Engineering assignment

( 1 )Applied Root Solving

Water Resources Engineering (CEE 333)

In water resources engineering, you will study the principles of water movement and management focusing on hydrology, water distribution, storm water management, and waste water collection. One application that will be studied is the flow of water in an open channel. Once the flow in an open channel is computed, it can be used to determine the proper size of the drainage ditches on the side of highways. Water runs into the ditch and flows in the channel. For a given volume of water flowing per unit of time (flowrate), the height of the water can be determined using Manning’s equation. The height of the water in the channel during steady state conditions is called the normal depth . Steady state is a condition in which none of the parameters of the problem are changing, e.g. flow rate, roughness of the channel, slope of the channel, etc.

Finding the normal depth in an open channel can be accomplished by using Manning’s equation . Manning’s equation relates the flow rate in the channel (discharge, Q) to the cross-sectional area (A), the hydraulic radius (R), the slope of the channel (So), and the channel roughness (N).

Eq. 1

where: Q – Channel Discharge (Cfs)

A – Channel Area (Ft2)

R – Hydraulic Radius (Ft)

So – Longitudinal Channel Slope (Ft/Ft)

N – Manning’s Roughness Value (see Figure 3) P – Wetted Perimeter (ft)

Eq. 1 can be turned into a root-solving problem and solved using root-solving techniques such as the Bisection Method, Newton-Raphson Method, etc. Depending upon how the problem is rewritten, you may introduce unwanted sensitivity problems. Use Eq. 2 for your root-solving problem to minimize numerical sensitivity problems.

Eq. 2

Trapezoidal Channel

Consider the trapezoidal channel shown in Figure 1 below. Note the bottom width of the channel (B) and the channel side-slopes (Z:1). The channel side-slope is not to be confused with the longitudinal slope of the channel (So). If the bottom width of the channel and the side slopes of the channel are known, then the cross-sectional area of flow and the wetted perimeter can be easily calculated using the equations below. If the left and right side slopes are identical, Eq. 4can be used; however, if the side slopes differ, Eq. 5 should be used for the computation of the area, wetted perimeter, and the hydraulic radius.

Eq. 3

		Eq. 4a
		Eq. 4b

		Eq. 5a
		Eq. 5b

Figure 1

VBA Application

Write a VBA program that computes the normal depth, y, using the Bisection Method for user supplied inputs for Q, B, ZL, ZR, So, and N. Also accept input for the starting lower and upper bounds, the maximum number of iterations, and the stopping criteria for the approximate relative error (e.g. 0.01%).

Example Input: Q = 100; B = 10; ZL = 2: ZR = 4; So = 0.01; N = 0.05; Imax = 100; es = 0.01; XL = 0; XU = 10

Example Solution: 1.819 ft

Circular Channel

The geometry of the circular channel shown in Figure 2 is controlled solely by the radius. The cross sectional area of the water and the wetted perimeter are governed by the depth of water in the channel. The area of water in circular pipe with radius, r, filled to a height, h, is

Eq. 6

The wetted perimeter, defined as the length of the channel that is wetted, is

Eq. 7

Figure 2

VBA Application

Write a VBA program that computes the normal depth, y, using the Bisection Method for user supplied inputs for Q, So, N, and r. Also accept input for the starting lower (XL) and upper bounds (XU), the maximum number of iterations (Imax), and the stopping criteria (es) for the approximate relative error (e.g. 0.01%).

Example Input: Q = 10; So = 0.01; N = 0.05; radius=1.5; Imax = 100; es = 0.01; XL = 0; XU = 1.92

Example Solution: 1.6315ft

Manning’s Roughness Coefficients

Reference: http://www.engineeringtoolbox.com/mannings-roughness-d_799.html

Surface Material	Manning's Roughness
	Coefficient
	- n -
Asbestos cement	0.011
Asphalt	0.016
Brass	0.011
Brick	0.015
Canvas	0.012
Cast-iron, new	0.012
Clay tile	0.014
Concrete - steel forms	0.011
Concrete (Cement) - finished	0.012
Concrete - wooden forms	0.015
Concrete - centrifugally spun	0.013
Copper	0.011
Corrugated metal	0.022
Earth, smooth	0.018
Earth channel - clean	0.022
Earth channel - gravelly	0.025
Earth channel - weedy	0.03
Earth channel - stony, cobbles	0.035
Floodplains - pasture, farmland	0.035
Floodplains - light brush	0.05
Floodplains - heavy brush	0.075
Floodplains - trees	0.15
Galvanized iron	0.016
Glass	0.01
Gravel, firm	0.023
Lead	0.011
Masonry	0.025
Metal - corrugated	0.022
Natural streams - clean and straight	0.03
Natural streams - major rivers	0.035
Natural streams - sluggish with deep pools	0.04
Natural channels, very poor condition	0.06
Plastic	0.009
Polyethylene PE - Corrugated with smooth inner walls	0.009 - 0.015
Polyethylene PE - Corrugated with corrugated inner walls	0.018 - 0.025
Polyvinyl Chloride PVC - with smooth inner walls	0.009 - 0.011
Rubble Masonry	0.017
Steel - Coal-tar enamel	0.01
Steel - smooth	0.012
Steel - New unlined	0.011
Steel - Riveted	0.019
Vitrified Sewer	0.013 - 0.015
Wood - planed	0.012
Wood - unplaned	0.013
Wood stove pipe, small diameter	0.011 - 0.012
Wood stove pipe, large diameter	0.012 - 0.013

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