
at location 1 and 2, *''z''1 and ''z''2 the vertical elevation – above some reference level – of the fluid particle, and *''g'' the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
. The first three terms, on either side of the
equal sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between two ...
are respectively the pressure, the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
density of the fluid and the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
density due to gravity. As can be seen, pressure acts effectively as a form of potential energy. In case of high-pressure pipe flows, when gravitational effects can be neglected, ''ΔE'' is equal to the loss ''Δ''(''p''+½''ρv''2): :\Delta E\, =\, \Delta \left( p\, +\, \, \rho\, v^2 \right). For
open channel flow In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but diff ...
s, ''ΔE'' is related to the
total head In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
loss ''ΔH'' as: :\Delta E\, =\, \rho\, g\, \Delta H, with ''H'' the total head:Chanson (2004), p. 22. H\, =\, h\, +\, \frac, where ''h'' is the
hydraulic head Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22. It is usually measured as a liquid surface elevation, expressed in units of length, ...
– the
free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
elevation above a reference
datum In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
: ''h'' = ''z'' + ''p''/(''ρg'').


Examples


Sudden expansion of a pipe

The Borda–Carnot equation is applied to the flow through a sudden expansion of a horizontal pipe. At cross section 1, the mean flow velocity is equal to ''v''1, the pressure is ''p''1 and the cross-sectional area is ''A''1. The corresponding flow quantities at cross section 2 – well behind the expansion (and regions of
separated flow In fluid dynamics, flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake. A boundary layer exists whenever there is relative movement between a fluid and a solid surface with viscous for ...
) – are ''v''2, ''p''2 and ''A''2, respectively. At the expansion, the flow separates and there are
turbulent In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
recirculating flow zones with mechanical energy losses. The loss coefficient ''ξ'' for this sudden expansion is approximately equal to one: ''ξ'' ≈ 1.0. Due to mass conservation, assuming a constant fluid
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
''ρ'', the
volumetric flow rate In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
through both cross sections 1 and 2 has to be equal: :A_1\, v_1\, = A_2\, v_2     so     v_2\, =\, \frac\, v_1. Consequently – according to the Borda–Carnot equation – the mechanical energy loss in this sudden expansion is: :\Delta E\, =\, \frac12\, \rho\, \left( 1\, -\, \frac \right)^2\, v_1^2. The corresponding loss of total head ''ΔH'' is: :\Delta H\, =\, \frac\, =\, \frac\, \left( 1\, -\, \frac \right)^2\, v_1^2. For this case with ''ξ'' = 1, the total change in kinetic energy between the two cross sections is dissipated. As a result, the pressure change between both cross sections is (for this horizontal pipe without gravity effects): :\Delta p\, =\, p_1\, -\, p_2\, =\, -\, \rho\, \frac \left( 1\, -\, \frac\right)\, v_1^2, and the change in hydraulic head ''h'' = ''z'' + ''p''/(''ρg''): :\Delta h\, =\, h_1\, -\, h_2\, =\, -\, \frac\, \frac \left( 1\, -\, \frac\right)\, v_1^2. The minus signs, in front of the
right-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.Bernoulli's principle In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
. According to this dissipationless principle, a reduction in flow speed is associated with a much larger increase in pressure than found in the present case with mechanical energy losses.


Sudden contraction of a pipe

In case of a sudden reduction of pipe diameter, without streamlining, the flow is not able to follow the sharp bend into the narrower pipe. As a result, there is
flow separation In fluid dynamics, flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake. A boundary layer exists whenever there is relative movement between a fluid and a solid surface with viscous fo ...
, creating recirculating separation zones at the entrance of the narrower pipe. The main flow is contracted between the separated flow areas, and later on expands again to cover the full pipe area. There is not much head loss between cross section 1, before the contraction, and cross section 3, the
vena contracta Vena contracta is the point in a fluid stream where the diameter of the stream is the least, and fluid velocity is at its maximum, such as in the case of a stream issuing out of a nozzle (orifice). (Evangelista Torricelli, 1643). It is a place wh ...
at which the main flow is contracted most. But there are substantial losses in the flow expansion from cross section 3 to 2. These head losses can be expressed by using the Borda–Carnot equation, through the use of the coefficient of contraction ''μ'': :\mu\, =\, \frac, with ''A''3 the cross-sectional area at the location of strongest main flow contraction 3, and ''A''2 the cross-sectional area of the narrower part of the pipe. Since ''A''3 ≤ ''A''2, the coefficient of contraction is less than one: ''μ'' ≤ 1. Again there is conservation of mass, so the volume fluxes in the three cross sections are a constant (for constant fluid density ''ρ''): :A_1\, v_1\, =\, A_2\, v_2\, =\, A_3\, v_3, with ''v''1, ''v''2 and ''v''3 the mean flow velocity in the associated cross sections. Then, according to the Borda–Carnot equation (with loss coefficient ''ξ''=1), the energy loss ''ΔE'' per unit of fluid volume and due to the pipe contraction is: :\Delta E\, =\, \frac12\, \rho\, \left( v_3\, -\, v_2 \right)^2\, =\, \frac12\, \rho\, \left( \frac\, -\, 1 \right)^2\, v_2^2\, =\, \frac12\, \rho\, \left( \frac\, -\, 1 \right)^2\, \left( \frac \right)^2\, v_1^2. The corresponding loss of total head ''ΔH'' can be computed as ''ΔH'' = ''ΔE''/(''ρg''). According to measurements by Weisbach, the contraction coefficient for a sharp-edged contraction is approximately: :\mu\, =\, 0.63\, +\, 0.37\, \left( \frac \right)^3.


Derivation from the momentum balance for a sudden expansion

For a sudden expansion in a pipe, see the figure above, the Borda–Carnot equation can be derived from mass- and
momentum conservation In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
of the flow., §5.15. The momentum flux ''S'' (i.e. for the fluid momentum component parallel to the pipe axis) through a cross section of area ''A'' is – according to the
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
: :S = A\, \left( \rho\, v^2 + p \right). Consider the conservation of mass and momentum for a
control volume In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fictitious region of a given v ...
bounded by cross section 1 just upstream of the expansion, cross section 2 downstream of where the flow re-attaches again to the pipe wall (after the flow separation at the expansion), and the pipe wall. There is the control volume's gain of momentum ''S''1 at the inflow and loss ''S''2 at the outflow. Besides, there is also the contribution of the force ''F'' by the pressure on the fluid exerted by the expansion's wall (perpendicular to the pipe axis): :F = ( A_2 - A_1 )\, p_1, where it has been assumed that the pressure is equal to the close-by upstream pressure ''p''1. Adding contributions, the momentum balance for the control volume between cross sections 1 and 2 gives: :S_1 - S_2 + F = A_1\, \left( \rho\, v_1^2 + p_1 \right) - A_2\, \left( \rho\, v_2^2 + p_2 \right) + ( A_2 - A_1 )\, p_1 = 0. Consequently, since by mass conservation : : \Delta p = p_1 - p_2 = -\rho\, \left( \frac\, v_1^2 - v_2^2 \right) = -\rho\, \frac\, \left( 1 - \frac \right)\, v_1^2, in agreement with the pressure drop Δ''p'' in the example above. The mechanical energy loss Δ''E'' is: : \begin \Delta E &= \left( p_1 + \tfrac12\, \rho\, v_1^2 \right ) - \left( p_2 + \tfrac12\, \rho\, v_2^2 \right ) = \Delta p + \tfrac12\, \rho\, \left( v_1^2 - v_2^2 \right) \\ &= -\rho\, v_2\, \left( v_1 - v_2 \right) + \tfrac12\, \rho\, \left( v_1^2 - v_2^2 \right) = \tfrac12\, \rho\, \left( v_1^2 - 2\, v_1\, v_2 + v_2^2 \right) \\ &= \tfrac12\, \rho\, \left( v_1 - v_2 \right)^2, \end which is the Borda–Carnot equation (with ξ = 1).


See also

*
Darcy–Weisbach equation In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation ...
*
Prony equation The Prony equation (named after Gaspard de Prony) is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe. It is an empirical equation developed by Frenchman Gaspard de Prony ...


Notes


References

*, 634 pp. *, 634 pp. *, 706 pp. {{DEFAULTSORT:Borda-Carnot equation Equations of fluid dynamics Fluid dynamics Hydraulics Piping