Manningcirc: Circular cross-section using the Gauckler-Manning-Strickler equation

Description

Manningcirc and Manningcircy solve for a missing variable for a circular cross-section. The uniroot function is used to obtain the missing parameter.

Usage

Manningcirc(
  Q = NULL,
  n = NULL,
  Sf = NULL,
  y = NULL,
  d = NULL,
  T = NULL,
  units = c("SI", "Eng")
)
Manningcircy(
  y = NULL,
  d = NULL,
  y_d = NULL,
  theta = NULL,
  Sf = NULL,
  Q = NULL,
  units = c("SI", "Eng")
)

Arguments

numeric vector that contains the discharge value [m^3/s or ft^3/s], if known.

numeric vector that contains the Manning's roughness coefficient n, if known.

numeric vector that contains the bed slope (m/m or ft/ft), if known.

numeric vector that contains the flow depth (m or ft), if known.

numeric vector that contains the diameter value (m or ft), if known.

numeric vector that contains the temperature (degrees C or degrees Fahrenheit), if known.

units

character vector that contains the system of units [options are SI for International System of Units and Eng for English units (United States Customary System in the United States and Imperial Units in the United Kingdom)]

y_d

numeric vector that contains the filling ration (y/d), if known.

theta

numeric vector that contains the angle theta (radians), if known.

Value

the missing parameter (Q, n, or Sf) & theta, area (A), wetted perimeter (P), velocity (V), top width (B), hydraulic radius (R), Reynolds number (Re), and Froude number (Fr) as a list for the Manningcirc function.

the missing parameter (d or y) & theta, area (A), wetted perimeter (P), top width (B), velocity (V), and hydraulic radius (R) as a list for the Manningcircy function.

Details

The Manningcirc function solves for one missing variable in the Gauckler- Manning equation for a circular cross-section and uniform flow. The possible inputs are Q, n, Sf, y, and d. If y or d are not initially known, then Manningcircy can solve for y or d to use as input in the Manningcirc function.

The Manningcircy function solves for one missing variable in the Gauckler- Manning equation for a circular cross-section and uniform flow. The possible inputs are y, d, y_d (ratio of y/d), and theta.

Gauckler-Manning-Strickler equation is expressed as

$$V = \frac{K_n}{n}R^\frac{2}{3}\sqrt{S}$$

V: the velocity (m/s or ft/s)
n: Manning's roughness coefficient (dimensionless)
R: the hydraulic radius (m or ft)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

This equation is also expressed as

$$Q = \frac{K_n}{n}\frac{A^\frac{5}{3}}{P^\frac{2}{3}}\sqrt{S}$$

Q: the discharge [m^3/s or ft^3/s (cfs)] is VA
n: Manning's roughness coefficient (dimensionless)
P: the wetted perimeter of the channel (m or ft)
A: the cross-sectional area (m^2 or ft^2)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

Other important equations regarding the circular cross-section follow: $$R = \frac{A}{P}$$

R: the hydraulic radius (m or ft)
A: the cross-sectional area (m^2 or ft^2)
P: the wetted perimeter of the channel (m or ft)

$$A = \left(\theta - \sin \theta\right) \frac{d^2}{8}$$

A: the cross-sectional area (m^2 or ft^2)
d: the diameter of the cross-section (m or ft)
$\theta$: see the equation defining this parameter

$$\theta = 2 \arcsin\left[1 - 2\left(\frac{y}{d}\right)\right]$$

$\theta$: see the equation defining this parameter
y: the flow depth (normal depth in this function) [m or ft]
d: the diameter of the cross-section (m or ft)

$$d = 1.56 \left[\frac{nQ}{K_n\sqrt{S}}\right]^\frac{3}{8}$$

d: the initial diameter of the cross-section [m or ft]
Q: the discharge [m^3/s or ft^3/s (cfs)] is VA
n: Manning's roughness coefficient (dimensionless)
S: the slope of the channel bed (m/m or ft/ft)
$K_n$: the conversion constant -- 1.0 for SI and 3.2808399 ^ (1 / 3) for English units -- m^(1/3)/s or ft^(1/3)/s

Note: This will only provide the initial conduit diameter, check the design considerations to determine your next steps.

$$P = \frac{\theta d}{2}$$

P: the wetted perimeter of the channel (m or ft)
$\theta$: see the equation defining this parameter
d: the diameter of the cross-section (m or ft)

$$B = d \sin\left(\frac{\theta}{2}\right)$$

B: the top width of the channel (m or ft)
$\theta$: see the equation defining this parameter
d: the diameter of the cross-section (m or ft)

$$D = \frac{A}{B}$$

D: the hydraulic depth (m or ft)
A: the cross-sectional area (m^2 or ft^2)
B: the top width of the channel (m or ft)

A rough turbulent zone check is performed on the water flowing in the channel using the Reynolds number (Re). The Re equation follows:

$$Re = \frac{\rho RV}{\mu}$$

Re: Reynolds number (dimensionless)
$\rho$: density (kg/m^3 or slug/ft^3)
R: the hydraulic radius (m or ft)
V: the velocity (m/s or ft/s)
$\mu$: dynamic viscosity (* 10^-3 kg/m*s or * 10^-5 lb*s/ft^2)

A critical flow check is performed on the water flowing in the channel using the Froude number (Fr). The Fr equation follows:

$$Fr = \frac{V}{\left(\sqrt{g * D}\right)}$$

Fr: the Froude number (dimensionless)
V: the velocity (m/s or ft/s)
g: gravitational acceleration (m/s^2 or ft/sec^2)
D: the hydraulic depth (m or ft)

References

Terry W. Sturm, Open Channel Hydraulics, 2nd Edition, New York City, New York: The McGraw-Hill Companies, Inc., 2010, page 2, 8, 36, 102, 120, 123-125, 153-154.
Dan Moore, P.E., NRCS Water Quality and Quantity Technology Development Team, Portland Oregon, "Using Mannings Equation with Natural Streams", August 2011, http://www.wcc.nrcs.usda.gov/ftpref/wntsc/H&H/xsec/manningsNaturally.pdf.
Gilberto E. Urroz, Utah State University Civil and Environmental Engineering - OCW, CEE6510 - Numerical Methods in Civil Engineering, Spring 2006 (2006). Course 3. "Solving selected equations and systems of equations in hydraulics using Matlab", August/September 2004, https://digitalcommons.usu.edu/ocw_cee/3.
Tyler G. Hicks, P.E., Civil Engineering Formulas: Pocket Guide, 2nd Edition, New York City, New York: The McGraw-Hill Companies, Inc., 2002, page 423, 425.
Wikimedia Foundation, Inc. Wikipedia, 26 November 2015, <U+201C>Manning formula<U+201D>, https://en.wikipedia.org/wiki/Manning_formula.
John C. Crittenden, R. Rhodes Trussell, David W. Hand, Kerry J. Howe, George Tchobanoglous, MWH's Water Treatment: Principles and Design, Third Edition, Hoboken, New Jersey: John Wiley & Sons, Inc., 2012, page 1861-1862.
Andrew Chadwick, John Morfett and Martin Borthwick, Hydraulics in Civil and Environmental Engineering, Fourth Edition, New York City, New York: Spon Press, Inc., 2004, page 133.
Robert L. Mott and Joseph A. Untener, Applied Fluid Mechanics, Seventh Edition, New York City, New York: Pearson, 2015, page 376, 377-378, 392.
Wikimedia Foundation, Inc. Wikipedia, 17 March 2017, <U+201C>Gravitational acceleration<U+201D>, https://en.wikipedia.org/wiki/Gravitational_acceleration.
Wikimedia Foundation, Inc. Wikipedia, 29 May 2016, <U+201C>Conversion of units<U+201D>, https://en.wikipedia.org/wiki/Conversion_of_units.

Examples

Run this code

# NOT RUN {
library("iemisc")
library(iemiscdata)

# Practice Problem 14.12 from Mott (page 392)
y <- Manningcircy(y_d = 0.5, d = 6, units = "Eng")

# See npartfull in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Corrugated Metal, 2) Stormdrain.

data(npartfull)

# We are using the culvert as a stormdrain in this problem
nlocation <- grep("Stormdrain",
npartfull$"Type of Conduit and Description")

n <- npartfull[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(d = 6, Sf = 1 / 500, n = n, y = y$y, units = "Eng")
# d = 6 ft, Sf = 1 / 500 ft/ft, n = 0.024, y = 3 ft, units = "Eng"
# This will solve for Q since it is missing and Q will be in ft^3/s



# Example Problem 14.2 from Mott (page 377-378)
y <- Manningcircy(y_d = 0.5, d = 200/1000, units = "SI")

# See npartfull in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Clay, 2) Common drainage tile.

data(npartfull)

nlocation <- grep("Common drainage tile",
npartfull$"Type of Conduit and Description")

n <- npartfull[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(Sf = 1/1000, n = n, y = y$y, d = 200/1000, units = "SI")
# Sf = 1/1000 m/m, n = 0.013, y = 0.1 m, d = 200/1000 m, units = SI units
# This will solve for Q since it is missing and Q will be in m^3/s



# Example 4.1 from Sturm (page 124-125)
Manningcircy(y_d = 0.8, d = 2, units = "Eng")

y <- Manningcircy(y_d = 0.8, d = 2, units = "Eng")
# defines all list values within the object named y

y$y # gives the value of y



# Modified Exercise 4.1 from Sturm (page 153)
# Note: The Q in Exercise 4.1 is actually found using the Chezy equation,
# this is a modification of that problem
# See nchannel in iemiscdata for the Manning's n table that the
# following example uses
# Use the normal Manning's n value for 1) Natural streams - minor streams
# (top width at floodstage < 100 ft), 2) Mountain streams, no vegetation
# in channel, banks usually steep, trees and brush along banks submerged at
# high stages and 3) bottom: gravels, cobbles, and few boulders.

data(nchannel)

nlocation <- grep("bottom: gravels, cobbles, and few boulders",
nchannel$"Type of Channel and Description")

n <- nchannel[nlocation, 3] # 3 for column 3 - Normal n

Manningcirc(Sf = 0.002, n = n, y = y$y, d = 2, units = "Eng")
# Sf = 0.002 ft/ft, n = 0.04, y = 1.6 ft, d = 2 ft, units = English units
# This will solve for Q since it is missing and Q will be in ft^3/s



# Modified Exercise 4.5 from Sturm (page 154)
library(NISTunits)

ysi <- NISTftTOmeter(y$y)

dsi <- NISTftTOmeter(2)

Manningcirc(Sf = 0.022, n = 0.023, y = ysi, d = dsi, units = "SI")
# Sf = 0.022 m/m, n = 0.023, y = 0.48768 m, d = 0.6096 m, units = SI units
# This will solve for Q since it is missing and Q will be in m^3/s


# }

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