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| In [[fluid dynamics]] the '''Borda–Carnot equation''' is an [[empirical]] description of the [[mechanical work#Mechanical energy|mechanical energy]] losses of the [[fluid]] due to a (sudden) [[fluid flow|flow]] expansion. It describes how the [[total head]] reduces due to the losses. This in contrast with [[Bernoulli's principle]] for [[dissipation]]less flow (without irreversible losses), where the total head is a constant along a [[streamline (fluid dynamics)|streamline]]. The equation is named after [[Jean-Charles de Borda]] (1733–1799) and [[Lazare Carnot]] (1753–1823).
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| This equation is used both for [[open channel flow]] as well as in [[pipe flow]]s. In parts of the flow where the irreversible energy losses are negligible, Bernoulli's principle can be used.
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| == Formulation ==
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| The Borda–Carnot equation is:<ref name=Chanson_231>Chanson (2004), p. 231.</ref><ref name=Massey_274>Massey & Ward-Smith (1998), pp. 274–280.</ref>
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| :<math>\Delta E\, =\, \xi\, {\scriptstyle \frac12}\, \rho\, \left( v_1\, -\, v_2 \right)^2,</math>
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| where
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| *''ΔE'' is the fluid's mechanical energy loss,
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| *''ξ'' is an empirical loss coefficient, which is [[dimensionless]] and has a value between zero and one, 0 ≤ ''ξ'' ≤ 1,
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| *''ρ'' is the fluid [[density]],
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| *''v''<sub>1</sub> and ''v''<sub>2</sub> are the mean [[flow velocity|flow velocities]] before and after the expansion.
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| In case of an abrupt and wide expansion the loss coefficient is equal to one.<ref name=Chanson_231/> In other instances, the loss coefficient has to be determined by other means, most often from [[empirical formula]]e (based on data obtained by [[experiment]]s). The Borda–Carnot loss equation is only valid for decreasing velocity, ''v''<sub>1</sub> > ''v''<sub>2</sub>, otherwise the loss ''ΔE'' is zero – without [[mechanical work]] by additional external [[force]]s there cannot be a gain in mechanical energy of the fluid.
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| The loss coefficient ''ξ'' can be influenced by [[Streamlines, streaklines, and pathlines|streamlining]]. For example in case of a pipe expansion, the use of a gradual expanding [[diffuser]] can reduce the mechanical energy losses.<ref>{{citation | title=Fluid Mechanics Through Problems | first=R. J. | last=Garde | publisher=New Age Publishers | year=1997 | isbn=81-224-1131-2 }}. See pp. 347–349.</ref>
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| == Relation to the total head and Bernoulli's principle ==
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| The Borda–Carnot equation gives the decrease in the constant of the [[Bernoulli's principle|Bernoulli equation]]. For an incompressible flow the result is – for two locations labelled 1 and 2, with location 2 downstream to 1 – along a [[Streamlines, streaklines, and pathlines|streamline]]:<ref name=Massey_274/>
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| :<math> | |
| p_1\, +\, {\scriptstyle \frac12}\,\rho\,v_1^2\, +\, \rho\,g\,z_1\,
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| =\,
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| p_2\, +\, {\scriptstyle \frac12}\,\rho\,v_2^2\, +\, \rho\,g\,z_2\,
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| +\, \Delta E,
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| </math>
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| with
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| *''p''<sub>1</sub> and ''p''<sub>2</sub> the [[pressure]] at location 1 and 2,
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| *''z''<sub>1</sub> and ''z''<sub>2</sub> the vertical elevation – above some reference level – of the fluid particle, and
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| *''g'' the [[Earth's gravity|gravitational acceleration]].
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| The first three terms, on either side of the [[equal sign]] are respectively the pressure, the [[kinetic energy]] density of the fluid and the [[potential energy]] density due to gravity. As can be seen, pressure acts effectively as a form of potential energy.
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| In case of high-pressure pipe flows, when gravitational effects can be neglected, ''ΔE'' is equal to the loss ''Δ''(''p''+½''ρv''<sup>2</sup>):
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| :<math>\Delta E\, =\, \Delta \left( p\, +\, {\scriptstyle\frac12}\, \rho\, v^2 \right).</math>
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| For [[open channel flow]]s, ''ΔE'' is related to the [[total head]] loss ''ΔH'' as:<ref name=Chanson_231/>
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| :<math>\Delta E\, =\, \rho\, g\, \Delta H,</math> with ''H'' the total head:<ref name=Chanson_22>Chanson (2004), p. 22.</ref> <math>H\, =\, h\, +\, \frac{v^2}{2g},</math>
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| where ''h'' is the [[hydraulic head]] – the [[free surface]] elevation above a reference [[datum (geodesy)|datum]]: ''h'' = ''z'' + ''p''/(''ρg'').
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| == Examples ==
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| === Sudden expansion of a pipe ===
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| [[File:Flow expansion.svg|thumb|right|A sudden flow expansion.]]
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| 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''<sub>1</sub>, the pressure is ''p''<sub>1</sub> and the cross-sectional area is ''A''<sub>1</sub>. The corresponding flow quantities at cross section 2 – after the expansion – are ''v''<sub>2</sub>, ''p''<sub>2</sub> and ''A''<sub>2</sub>, respectively. The loss coefficient ''ξ'' for this sudden expansion is equal to one: ''ξ'' = 1.0. Due to mass conservation, assuming a constant fluid [[density]] ''ρ'', the [[volumetric flow rate]] through both cross sections 1 and 2 has to be equal:
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| :<math>A_1\, v_1\, = A_2\, v_2</math> so <math>v_2\, =\, \frac{A_1}{A_2}\, v_1.</math>
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| Consequently – according to the Borda–Carnot equation – the mechanical energy loss in this sudden expansion is:
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| :<math>\Delta E\, =\, \frac12\, \rho\, \left( 1\, -\, \frac{A_1}{A_2} \right)^2\, v_1^2.</math>
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| The corresponding loss of total head ''ΔH'' is:
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| :<math>\Delta H\, =\, \frac{\Delta E}{\rho\,g}\, =\, \frac{1}{2\,g}\, \left( 1\, -\, \frac{A_1}{A_2} \right)^2\, v_1^2.</math>
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| 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):
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| :<math>\Delta p\, =\, p_1\, -\, p_2\, =\, -\, \rho\, \frac{A_1}{A_2} \left( 1\, -\, \frac{A_1}{A_2}\right)\, v_1^2,</math>
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| and the change in hydraulic head ''h'' = ''z'' + ''p''/(''ρg''): | |
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| :<math>\Delta h\, =\, h_1\, -\, h_2\, =\, -\, \frac{1}{g}\, \frac{A_1}{A_2} \left( 1\, -\, \frac{A_1}{A_2}\right)\, v_1^2.</math>
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| The minus signs, in front of the [[right-hand side]]s, mean that the pressure (and hydraulic head) are larger after the pipe expansion.
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| That this change in the pressures (and hydraulic heads), just before and after the pipe expansion, corresponds with an energy loss becomes clear when comparing with the results of [[Bernoulli's principle]]. 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.
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| ===Sudden contraction of a pipe===
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| [[File:Flow contraction.svg|thumb|right|Flow through a sudden contraction of the pipe diameter, with [[flow separation]] bubbles near cross section 3.]]
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| In case of a sudden reduction of pipe diameter, without streamlining, the flow is not be able to follow the sharp bend into the narrower pipe. As a result, there is [[flow separation]], 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.
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| There is not much head loss between cross section 1, before the contraction, and cross section 3, the [[vena contracta]] 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'':<ref>Garde (1998), pp. 349–350.</ref>
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| :<math>\mu\, =\, \frac{A_3}{A_2},</math>
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| with ''A''<sub>3</sub> the cross-sectional area at the location of strongest main flow contraction 3, and ''A''<sub>2</sub> the cross-sectional area of the narrower part of the pipe. Since ''A''<sub>3</sub> ≤ ''A''<sub>2</sub>, 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''):
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| :<math>A_1\, v_1\, =\, A_2\, v_2\, =\, A_3\, v_3,</math>
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| with ''v''<sub>1</sub>, ''v''<sub>2</sub> and ''v''<sub>3</sub> 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:
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| :<math>\Delta E\, =\, \frac12\, \rho\, \left( v_3\, -\, v_2 \right)^2\,
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| =\, \frac12\, \rho\, \left( \frac{1}{\mu}\, -\, 1 \right)^2\, v_2^2\,
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| =\, \frac12\, \rho\, \left( \frac{1}{\mu}\, -\, 1 \right)^2\, \left( \frac{A_1}{A_2} \right)^2\, v_1^2.
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| </math>
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| The corresponding loss of total head ''ΔH'' can be computed as ''ΔH'' = ''ΔE''/(''ρg'').
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| According to measurements by [[Julius Weisbach|Weisbach]], the contraction coefficient for a sharp-edged contraction is approximately:<ref>{{citation | title=Prandtl's Essentials of Fluid Mechanics | first1=Herbert | last1=Oertel | first2=Ludwig | last2= Prandtl | first3=M. | last3=Böhle | first4=Katherine | last4=Mayes | publisher=Springer | year=2004 | isbn=0-387-40437-6 }}. See pp. 163–165.</ref>
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| :<math>\mu\, =\, 0.63\, +\, 0.37\, \left( \frac{A_2}{A_1} \right)^3.</math>
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| == See also ==
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| *[[Darcy–Weisbach equation]]
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| *[[Prony equation]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| *{{citation
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| | title=Hydraulics of Open Channel Flow: An Introduction
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| | first=Hubert
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| | last=Chanson
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| | authorlink=Hubert Chanson
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| | publisher=Butterworth–Heinemann
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| | year=2004
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| | isbn=0-7506-5978-5
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| | edition=2<sup>nd</sup>
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| }}, 650 pp.
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| *{{citation
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| | title=Mechanics of Fluids
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| | first1=Bernard Stanford
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| | last1=Massey
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| | first2=John
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| | last2=Ward-Smith
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| | publisher=Taylor & Francis
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| | year=1998
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| | isbn=0-7487-4043-0
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| | edition=7<sup>th</sup>
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| }}, 744 pp.
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| {{DEFAULTSORT:Borda-Carnot equation}}
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| [[Category:Equations of fluid dynamics]]
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| [[Category:Fluid dynamics]]
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| [[Category:Hydraulics]]
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| [[Category:Piping]]
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They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. It is very easy to customize plugins according to the needs of a particular business. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. If a newbie missed a certain part of the video then they could always rewind. Which is perfect for building a mobile site for business use. You can up your site's rank with the search engines by simply taking a bit of time with your site. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business.
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If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. The company gains commission from the customers' payment. See a product, take a picture, and it gives you an Amazon price for that product, or related products.
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