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aga
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Just finished filling this newly-built water tank from a bit of 100mm dia pipe and a 3 bar pump.
Took 2.71 hours to fill.
Where Bernoulli ballsed up was in the calculation of how long it took to get the pump running again after some idiot wired it up wrong.
needs to add Ki^d
where K is the idiocy constant, i is the idiot and d is the idiot density.
This is/was an excellent brain teaser, and highly informative.
Post more !
[Edited on 6-3-2016 by aga]
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blogfast25
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Quote: Originally posted by aga  | Just finished filling this newly-built water tank from a bit of 100mm dia pipe and a 3 bar pump.
Took 2.71 hours to fill.
Where Bernoulli ballsed up was in the calculation of how long it took to get the pump running again after some idiot wired it up wrong.
needs to add Ki^d
where K is the idiocy constant, i is the idiot and d is the idiot density.
This is/was an excellent brain teaser, and highly informative.
Post more !
[Edited on 6-3-2016 by aga] |
I doubt that: Bernoulli was an excellent electrician, although they do say he didn't have the edge on Aristotle!
Maybe a basic course in calculus could be useful?
Ah, C-A-L-C-U-L-U-S!!! Succulent abracadabra, bish, bosh, bash magic of numbers!   
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aga
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Some maths would be immensely helpful.
Perhaps the B&D Uni could offer such a course ?
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Magpie
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@blogfast:
I have a couple comments:
| Quote: Originally posted by blogfast25 |
Start from Bernoulli's continuity equation. Applied here:
$$\frac{v_z^2}{2}+gz+\frac{p_z}{\rho}=\frac{v_0^2}{2}+gz_0+\frac{p_0}{\rho}$$
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minor comment: The Bernoulli equation is an energy conservation equation, not a continuity equation. The equation that defines Q is the
continuity equation.
It looks like you switched signs under the radical. Did I miss something?
The single most important condition for a successful synthesis is good mixing - Nicodem
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Metacelsus
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Yes, my equation also had the signs switched from what Blogfast wrote (but that was also the equation that gave me imaginary numbers).
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Magpie
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| Quote: Originally posted by Metacelsus | | Yes, my equation also had the signs switched from what Blogfast wrote (but that was also the equation that gave me imaginary numbers).
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Yes, when I tried to correct my solution by removing my entrance loss term and adding the velocity term of the incoming stream I also ended up with a
negative quantity under the radical.
The single most important condition for a successful synthesis is good mixing - Nicodem
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blogfast25
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You are the most apprehensive reader of that derivation, Magpie! Kudos.
Yes, there's been a switcheroo of signs and let me explain why.
You are correct that this form of the Bernoulli equation:
$$\frac{v^2}{2}+gz+\frac{p}{\rho}=\text{Constant}$$
... if you multiply it with the mass flow:
$$\dot{m}$$
... is an energy (per unit of time) conservation equation and if we assume there no mass flow loss or increase and no loss or addition of energy, the
equation represents transformations of energy (kinetic, potential, internal) between one point on the streamline and another.
However, kinetic energy is direction invariant because of the squares. Squares give the same (positive) outcome regardless whether the
argument is positive or negative.
Here we have a situation where the velocity vector:
$$\vec{v_z}$$
points in the opposite direction of gravimetric acceleration:
$$\vec{g}$$
That's why a change in sign is needed because otherwise the expression for
$$Q(z)$$
makes no sense.
I noticed the problem a few days ago when modelling a sinking ship: with a hole in the bottom the ship takes on water due to outside pressure but
hydrostatic pressure of water already taken on board actually slows down that flow. There too a change in sign in the argument of Q(z) is needed to
represent reality.
[Edited on 7-3-2016 by blogfast25]
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Magpie
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I've never run across anything like this before when using the Bernoulli equation. I'm going to have to continue thinking about it for awhile.
Most textbook problems incorporate friction loss terms due to wall friction as well as "minor losses." Minor losses are due to ells, tees, and other
fittings. Maybe that's why I've never seen this issue before.
The single most important condition for a successful synthesis is good mixing - Nicodem
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Oscilllator
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Wait, I don't get it. If the pressure in the tank is a constant 1.3 bar and the pressure outside is a constant 3bar, then surely there is a constant
pressure differential of 1.7bar.
So we have a constant pressure differential from which it is easy to get a flow rate using the cross-sectional area, and then the fill time using the
volume.
Where did I go wrong in this reasoning?
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blogfast25
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| Quote: Originally posted by Magpie | I've never run across anything like this before when using the Bernoulli equation. I'm going to have to continue thinking about it for awhile.
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If you are, then think 'Hamiltonian' and how the first time derivative of it gives you the Equation of Motion (Newtonian).
Alternatively, determine the time needed for the tank to empty from full (take away the pump, of course! ), using Bernoulli. You'll see no sneaky sign switcheroo is needed then.
Another reason why you may never have encountered is the following. In my own Uni textbook I haven't found a single application of Bernoulli
where flow wasn't somewhat downward or simply plain horizontal. In all these cases no sign change is needed to get a Qv(z) equation.
[Edited on 7-3-2016 by blogfast25]
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blogfast25
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| Quote: Originally posted by Oscilllator | Wait, I don't get it. If the pressure in the tank is a constant 1.3 bar and the pressure outside is a constant 3bar, then surely there is a constant
pressure differential of 1.7bar.
So we have a constant pressure differential from which it is easy to get a flow rate using the cross-sectional area, and then the fill time using the
volume.
Where did I go wrong in this reasoning? |
As the tank fills, the water already in the tank exerts hydrostatic pressure:
$$p_{hydr}= \rho g z$$
with z the height of the water.
The 'effective' pressure is thus:
$$\Delta p_{eff}= \Delta p - \rho g z$$
Flow rate into the tank thus slows down as it fills up.
[Edited on 7-3-2016 by blogfast25]
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blogfast25
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If you ask me very, very nicely I'll put it to the Chairman of the Bored.
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