Papyrus 36

In fluid dynamics the Borda–Carnot equation is an empirical description of the mechanical energy losses of the fluid due to a (sudden) flow expansion. It describes how the total head reduces due to the losses. This in contrast with Bernoulli's principle for dissipationless flow (without irreversible losses), where the total head is a constant along a streamline. The equation is named after Jean-Charles de Borda (1733–1799) and Lazare Carnot (1753–1823).

This equation is used both for open channel flow as well as in pipe flows. In parts of the flow where the irreversible energy losses are negligible, Bernoulli's principle can be used.

Formulation

The Borda–Carnot equation is:

$\Delta E\,=\,\xi \,{\scriptstyle {\frac {1}{2}}}\,\rho \,\left(v_{1}\,-\,v_{2}\right)^{2},$

where

• ΔE is the fluid's mechanical energy loss,
• ξ is an empirical loss coefficient, which is dimensionless and has a value between zero and one, 0 ≤ ξ ≤ 1,
• ρ is the fluid density,
• v₁ and v₂ 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 formulae (based on data obtained by experiments). The Borda–Carnot loss equation is only valid for decreasing velocity, v₁ > v₂, otherwise the loss ΔE is zero – without mechanical work by additional external forces 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 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. For an incompressible flow the result is – for two locations labelled 1 and 2, with location 2 downstream to 1 – along a streamline:

$p_{1}\,+\,{\scriptstyle {\frac {1}{2}}}\,\rho \,v_{1}^{2}\,+\,\rho \,g\,z_{1}\,=\,p_{2}\,+\,{\scriptstyle {\frac {1}{2}}}\,\rho \,v_{2}^{2}\,+\,\rho \,g\,z_{2}\,+\,\Delta E,$

with

• p₁ and p₂ the pressure at location 1 and 2,
• z₁ and z₂ the vertical elevation – above some reference level – of the fluid particle, and
• g the gravitational acceleration.

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.

In case of high-pressure pipe flows, when gravitational effects can be neglected, ΔE is equal to the loss Δ(p+½ρv²):

$\Delta E\,=\,\Delta \left(p\,+\,{\scriptstyle {\frac {1}{2}}}\,\rho \,v^{2}\right).$

For open channel flows, ΔE is related to the total head loss ΔH as:

$\Delta E\,=\,\rho \,g\,\Delta H,$ with H the total head: $H\,=\,h\,+\,{\frac {v^{2}}{2g}},$

where h is the hydraulic head – the free surface elevation above a reference datum: h = z + p/(ρg).

Examples

Sudden expansion of a pipe

A sudden flow expansion.

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 v₁, the pressure is p₁ and the cross-sectional area is A₁. The corresponding flow quantities at cross section 2 – after the expansion – are v₂, p₂ and A₂, 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:

$A_{1}\,v_{1}\,=A_{2}\,v_{2}$     so     $v_{2}\,=\,{\frac {A_{1}}{A_{2}}}\,v_{1}.$

Consequently – according to the Borda–Carnot equation – the mechanical energy loss in this sudden expansion is:

$\Delta E\,=\,{\frac {1}{2}}\,\rho \,\left(1\,-\,{\frac {A_{1}}{A_{2}}}\right)^{2}\,v_{1}^{2}.$

The corresponding loss of total head ΔH is:

$\Delta H\,=\,{\frac {\Delta E}{\rho \,g}}\,=\,{\frac {1}{2\,g}}\,\left(1\,-\,{\frac {A_{1}}{A_{2}}}\right)^{2}\,v_{1}^{2}.$

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):

$\Delta p\,=\,p_{1}\,-\,p_{2}\,=\,-\,\rho \,{\frac {A_{1}}{A_{2}}}\left(1\,-\,{\frac {A_{1}}{A_{2}}}\right)\,v_{1}^{2},$

and the change in hydraulic head h = z + p/(ρg):

$\Delta h\,=\,h_{1}\,-\,h_{2}\,=\,-\,{\frac {1}{g}}\,{\frac {A_{1}}{A_{2}}}\left(1\,-\,{\frac {A_{1}}{A_{2}}}\right)\,v_{1}^{2}.$

The minus signs, in front of the right-hand sides, mean that the pressure (and hydraulic head) are larger after the pipe expansion. 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.

Sudden contraction of a pipe

Flow through a sudden contraction of the pipe diameter, with flow separation bubbles near cross section 3.

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.

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:

$\mu \,=\,{\frac {A_{3}}{A_{2}}},$

with A₃ the cross-sectional area at the location of strongest main flow contraction 3, and A₂ the cross-sectional area of the narrower part of the pipe. Since A₃ ≤ A₂, 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 v₁, v₂ and v₃ 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:

$\Delta E\,=\,{\frac {1}{2}}\,\rho \,\left(v_{3}\,-\,v_{2}\right)^{2}\,=\,{\frac {1}{2}}\,\rho \,\left({\frac {1}{\mu }}\,-\,1\right)^{2}\,v_{2}^{2}\,=\,{\frac {1}{2}}\,\rho \,\left({\frac {1}{\mu }}\,-\,1\right)^{2}\,\left({\frac {A_{1}}{A_{2}}}\right)^{2}\,v_{1}^{2}.$

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:

$\mu \,=\,0.63\,+\,0.37\,\left({\frac {A_{2}}{A_{1}}}\right)^{3}.$

See also

• Darcy–Weisbach equation
• Prony equation

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, 744 pp.