Hydraulic pressure/flow -- friction loss question

[magneticanomaly]

Flow delivered by a pipe can be calculated from input pressure, inlet geometry, pipe length, and a flow coefficient related to pipe cross-section, shape, and roughness. If we create an example of a pipe attached at the bottom of a self-refilling standpipe, which automatically supplies a constant inlet pressure related to its height, let the height of the standpipe be 100 ft, and the length of the horizontal pipe be 1000 feet.

Inlet pressure will be 100 feet of water column. Static pressure along the pipe will vary, linearly I suppose, from 100 ft w/c at inlet to zero at outlet. Delivery will be x gallons per minute.

How will the delivery rate be different if the 1000 feet of pipe is arranged on a 10% grade, so that there is 100 ft w/c of pressure head available over the length of the pipe, but inlet pressure is zero, or negligible, because the pipe inlet is at the level of the surface of the reservoir feeding it?

Thanks!

 

[Chestermiller]

Are you able to perform a force balance in the direction along the pipe over a differential section of pipe length for this case? What do you get, taking into account the gravitational component of force along the length?

 

[magneticanomaly]

Chestermiller, your suggestion sounds like the right approach, but I am only familiar with the simplistic empirical equation I describe. Intuitively, since in the first example the force (pressure) available to push the water through the pipe is maximum at the inlet and zero at outlet, while in the second the force is zero at both ends and maximum (but half the initial pressure in the first example) in the middle, I would think the flow in the second condition is 25% that in the first.

Can you suggest how to set up the force balance equations?

 

[BvU]

Is this a homework assignment ? Could you provide the complete problem description ?

 

[Chestermiller]

The axial force balance on the tilted pipe will look like this: $$\frac{\pi D^2}{4}L\rho g \sin{\beta}=\pi DL\tau_w$$ or $$\tau_w=\rho g \sin{\beta}\frac{D}{4}$$where $\tau_w$ is the shear stress at the wall of the pipe.

For the horizontal pipe, the axial force balance will be $$\frac{\pi D^2}{4}\rho g h=\pi D L \tau_w$$ or $$\tau_w=\rho g \frac{h}{L}\frac{D}{4}$$
But, in the two cases, as you described it, $$\sin{\beta}=\frac{h}{L}$$

What does this tell you about the shear stresses at the wall and the flow rates?

 

[magneticanomaly]

Not a homework assignment, relevant to designing springwater delivery and hydropower systems.

Thanks very much , Chester, for laying out the equations. I assume D is pipe inside diameter, g is acceleration of gravity, L length of pipe. I assume beta is incline angle of pipe..do you mean angle above horizontal or below vertical? Is rho, viscosity? What units do you recommend?

I suppose shear stress is proportional to flow rate.

Grinding my doubtful way through your formulae, it appears to me that none of my questions above need answers, and since sin beta = h/L, the shear stress and thus the flow are the same in both situations. This makes sense to me from energetic considerations, since the same potential energy is dissipated over the same length of pipe with the same unit resistance in both cases.

THANKS!

 

[Chestermiller]

Not a homework assignment, relevant to designing springwater delivery and hydropower systems.

Thanks very much , Chester, for laying out the equations. I assume D is pipe inside diameter, g is acceleration of gravity, L length of pipe. I assume beta is incline angle of pipe..do you mean angle above horizontal or below vertical? Is rho, viscosity? What units do you recommend?

beta is the angle of downward tilt relative to the horizontal. rho is the density. The choice of units is at your discretion. I would use imperial units myself.

I suppose shear stress is proportional to flow rate.

Yes, for a Newtonian fluid in laminar flow. Otherwise it is just a function of flow rate.

Grinding my doubtful way through your formulae, it appears to me that none of my questions above need answers, and since sin beta = h/L, the shear stress and thus the flow are the same in both situations. This makes sense to me from energetic considerations, since the same potential energy is dissipated over the same length of pipe with the same unit resistance in both cases.

THANKS!

The first formula says that the axial component of the weight of the fluid is balanced by the shear force along the wall of the tilted pipe. The second formula says that the force of the pressure difference between the ends of the pipe is balanced by the shear force among the wall of the horizontal pipe.