Mechanical Engineering - Hydraulics and Fluid Mechanics - Discussion

Here is solution to the contradictions, please read this carefully.

Head loss due to viscosity in case of laminar flow in pipe = 32μvl/wd^2

In this question, Head loss due to friction in pipe = 4flv^2/2gd will not be considered because this equation holds for head loss due to friction in the pipe(for both, laminar and turbulent), not for head loss due to 'viscosity in case of laminar flow' in pipe.

Since, for laminar flow condition viscosity is dominant over inertia of fluid flowing, hence head loss due to viscosity is being considered and i.e. 32μvl/wd^2.

In fact, head loss due to friction will also act in case of laminar flow but in this question head loss due to laminar flow(i.e. head loss due to viscosity) is being asked only.
So, in this question, Head loss is proportional to Velocity.

For Laminar flow
hL =(32μLVavg)/(ρgD^2 ) -------------> (i)

For Laminar or turbulent flows, circular or noncircular pipes, smooth or rough surfaces, horizontal or inclined pipes.
hL =f L/D (Vavg^2)/2g -------------> (ii)

Now, value of friction factor, f is different for different flows
In case of laminar flow
friction factor, f = 64μ/(ρDVavg )=64/Re ------------> (iii)

By putting the value of friction factor (laminar given above) in equation (ii) we will get the equation (i) which conclude that loss of pressure head (hL) is directly proportional to velocity, Vavg (Not V^2).

4flv2/2gd is a loss of pressure head and velocity is taken.

Whereas 32vl/d2 is a pressure drop and here average velocity is calculated.

Hence the answer given is contradictory.

Actually Hf= 4flVsq/2gd.

But friction factor 4f = 64/Re.
And Re = RoVD/Mu.
Thus Vsq and V cancel each other gives V on vthe numerator. Hence Hf is proportional to V.

Laminar directly proportional to velocity (hagen poisellea eqn} and in case turbulent proportional to square of velocity (darcy formula).

I think answer is (velocity)2.
As change(pressure) = row*g*h.

h=flv2/2gd.
So change(pressure) directly proportional to v2.

The question is about the loss of PRESSURE HEAD, not HEADLOSS (Hf).

So the correct answer is PRESSURE HEAD = 32μvl/wd^2.

Why so confusing. It's simply asking head loss and is directly proportional to the square of the velocity that's it.

For Laminar flow :

Loss of head = hF = [32*u*V(average)L]/r*g*d^2.

For turbulent flow:

HF = (flv^2)/(2*g*d).

Answer is right.

Loss of head = [32u*V(average)L]/r*g*d^2.

V(Average) = Average velocity.