Darcy weisbach equation derivation || fluid mechanics ||

DARCY WEISBACH EQUACTION DERIVATION || fluid mechanics ||
In fluid dynamics, the Darcy-Weisbach equation is an empirical 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 for an incompressible fluid. 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 variously called the Darcy-Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.
Pressure-loss form
In a cylindrical pipe of uniform diameter D, flowing full, the pressure loss due to viscous effects Δp is proportional to length L and can be characterized by the Darcy-Weisbach equation:[2]
{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {
ho }{2}}\cdot {\frac {{\langle v
angle }^{2}}{D}},} {\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {
ho }{2}}\cdot {\frac {{\langle v
angle }^{2}}{D}},}
where the pressure loss per unit length
Δp
/
L
(SI units: Pa/m) is a function of:
ρ, the density of the fluid (kg/m3);
D, the hydraulic diameter of the pipe (for a pipe of circular section, this equals the internal diameter of the pipe; otherwise D ≈ 2√A/π for a pipe of cross-sectional area A) (m);
⟨v⟩, the mean flow velocity, experimentally measured as the volumetric flow rate Q per unit cross-sectional wetted area (m/s);
fD, the Darcy friction factor (also called flow coefficient λ[3][4]).
For laminar flow in a circular pipe of diameter {\displaystyle D_{c}} D_{c}, the friction factor is inversely proportional to the Reynolds number alone (fD =
64
/
Re
) which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy-Weisbach equation is rewritten as
{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},} {\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}
where
μ is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s));
Q is the volumetric flow rate, used here to measure flow instead of mean velocity according to Q =
π
/
4
Dc2⟨v⟩ (m3/s).
Note that this laminar form of Darcy-Weisbach is equivalent to the Hagen-Poiseuille equation, which is analytically derived from the Navier-Stokes equations
Head-loss form
The head loss Δh (or hf) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the pressure drop is
{\displaystyle \Delta p=
ho g\,\Delta h,} {\displaystyle \Delta p=
ho g\,\Delta h,}
where
Δh is the head loss due to pipe friction over the given length of pipe (SI units: m);[b]
g is the local acceleration due to gravity (m/s2).
It is useful to present head loss per length of pipe (dimensionless):
{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{
ho g}}\cdot {\frac {\Delta p}{L}},} {\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{
ho g}}\cdot {\frac {\Delta p}{L}},}
where L is the pipe length (m).
Therefore, the Darcy-Weisbach equation can also be written in terms of head loss:[5]
{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v
angle }^{2}}{D}}.} {\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v
angle }^{2}}{D}}.}
In terms of volumetric flow
The relationship between mean flow velocity ⟨v⟩ and volumetric flow rate Q is
{\displaystyle Q=A\cdot \langle v
angle ,} {\displaystyle Q=A\cdot \langle v
angle ,}
where:
Q is the volumetric flow (m3/s),
A is the cross-sectional wetted area (m2).
In a full-flowing, circular pipe of diameter {\displaystyle D_{c}} D_{c},
{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v
angle .} {\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v
angle .}
Then the Darcy-Weisbach equation in terms of Q is
{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.} {\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}
Shear-stress form
The mean wall shear stress τ in a pipe or open channel is expressed in terms of the Darcy-Weisbach friction factor as[6]
{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}
ho {\langle v
angle }^{2}.} {\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}
ho {\langle v
angle }^{2}.}
The wall shear stress has the SI unit of pascals (Pa)
#DARCY #DARCYWEISBACH #DARCYWEISBACHEQUATION

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[@ravipendra7148]

your explanation is very very nice thanks for providing this type of video

[@mohdsajid1571]

THANKS MAM FOR YOUR EASY UNDERASTANDABLE EXPLANATION

[@Focuss09]

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may god bless U and be happy 100 years teaching like this only

[@musicalupdate4580]

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[@vemamalathi3799]

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[@moinakter4708]

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[@theboyofjoyy]

Thanks for your work!! begins at 1:50

[@user-oy8od7ru8k]

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[@siddarthaveerla1060]

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[@akashkumarkumar1198]

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[@shivangitripathi520]

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[@sadhguruphilosophy2065]

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[@ritikaverma8797]

Thank you mam

[@Dhivakar_Marimuthu_knm-99]

Explanation good

[@MSD-es1hw]

Good Madum but in friction.... There is one correction that( pie.d.l.....)means wetted surface area.... Thanq really helpful.....

[@Mediation4Healing_Calm_Peace.]

Nice explaination mam

[@newstop6230]

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[@lavanyas.b8389]

Tq mam .i easily understood it

[@caleb461]

Well done!

[@Pradeep-ri6dc]

In last step 4÷2 =2 not 1/2

[@HANVIKHA_JOY_MANAR]

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[@collinskyama2300]

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though the language am not getting it. but thanks

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[@kalisettinagamanikanta4087]

Dhanyavaad

[@tracysara6754]

Still confused on friction factor and coefficient of friction

[@TheLokhande]

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[@chanduchandu6893]

Thanks

[@neerajsetty7119]

Thank you

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[@nithinraghavendra9218]

Perfect!

[@dgn6283]

Thank you teacher

[@poonamshukla2028]

Thankyou

[@spectrum_gaming9382]

Madam. Since we have two pressure heads P1 and P2 so then why don't we have frictional forces F1 and F2

[@harleylord2752]

Bro friction is same with surface

[@vickyvk8876]

Friction is always acts opposite to the flow of the fluid.....

[@jayavijyacreations3774]

Thank you very much for your brief and straight to point explanation madam

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[@user-yp9mo6dl5h]

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[@podakoppe5535]

Got the concept :)))

[@IOSDINEGAMING12308]

Thanks mam❤

[@diliptamata3034]

Mam we can cancel out 4/2 so i.e answer will 2flv^2/gd ❤️

[@Sam-du5fb]

May I ask why f'/p = f/2?

[@Janufoodtravelvlogs]

She tell know it's constant

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Other wise

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[@masinisudheerreddy5817]

Thanku mam

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tq mam

[@mujahidshaikh7]

in last line some mistake may be please you see and do correct this
i do not understand last line how came in the position put f=f'

[@saiteja2928]

Thank u madam

[@jmfchittu8419]

mam why pressure p2 is opposite to flow

[@lakshmiarepaka4020]

How does row turns into 2 in the last step

[@pratikparbat341]

Its actually rho(density)

[@vickyvk8876]

Same doubt

[@vickyvk8876]

@@pratikparbat341 no its 2 only... if it is rho where is that 4

[@booraraghavendralokesh2373]

@@vickyvk8876 f'/rho =f/2 ....so in the place of f'/rho we have taken f/2

[@booraraghavendralokesh2373]

@@vickyvk8876 and 4 is came from p/A =πd/π/4 d square ....so numerator d and dinominator d gets cancel and π also gets cancel in both numerator and dinominator....and we get 4/d

[@ravirowdy828]

what are the minor losses mam

[@karthikkarthikkn4615]

10:12

[@Janufoodtravelvlogs]

Boundary layer chapter

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[@sadeek2009]

لكل زول بقرى عند ساتي يدعي للمرة تجي تقرينا

[@kazimd.mehedihasan399]

never hf=(4fLV2/2gd) It is hf=(4fLV2/8gd)

[@SandeepYadav-bj1zl]

Madam why u take P2 as negative

[@vinayakhuracan5182]

Velocity at inlet is high (low pressure), as moving on through the pipe velocity decreases due to friction so at outlet velocity is relatively low to inlet (high pressure), always pressure flows from high to low ( fancy term:back flow)

[@vijay-sg3bb]

It is in opposite direction

[@mr.ramanbhagat1627]

jo samjana nhi aaya bahut jldi skip kiya (f/2)

[@shashidharmadar2964]

Thank you madam