Help with question: An open channel energy loss problem.

[ptatohed]

Water Resource guys and gals:

I am helping my friend study for the 8hr PE exam. We have a question about the following problem:

An open channel has normal flow with characteristics S = 0.003 , n = 0.013 , V = 10 ft/sec and R = 2.25ft . What is the energy loss per 1000 ft ?



a. 5.0 ft

b. 2.0 ft

c. 3.0 ft

d. 2.5 ft



In the CERM (sorry, I only have an old 10th edition in front of me) and in the All-In-One (not in front of me), the authors equate

hf = LS = Ln2v2/2.208R4/3

CERM Eqns 19.29 and 19.30(b) in my 10th edition.

hf = head/energy loss, ft. L = Length of channel, ft. S = Channel/Water surface slope, decimal. n = Manning coefficient, unitless. v = flow velocity, ft/s. R = Hydraulic radius, ft.

CERM Example 19.4 of the 10th Ed is a near-exact problem to that above, with only some of the numbers different. The CERM works out the problem using both equations (hf = LS and hf = Ln2v2/2.208R4/3 ). Of course, they get the same answer using either equation.

So, here's my question. When I solve the above problem, I get two different answers with the two different hf equations.

hf = LS = 3.0 ft

hf = Ln2v2/2.208R4/3 = 2.6 ft

If the two energy loss equations are equal, why do I get two different answers?

For the record, the solution as provided by the source is hf = LS = 3.0 ft, Answer C.

Thanks!

 

[IlPadrino]

Off the top of my head... two things come to mind.

1) S=n2v2/2.208R4/3 is just Manning's equation solved in terms of V rather than the normal Q and A. [Mannings in normally used as Q=(1.49/n)A2/3S1/2]

2) Applying the energy equation,


with P1=P2, V1=V2, and no pump yields hL=Z1-Z2, which is the same as hL=LS

Does this help?

 

[ptatohed]

Thanks IlP. This does help show me how the two equations are equated to each other. I'm still confused why I get two different answers when using the two different equations though.

 

[bradlelf]

I would have also given the answer as 2.6 ft.

There seems to be an error in the problem. Utilizing the mannings velocity equation V=1.49/n*R^(2/3)*S^(1/2) the velocity would come out to 7.23 fps not 10 fps.

 

[IlPadrino]

^ There you have it... With the values given, S != n2v2/2.208R4/3. Had they given S=0.0026 or n=0.014, the two methods would give you the same answer.

Still, 2.6 is the *WRONG* answer because Manning's equation is empirical, not theoretical. The energy equation must be obeyed!

 

[ptatohed]

Very good, thanks so much guys.

P.S. Brad, I get V = 10.78 fps using Manning's eqn (not 7.23 fps).

 

[bradlelf]

Very good, thanks so much guys.

P.S. Brad, I get V = 10.78 fps using Manning's eqn (not 7.23 fps).

You are right ... i used 1/n instead of 1.49/n ... good catch

 

[ptatohed]

Because this problem is so close to the problem in the CERM and comes from a questionable site, I suspect they simply took the CERM problem, changed some values and called it good. They did not check to see if their changed values actually work with the Manning's eqn. Thanks again guys.

 

[IlPadrino]

I'd use this problem as an opportunity to emphasize the importance of understanding the energy relationships for open channel flow, including the energy grade line and the hyrdaulic grade line.