Flow under an angle

Klavoir · September 20, 2022, 03:37

Hi,

I have a basic fluid dynamics problem, someone might know the answer right away.
I have a horizontal tank placed under a small angle. At the lowest point water is pumped out from the bottom with a certain
flow rate. On the higher side water the same amount is entering. So the velocity of the water through the tank is determined by
by the flow rate of the pump. I wonder what will be the height of the water level measured to the bottom of the tank over the length of the tank.
When the water is stationary the height at the lowest end of the tank will obviously be higher than at the highest point
(the water surface is then horizontal).
When the water is flowing will the height of the water layer over the length of the tank be dependent on the flow rate?
I would think the more flow, the more the surface of the water will be parallel to the bottom of the tank?

LuckyTran · September 20, 2022, 14:23

The analysis can become rather cumbersome when you take everything into account and get complicated for a rather trivial result. There is a water level decrease even for a flat bottom'd tank. In 1D you can use a simple mechanical energy balance to determine the surface deflection for a given flow velocity (i.e. bernoulli). You'll quickly find you need very large velocities or large horizontal distances (i.e. km in the horizontal direction) to see meaningful differences in water level height. And then you can take into account 2D/3D effects if you want. And you can consider that the height of the air column above the water also needs to move with the water column.

But yes, the more the velocity, the more it wants to follow the shape of the tank bottom.

Klavoir · September 21, 2022, 08:32

Good to hear you agree with my gut feeling about it. To quantify this a little more. The tank is 5m long, the angle is such that the bottom of the tank has a height difference of 20 mm about the length. The velocity is 0.5 m/s. I guess with this velocity the difference in height water layer relative to the bottom of the tank at the front and the back of the tank is close to 20 mm?

LuckyTran · September 21, 2022, 09:58

Clarification: I did not mean Bernoulli but rather to use the Darcy-Weisbach relation for open channel flows. There is an open channel equivalent of friction factor for open channels just like in pipes. You can use this to determine the difference in elevation head or slope of water surface for a given flowrate.

$\Delta z = f \frac{L}{D_h} \frac{V^2}{2g}$

$\frac{\Delta z}{L}$ is the slope of the surface for a flat bottomed tank or tank placed horizontally. This slope will most likely be less than the slope of your tank (20 mm / 1 m) once you put in all the other numbers (and that's why I say the flow tends to follow the tank shape, although it could be the other way around for atypical parameters). Note that the friction factor is Reynolds number dependent but the slope is for sure more with more velocity. The influence of tilting your tank is to alter the local hydraulic diameter at each section, but since you drain from the bigger part of the tank, the tilt of your tank will counteract the previous slope.

Traditionally it is Darcy friction factor if it only accounts for wall shear stress from surface roughnes (i.e. Nikuradse's work). If you look in handbooks you can find friction factors (with lots of more name variants) which account for surface water effects.

September 20, 2022, 03:37	Flow under an angle	#1
Klavoir New Member Join Date: Sep 2021 Posts: 22 Rep Power: 5	Hi, I have a basic fluid dynamics problem, someone might know the answer right away. I have a horizontal tank placed under a small angle. At the lowest point water is pumped out from the bottom with a certain flow rate. On the higher side water the same amount is entering. So the velocity of the water through the tank is determined by by the flow rate of the pump. I wonder what will be the height of the water level measured to the bottom of the tank over the length of the tank. When the water is stationary the height at the lowest end of the tank will obviously be higher than at the highest point (the water surface is then horizontal). When the water is flowing will the height of the water layer over the length of the tank be dependent on the flow rate? I would think the more flow, the more the surface of the water will be parallel to the bottom of the tank?

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September 20, 2022, 14:23		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	The analysis can become rather cumbersome when you take everything into account and get complicated for a rather trivial result. There is a water level decrease even for a flat bottom'd tank. In 1D you can use a simple mechanical energy balance to determine the surface deflection for a given flow velocity (i.e. bernoulli). You'll quickly find you need very large velocities or large horizontal distances (i.e. km in the horizontal direction) to see meaningful differences in water level height. And then you can take into account 2D/3D effects if you want. And you can consider that the height of the air column above the water also needs to move with the water column. But yes, the more the velocity, the more it wants to follow the shape of the tank bottom.

September 21, 2022, 08:32		#3
Klavoir New Member Join Date: Sep 2021 Posts: 22 Rep Power: 5	Good to hear you agree with my gut feeling about it. To quantify this a little more. The tank is 5m long, the angle is such that the bottom of the tank has a height difference of 20 mm about the length. The velocity is 0.5 m/s. I guess with this velocity the difference in height water layer relative to the bottom of the tank at the front and the back of the tank is close to 20 mm?

September 21, 2022, 09:58		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	Clarification: I did not mean Bernoulli but rather to use the Darcy-Weisbach relation for open channel flows. There is an open channel equivalent of friction factor for open channels just like in pipes. You can use this to determine the difference in elevation head or slope of water surface for a given flowrate. $\Delta z = f \frac{L}{D_h} \frac{V^2}{2g}$ $\frac{\Delta z}{L}$ is the slope of the surface for a flat bottomed tank or tank placed horizontally. This slope will most likely be less than the slope of your tank (20 mm / 1 m) once you put in all the other numbers (and that's why I say the flow tends to follow the tank shape, although it could be the other way around for atypical parameters). Note that the friction factor is Reynolds number dependent but the slope is for sure more with more velocity. The influence of tilting your tank is to alter the local hydraulic diameter at each section, but since you drain from the bigger part of the tank, the tilt of your tank will counteract the previous slope. Traditionally it is Darcy friction factor if it only accounts for wall shear stress from surface roughnes (i.e. Nikuradse's work). If you look in handbooks you can find friction factors (with lots of more name variants) which account for surface water effects.