Mannings versus Moody's Equation 3

[CompEngr]

Hi,

I am looking over a Civil Engineer's calculations for pipe sizing. He uses the Rational Method to calculate rain fall. Then he uses the Manning Equation to size the 24" diameter pipe assuming the pipe is 94% full and a 1% slope.

As a mechanical engineer, I've sized smaller diameter (not 24" diameter) Schedule 40 or 60 stainless pipe using Bernoulli's law and the Moody chart. My understanding the Manning's equation is empirical equation for open channel flow that does not consider static head.

Why do Civil Engineers use Manning versus Bernoulli and the Moody chart? Does the Moody chart assume a pipe that is full while Civil Engineers are sizing larger size pipe where the pipe is not full? As such, the pipe could be treated as a open channel for larger diameters?

Thank you,
A mechanical engineer trying to figure a civil engineers calcs

 

[1503-44]

Bernouli still applies.

Manning is for open channel, uniform flow where there is a free loquid surface, hence there is no pressure differential available to drive the flow. Bernouli's DeltaP=0. The slope of the channel dictates the gain or loss of energy due to gravity, Bernouli's differential head Delta Z. Bernouli's Velocity head, V^2/2/g is balanced against Delta Z - channel friction f.

Delta Z per unit length of channel is the channel's slope S.

V^2/2/g = S -f

Keeping S constant, you see that there is a corresponding velocity which balances the energy gain, or loss due to slope of the channel minus the friction factor of the channel/liquid interface at that velocity.

So, pretty much the same except velocity is not a function of pressure differential, only slope and channel friction. Manning drops the pressure term.

The Manning friction factor is not based on full circular channel flow. You calculate the hydraulic radius, which varies with depth, which in turn is a function of velocity. Using full flowing circular pipes with a full wetted perimeter also allows "us" to eliminate some variables in the determination of "hydraulic radius" and depth is constant, so we kind of ignore that aspect, since that is already assumed in "our" friction factor calculation. If you had full flowing square pipes, you would have to calculate that differently and modify your usual friction factors.

 

[pierreick]

Hi,
Consider the link attached to support your work :

https://h2ometrics.com/manning-equation/

Good luck
Pierre

 

[1503-44]

Let's try that.

Q = 1.49/n A R^(2/3) S^(1/2)

V = 1.49/n R^(2/3) S^(1/2)
V^2/1.49^2 = (1/n)^2 * (R^(2/3))^2 * S
V^2/2.22 = n^0.5 * R^1.33 * S
V^2 = 2.22 * n^0.5 * R^1.33 * S
Divide by 2g
V^2 /2g= 2.22 / 2g * n^0.5 * R^1.33 * S
V^2/2g= 1.11 / g * n^0.5 * R^1.33 * S
g = 32.2
V^2/2g= 1.11 / 32.2 * n^0.5 * R^1.33 * S
V^2/2g = 0.035 * n^0.5 * R^1.33 * S
f = 0.035 * n^0.5 * R^1.33

V^2/2g = f * S Bernouli V^2/2g = S - F
Manning seems to have not entirely separated friction factor from slope, or
I made an error. I think I am correct, as in the Maning equation, Q is a function of slope, whereas Bernouli indicates it is function of the difference in friction and slope. That probably explains some complaints about Manning not being accurate at large slopes.

 

[MortenA]

@CompEngr: "Why do Civil Engineers use Manning versus Bernoulli and the Moody chart? Does the Moody chart assume a pipe that is full while Civil Engineers are sizing larger size pipe where the pipe is not full? As such, the pipe could be treated as a open channel for larger diameters?" - I think you answered this one yourself but yes:

Manning is for open channels
Moody is for full flowing pipes

If a civil engineer uses Manning for sizing full flowing (pressurized) pipes then hes wrong! But if you are sizing a pipe for draining rain fall he could be right.

Best regards, Morten

 

[jgailla]

As a civil engineer, Manning's is generally used for all storm piping and culverts. It's the custom. If a reviewer asked me for Moody's to get a pipe size, I would assume they were not qualified to review my work, or perhaps overqualified .
You already have a large margin of error using the Rational Method.
There's simply no reason to overcomplicate storm pipe sizing.
One mistake I frequently see is contractors using the slope of the pipe for S instead of the hydraulic grade line for submerged piping, but that is a different issue.

 

[bimr]

The Manning equation is widely used in the United States for open channel flow, such as pipes that are partly full.

There are many formulas for flow in pipes, but none is easier to use than the Hazen-Williams, and none is more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or the Colebrook equation.

The limitation of the accuracy of all pipe formulas lies in the estimation of the proper coefficient of friction, a value that cannot be physically measured and, hence, is subject to large error.

From Pump Station Design:

The first well-known formula for flow in pipes was proposed by deChezy. The deChezy friction coefficient was given by a complicated equation developed by Kutter. These formulas are no longer in common use. The Hazen-Williams formula has been used in the United States for 90 years. It is simple and easy to use, it has been verified by many field observations for common sizes of pipes at conventional flow rates. It has, however, some serious limitations.

The Manning equation is somewhat similar to the H-W formula and is subject to the same limitations. It is widely used in the United States for open channel flow, such as pipes that are partly full. Sometimes it is used for full pipes, but for that application it has no advantage over the Hazen-Williams formula.

The Colebrook-White equation is more accurate than the H-W formula and is applicable to a wider range of flow, pipe size, and temperature. It is widely used in the United Kingdom and elsewhere in Europe.

The Darcy-Weisbach equation is the only rational formula, and it is applicable to turbulent, laminar, or transitional flow, all sizes of pipe, and any incompressible Newtonian fluid at any temperature. It has not been popular because, being an implicit equation, it must be solved by successive trials. It was therefore inconvenient to use, but now, modern computers (even programmable pocket calculators) can be used to solve the equation in seconds.

The Hazen-Williams equation, developed from extensive reviews of data on pipes installed all over the world, was made public in 1905. The appeal of the equation is due partly to its simplicity and ease of use, partly to the source of the data (real pipes in the field—not just laboratory pipes), and, by now, to a tradition of nearly a century of use and a blind faith in the results. Unfortunately, the formula is irrational; it is valid only for water at or near room temperature and flowing at conventional velocities; the flow regime must be in the transition zone; the C factor varies with pipe size; and 92% of the pipes studied were smaller than 1500 mm (90 in.) in diameter. These disadvantages seem to be generally ignored, but errors can be appreciable (up to 40%) for pipes less than 200 mm (8 in.) and larger than, say, 1500 mm (60 in.), for very cold or hot water, and for unusually high or low velocities.

 

[fel3]

MortenA...

Mannings can certainly be used for full-flowing, pressurized pipe. It is commonly used (and I have used it) for pipeline hydraulics for large-diameter storm drainage and irrigation pipes flowing full. Hazen-Williams is usually reserved for smaller diameter, smooth-wall pipe, such as are used in water distribution systems.

I have attached a comparison of Darcy-Weisbach, Hazen-Williams, and Mannings. This .zip file includes an SMathStudio worksheet and a .pdf file of the worksheet. My main purposes for creating this worksheet were to investigate the error bars associated with each equation, provide a means to convert the friction factors from one equation to another, and provide worksheet elements related to each equation for use in other worksheets.

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bimr...

It's not Darcy-Weisbach that is implicit. Instead it's Colebrook-White that is implicit, but just for the friction factor. And, it's not Colebrook-White that is more accurate than Hazen-Williams. Instead it's Darcy-Weisbach that is more accurate than Hazen-Williams. Frictions factors are always empirical, whether estimated according to Colebrook-White or back-calculated using Hazen-Williams. Regardless, because Darcy-Weisbach is rational and Colebrook-White matches laboratory results pretty well (but certainly not exactly), Darcy-Weisbach will always be the "gold standard."

Fortunately, there are many good to excellent explicit approximations to Colebrook-White that can be used instead being forced to iterate or to work with the Moody Diagram. I analyzed about two dozen of these explicit approximations back in 2014 using Mathcad Prime 3.0 and posted the worksheets and .pdf files of them on the PTC Mathcad forums. You can find my analysis here:

https://community.ptc.com/t5/PTC-Mathcad/Lusk-Darcy-Friction-Factors-v2-zip/m-p/450068#M175864

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"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

 

[1503-44]

Using Manning for full pipe pressurized flow would only need the addition of pressure heads and calculating R using the full circumference.

 

[bimr]

It's not Darcy-Weisbach that is implicit. Instead it's Colebrook-White that is implicit, but just for the friction factor. And, it's not Colebrook-White that is more accurate than Hazen-Williams. Instead it's Darcy-Weisbach that is more accurate than Hazen-Williams. Frictions factors are always empirical, whether estimated according to Colebrook-White or back-calculated using Hazen-Williams. Regardless, because Darcy-Weisbach is rational and Colebrook-White matches laboratory results pretty well (but certainly not exactly), Darcy-Weisbach will always be the "gold standard."

Fortunately, there are many good to excellent explicit approximations to Colebrook-White that can be used instead being forced to iterate or to work with the Moody Diagram. I analyzed about two dozen of these explicit approximations back in 2014 using Mathcad Prime 3.0 and posted the worksheets and .pdf files of them on the PTC Mathcad forums. You can find my analysis here:

https://community.ptc.com/t5/PTC-Mathcad/Lusk-Darc...

If you have implicit criticism of Pump Station Design, then you should contact the book editor's for the fourth edition.

However, there are plenty of other references that agree with Pump Station Design's statement "The Darcy-Weisbach equation is the only rational formula, and it is applicable to turbulent, laminar, or transitional flow, all sizes of pipe, and any incompressible Newtonian fluid at any temperature. It has not been popular because, being an implicit equation, it must be solved by successive trials.

For example:

Link

The point that I was making is that all of these equations have underlying assumptions and it is best not to use them blindly. Pump Station Design did state that the "Darcy-Weisbach equation is the only "rational formula" which is probably more accurate than being described as a "gold standard".

 

[1503-44]

Not that it matters. I've never seen any piping system design that I could say was completed within a hydraulic tolerance of error less than the difference between any popular liquid pipe flow equation. In fact probably by an order of magnitude ... or two.

 

[fel3]

bimr...

The pumping station design book is clearly in error here, which I will explain below. I have the second edition, so I don't know if they cleared this up later.

A few key statements from Pumping Station Design, Second Edition (Sanks, et al; 1998), pp. 36-37:

“Unfortunately, the formula [Hazen-Williams] is irrational.”
This statement is correct. H-W is also empirical, has limited applicability, and is explicit for all variables.

“The Colebrook-White equation is more accurate than the H-W [Hazen-Williams) formula…”
This statement is obviously incorrect because C-W and H-W do not solve the same things and therefore cannot be compared this way. This statement may be a backhanded way to claim (correctly) that friction factors calculated via C-W more accurately reflect laboratory test data than does the C factor used in H-W. Regardless, this is a terribly muddled explanation. Also, C-W is empirical, is implicit for f, and is explicit for the relative roughness and the Reynolds Number.

“The Darcy-Weisbach equation is the only rational formula…”
This statement is correct.

“It [Darcy-Weisbach] has not been popular because, being an implicit equation, it must be solved by successive trials.”
This statement is obvious incorrect because D-W is explicit for all variables. However, D-W appears implicit in practice when friction factors are estimated using the implicit C-W equation. Again, a terribly muddled explanation.

“An explicit, empirical equation for f was developed by Swamee and Jain…the value of f…differs from f calculated from the Colebrook equation by less than 1%.”
I commend the authors for providing an easier way to estimate the friction factor. However, my calculations show that Swamee and Jain deviates from C-W by as much as 3.4%. Not only that, but Jain published a modified version of S-J at the same time that is slightly better. Also, when this book was published (1998), there were at least 11 published explicit approximations for f that were as good as or better than Swamee and Jain’s equation in approximating C-W and some of these were of a similar level of complexity, including Churchill #1 (1973), Churchill #2 (1977), Jain (1976), Manadilli (1997), Haaland (1983), Zigrang & Sylvester #1 (1982), Chen (1979), Serghides #1 (1984), Monzon Romeo & Royo (1997), Zigrang & Sylvester #2 (1982), and Serghides #2 (1984).

BTW, I used to work with one of the authors. I will attempt to track him down and see if he is interested.


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"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

 

[bimr]

There doesn't appear to be any significant difference between the 2nd edition or the 3rd edition (August 6, 2008) of Pumping Station Design in regard to friction factors.

 

[fel3]

bimr...

Thanks for the pages form the 3rd edition. At first glance I didn't see any difference.

============
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill