1 There are various techniques for estimating discharge for small watersheds. If you know the maximum discharge that you need to convey, how do you determine the size of the culvert that will carry that discharge, e.g. under a road? Prepared for SSAC by Paul Butler The Evergreen State College, Olympia, WA © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved Quantitative Skills and Concepts Forward modeling Inverse problem Rearranging equations Iterative solutions Power functions Graphs, XY scatter plots SSAC2005.TC401.PB1.1 The Manning Equation Finding the size of a culvert to carry a specified discharge

2 Slides 3-5 introduce the Manning equation and allow you to calculate discharge based on culvert diameter, type of material used, and slope. This is forward modeling. Slides 6-8 allow you to calculate culvert diameter, given a specific discharge value, and to explore the relationships between variables. This is a backwards calculation, the opposite of forward modeling. It is the inverse problem. Slides give the assignment to hand in. Overview of Module Culverts are designed to carry a specified discharge and are based on estimates of runoff. The runoff estimates, in turn, are often based on the magnitude of precipitation events e.g., the 50-year or 100-year flood; basin characteristics, e.g., drainage area, topography; and land use. With the estimates of runoff, engineers can determine an appropriate culvert size. Their goal is to select a diameter that will be just large enough to carry the required discharge. If the culvert is too small, flooding is possible. If it is too large, money is wasted.

3 Variables v = velocity R = hydraulic radius s = slope n = Manning roughness coefficient Hydraulic Radius: The hydraulic radius for flow in a pipe or open channel is defined as the cross-sectional area divided by the wetted perimeter. It is not directly measurable, but it is used frequently in calculations. For pipes or culverts, the cross-sectional area is equal to the area of a circle (  r 2 ), while the wetted perimeter is equal to the circumference (2  r). Therefore, for full pipes, hydraulic radius R = r/2. Large pipes or culverts are sized by their inside dimension. The pipes are commonly available with diameters that increase by ½-foot increments. The Manning equation was derived empirically. It has a long history using English units. As the relevant data and culvert sizes are commonly collected in those units, we will use them here. For metric applications, 1.49 is replaced by 1. The Manning Equation

4 Discharge (Q) versus Velocity (v): Discharge is volume per unit of time (cubic feet per second in English units). Volume has dimensions of L 3, and so discharge has dimensions of L 3 / T. Velocity is distance per unit of time (e.g., feet per second), so velocity has dimensions of L / T. The velocity (v) times the cross-sectional area (A) equals the discharge (Q), i.e., Q = v A. Slope (s): The slope or gradient for a pipe or culvert is commonly set at the same angle as the natural stream channel. As the gradient is defined as the amount of elevation loss over some distance, it is dimensionless, i.e., L / L = 1, as long as both numerator and denominator are in the same units. Manning Roughness Coefficient (n): The roughness variable was initially derived empirically. In the original experiments, velocity, hydraulic radius, and slope were measured, and then roughness (n) was calculated. An increase in n indicates an increase in resistance to flow. In order to make the equation dimensionally correct, the dimensions of n are T L -1/3, but are seldom used. As a side exercise, you might figure out what 1.49 sec- ft -1/3 is in sec-m -1/3. Combining the formulas for Q and v : The Manning Equation

5 Recreate the spreadsheet below to answer the following question: What is the discharge for a culvert that is 2.5 feet in diameter, made of cast iron (n = 0.015), and set at a slope of 0.01? The Problem Given values. Cell with a number in it. Change one of these numbers and other numbers will change. Answers. Cell with equation in it. If you are given the culvert diameter, the culvert material, and the slope (gradient) of the culvert, you can use the Manning equation to determine velocity and then discharge.

6 Although this strategy works when the culvert is full, it does not work (the equation is not correct) when the culvert is not full. Why not? In general, the Manning equation cannot be inverted when it is applied to open-channel flow. Why not? For this exercise, we will use trial and error for a full pipe to illustrate the technique and to explore the relationships between the variables. Selecting the appropriate culvert diameter Now you are ready to determine how to select the appropriate culvert diameter given the discharge that it needs to carry. This means turning the calculation around, or solving the inverse problem. There are several possible approaches to this task. For a simple geometry such as a full, circular pipe one can invert the equation algebraically: substitute  r 2 / 2  r for R, and then solve for r. How would that look? To start we eliminate the R and find Q as a function of r, s, and n. Algebra Note: To solve this equation for r, you need to simplify it by combining the r’s. You will get an expression with r 8/3. That exponent shows up in the equation for the relationship between discharge and culvert diameter on Slide 8.

7 How large should a culvert be to handle a discharge of 12 ft 3 /s? Trial and error: We can answer the question by setting up a spreadsheet, trying a variety of culvert sizes that are commonly available, and checking the discharge against the desired amount.. Recreate this spreadsheet. Closest value that will handle the design flow. Remember, culverts come in set sizes. Select first size that is greater than given Q. Design discharge Selecting the appropriate culvert diameter

8 Recreate the graph below by plotting discharge versus culvert diameter (from slide 7) using the XY (scatter plot) option. Be sure to add the trendline and trendline equation into the graph. Note the coefficient – look familiar? What is the relationship between discharge and culvert diameter? Because the exponent (2.6667) is larger than 1, you know that the discharge increases at a faster rate than the culvert diameter.

9 1. What is the relationship between velocity and culvert diameter? Using the data that you reproduced in your spreadsheet to determine the culvert diameter needed to handle 12 cubic feet per second (Slide 7), plot the relationship between velocity and pipe diameter. Does velocity have the same relationship with culvert diameter as discharge? Explain your answer. 2. Look at the combined equation on the bottom of slide 4. What happens to discharge when the slope of the culvert is increased? What happens to discharge when the roughness coefficient (n) is increased? Begin by answering qualitatively. Now explore these relationships using your spreadsheet. Which has a bigger effect on Q: doubling the slope from to 0.01, or doubling the roughness coefficient from to 0.022? Express your answer in absolute terms, and as a proportion. When doubling slope, keep n = When doubling roughness, keep s = Use a culvert diameter of 2.0 feet. Compare these results to your qualitative answers. End of Module Assignment

10 3. You’ve been asked by your boss to install a culvert to handle 25 cfs. She tells you that the stream gradient is 0.002, and the pipe available is ordinary concrete (n = 0.013). How big should the culvert be to handle this flow? Use trial and error, with 0.5-foot increments of change in pipe diameter. 4. Just as you and your crew are loading up to head for the job site, your cell phone rings, and the supply office tells you that all the concrete culvert was used on another job. All they have left is corrugated metal culvert with a diameter of 2.5 feet (n = 0.022). At what slope should you install the 2.5-foot corrugated metal culvert so that it will handle the discharge of 25 cfs? Again, you can use trial and error. End of Module Assignment