Partially Full Pipe Flow Calculations Using Excel Spreadsheets 7-18-14

Partially Full Pipe Flow Calculations Using Excel Spreadsheets Harlan H. Bengtson, PE, PhD Emeritus Professor of Civil Engineering Southern Illinois University Edwardsville

Table of Contents Introduction Section 1 – Manning Equation Review Section 2 – Hydraulic Radius Calculation – Less than Half Full Pipes Section 3 – Hydraulic Radius Calculation – More than Half Full Pipes Section 4 – Is the Manning Roughness Coefficient, n, Constant for Partially Full Pipe Flow Section 5 – Equation for Manning Roughness Coefficient Section 6 – Calculation of Flow Rate and Average Velocity Section 7 – Calculation of Required Diameter for a Target Value of y/D Section 8 – Calculation of Normal Depth Summary References

Introduction The Manning equation is used for a wide variety of uniform open channel flow calculations, including gravity flow in pipes, the topic of this book. Gravity flow occurs in pipes for partially full flow, up to and including full pipe flow, as long as the pipe isn’t pressurized. Equations for calculating area, wetted perimeter and hydraulic radius for partially full pipe flow are included in this book along with a brief review of the Manning equation and discussion of its use to calculate a) the flow rate in a given pipe (diameter, slope, & Manning roughness) at a specified depth of flow, b) the required diameter for a specified flow rate at a target percent full in a given pipe, and c) the normal depth (depth of flow) for a specified flow rate in a given pipe. This includes presentation and discussion of the equations for the calculations, example calculations, and spreadsheets to facilitate the calculations.

Section 1: Manning Equation Review A brief review of the Manning equation is included here. For more details see the first book in this series, “The Manning Equation for Open Channel Flow Calculations.” The Manning equation, given below as equations (1) and (2), is the most widely used equation for uniform open channel flow calculations. Note that the Manning equation is a dimensional equation. That is, the U.S. units given below must be used for the U.S. version of the equation and the S.I. units given below must be used for the S.I. version. U.S. version: Q = (1.49/n)A(Rh2/3)S1/2

(1)

S.I. version: Q = (1.00/n)A(Rh2/3)S1/2

(2)

The parameters in these equations for gravity pipe flow are as follows: Q is the volumetric flow rate through the pipe in cfs (U.S.) or m3/s (S.I.) n is the Manning roughness coefficient, an empirical constant (See typical n values in Table 1 below) S is the slope of the pipe expressed as a dimensionless fraction A is the cross-sectional area of flow in ft2 (U.S.) or m2 (S.I.) P is the wetted perimeter of the flow through the pipe in ft (U.S.) or m (S.I.) Rh is the hydraulic radius = A/P Uniform open channel flow is necessary for use of the Manning equation. For gravity pipe flow this means there must be a constant flow rate through a section of pipe with constant diameter, slope, and surface roughness. In this case the depth of flow in the pipe will be the same throughout that section of pipe.

The source for the n values in Table 1 is: Stormwater Collection Systems Design Handbook, Sec. 3.8. Values of the Roughness Coefficient, n

Section 2: Hydraulic Radius Calculation – Less than Half Full Pipes The equations for calculating the area of flow and wetted perimeter for pipes flowing less than half full are covered in this chapter. The slightly different equations for more than half full are covered in the next chapter. Figure 2 below shows a diagram with the parameters for a pipe flowing less than half full. The equations for calculating A and P are shown alongside the diagram.

Note that the parameters used to calculate A and P are h and r, with the central angle, θ, used as an intermediate value for the calculations. The parameter r is the pipe radius (D/2) and for less than half full flow, h is equal to the depth of flow, y. The following examples illustrate the calculations. Example #1: What would be the hydraulic radius in feet for water flowing 10 inches deep in a 36 inch diameter pipe? Solution: From the given information: r = 36/2 = 18 in = 1.5 ft; h = y = 10 in = 10/12 ft

θ = 2 arccos [ (r – h)/r ] = 2 arccos [ (1.5 – 10/12)/1.5 ] = 2.22 radians A = r2(θ – sinθ)/2 = [ 1.52(2.22 – sin(2.22)) ]/2 = 1.60 ft2 P = rθ = (1.5)(2.22) = 3.33 ft Rh = A/P = 1.60/3.33 = 0.481 ft This type of calculation can be conveniently done with a spreadsheet. The screenshot in Figure 3, on the next page, shows a possible layout for a spreadsheet to calculate hydraulic radius for less than half full pipe flow, and shows the solution to Example #1. Once the appropriate formulas are entered into the blue cells, all one must do to calculate a hydraulic radius is enter a pipe diameter in inches and depth of flow in feet. The source for the screenshot in Figure 3 is the spreadsheet workbook at: http://accessengineeringlibrary.com/browse/spreadsheet_S0009_Partially_Full_Pipe_Flow_Calc ulations. This screenshot shows part of a worksheet set up to calculate the flow rate and velocity for partially full pipe flow, which will be discussed and illustrated with examples in Chapters 6 and 7. Example #2 below illustrates a hydraulic radius calculation in S.I. units for less than half full pipe flow. Example #2: What would be the hydraulic radius in meters for water flowing 50 mm deep in a 200 mm diameter pipe? Solution: For the given information: r = 200/2 = 100 mm = 0.1 m; h = y = 50 mm = 0.05 m θ = 2 arccos [ (r – h)/r ] = 2 arccos [ (0.1 – 0.05)/0.1 ] = 2.09 radians A = r2(θ – sinθ)/2 = [ 0.12(2.09 – sin(2.09)) ]/2 = 0.00611 m2 P = rθ = (0.1)(2.09) = 0.209 m Rh = A/P = 0.00611/0.209 = 0.0292 m

Fig 3. A Spreadsheet for Hydraulic Radius Calculation

Section 3: Hydraulic Radius Calculation – More than Half Full Pipes The equations for calculating the area of flow and wetted perimeter for pipes flowing more than half full are covered in this chapter. These equations are slightly different than those just presented and discussed for less than half full pipe flow. In this case h is calculated as 2r –y instead of simply y = r as it was for less than half full pipe flow. The wetted perimeter and area of flow are also calculated a bit differently. The wetted perimeter is calculated by subtracting the length of the dry perimeter at the top from the total perimeter and the area of flow is calculated by subtracting the area of the space above the water from the total cross-sectional area of the pipe. Figure 4 below shows a diagram of more than half full pipe flow with the parameters and equations used for more than half full pipe flow calculations with the Manning equation.

Example #3: What would be the hydraulic radius for water flowing 24 inches deep in a 36 inch diameter pipe? Solution: From the given information: r = 36/2 = 18 in = 1.5 ft; h = 2r - y = (36 – 24) in = 1 ft θ = 2 arccos [ (r – h)/r ] = 2 arccos [ (1.5 – 1)/1.5 ] = 2.46 radians A = πr2 - r2(θ – sinθ)/2 = π1.52 [ 1.52(2.46 – sin(2.46)) ]/2 = 5.01 ft2

P = 2πr - rθ = 2π(1.5) - (1.5)(2.46) = 5.73 ft Rh = A/P = 5.01/5.73 = 0.87 ft Calculations like this for more than half full pipe flow can also be conveniently completed with a spreadsheet very similar to the one shown in Fig. 3, just using the “more than half full” equations for h, A, & P.

Section 4: Is the Manning Roughness Coefficient, n, Constant for Partially Full Pipe Flow? The Manning roughness coefficient, n, is considered to be a constant, independent of depth of flow, for most Manning equation/open channel flow calculations. For partially full pipe, however, this is not the case. The area of flow, A, and hydraulic radius, Rh, can be calculated for any specified pipe diameter and depth of flow, as described in the previous two chapters. The seemingly logical approach for partially full pipe flow Manning equation calculations would be to use the Manning roughness coefficient as a constant for a given pipe material. It had been observed as early as the mid twentieth century, however, that measured values of flow rate in partially full pipes do not agree with values calculated using the Manning equation with normal open channel flow calculation procedures. T.R. Camp described an improved procedure for making Manning Equation calculations for partially full pipe flow in a 1946 article, “Design of Sewers to Facilitate Flow” (reference #3). In that article, Camp described the use of a variable Manning roughness coefficient, n, as a function of depth of flow divided by pipe diameter. In other words, he described a variation in n/nfull as a function of y/D. Camp’s procedure, described in his 1946 article, led to development of graphs like that in Figure 5 below, which gives Q/Qfull, V/Vfull, and n/nfull as functions of y/D. The graph shown in Figure 4 was developed by Camp and has since appeared in publications of the Water Pollution Control Federation, publications of the American Society of Civil Engineers, and in many environmental engineering textbooks. The graph in Figure 5 was drawn using values taken from a similar graph in Steel and McGhee’s textbook (reference #5). Partially full pipe flow calculations using the trigonometric/geometric equations for A, P, & Rh can be done conveniently through the use of a spreadsheet as described in Chapters 2 and 3. Before spreadsheets and personal computers became widely available, however, graphs like that in Figure 5 provided a convenient way to determine flow rate and velocity for partially full, gravity flow in pipes. V/Vfull and Q/Qfull can be found from the graph for any specified value of the fraction full, y/D. Vfull and Qfull can be calculated with the Manning equation, for known diameter, D, using Rh = D/4 and A = πD2/4.

There is no physical explanation for the variation in Manning roughness coefficient, n, with y/D, as shown in the graph. It is simply a calculation procedure to make Manning equation calculations for partially full pipe flow agree with experimental measurements. The Manning equation was developed for open channel flow in shapes similar to rectangular or trapezoidal and it works well for those channel shapes with the Manning roughness coefficient, n, constant for a given channel surface. Use of variation in n with y/D as shown in Fig. 5, however, works better for gravity flow in circular pipes. Example #4: Use Figure 5 to find the flow rate and average velocity in a 24 inch diameter storm sewer when it is flowing at a depth of 18 inches. The slope of the storm sewer is 0.0013 and has Manning n = 0.012. Solution: y/D = 18/24 = 0.75; From Fig. 5: Q/Qfull = 0.79 and V/Vfull = 0.98 Qfull and Vfull can be calculated with the Manning Equation: Qfull = (1.49/n)A(Rh2/3)(S1/2)

= (1.49/0.012)(π22/4)((2/4)2/3)(0.00131/2) = 8.86 cfs Vfull = (1.49/n)(Rh2/3)(S1/2) = (1.49/0.012)*((2/4)2/3)(0.00131/2) = 2.82 ft/sec For the 18 inch deep flow: Q = (Q/Qfull)(Qfull) = 0.78*8.86 = 7.0 cfs V = (V/Vfull)(Vfull) = 0.97*2.82 = 2.8 ft/sec

Section 5: Equation for Manning Roughness Coefficient Since spreadsheets are now so widely used, they provide a convenient alternative to the use of a graph like Fig. 5 for partially full pipe flow calculations. Spreadsheet calculation of A, P, and Rh has already been illustrated in chapters 2 and 3. The remaining variable parameter, for which values must be calculated, is the Manning roughness coefficient, n, for a specified y/D. The source for the following empirical equation for n/nfull as a function of y/D is: Civil Engineering All-in-One PE Exam Guide: Breadth and Depth, 2nd Ed. Equation 303.32.

Equation (3)

Example #5: Calculate the Manning roughness coefficient for water flowing 8 inches deep in an 18 inch diameter corrugated metal storm drain pipe Solution: From Table 1, the average value of nfull is 0.024 for corrugated metal pipe. From the problem statement y = 8 inches and D = 18 inches, thus: y/D = 8/18 = 0.4444 Substituting this value into equation (3) gives: n/nfull = 1 + 0.4440.54 – 0.4441.20 = 1.27 n = (n/nfull)nfull = (1.27)(0.024) = 0.0305

Use of a spreadsheet for this type of calculation will be discussed in the next chapter, along with the discussion of calculating flow rate, Q, and average velocity, V, in partially full pipe flow.

Section 6: Calculation of Flow Rate and Average Velocity In this chapter, we will just put together the components discussed in previous chapters. That is, For specified values of D, y, nfull, and pipe slope, S: a) calculate A, P and Rh, using the equations in chapter 2; b) calculate n as just discussed in chapter 5; and c) calculate Q and V using the Manning equation as discussed in chapter 1. These calculations and use of spreadsheets to make the calculations are illustrated in the following examples. Example #6: What would be the flow rate and average velocity of water flowing 28 mm deep in a 200 mm diameter vitrified clay sewer pipe (average value for nfull = 0.014 from Table 1) with a slope of 0.005? Solution: Using equations from chapter 2: r = D/2 = 200/2 mm = 100 mm = 0.1 m h = y = 28 mm = 0.028 m θ = 2 arccos [ (r - h)/r ] = 2 arccos [ (0.1 - 0.028)/0.1 ] = 1.534 radians A = r2(θ - sinθ)/2 = (0.12)[1.534 - sin(1.534)]/2 = 0.00267 m2 P = rθ = (0.1)(1.534) = 0.153 m Rh = A/P = 0.00267/0.153 = 0.0174 m For y/D = 28/200 = 0.14, from Eqn (3): n/nfull = 1 + 0.140.54 - 0.141.20 = 1.251 n = (n/nfull)(nfull) = (1.251)(0.014) = 0.018

Now the Manning equation can be used to calculate Q and V: Q = (1.00/n)A(Rh2/3)S1/2 = (1.00/0.014)(0.00267)(0.01742/3)(0.0051/2) Q = 0.000725 m3/s

V = Q/A = 0.000725/0.00267 = 0.271 m/s This entire set of calculations can very conveniently be carried out with an Excel spreadsheet. Figure 6 on the next page shows the solution to Example #6 in a screen shot of a spreadsheet available at: http://accessengineeringlibrary.com/browse/spreadsheet_S0009_Partially_Full_Pipe_Flow_Calc ulations. In order to use the spreadsheet a user simply needs to enter values for D, y, nfull, and S in the yellow cells. The formulas presented and discussed in Chapters 1, 2, & 5 are in the various blue and pink cells and make all the calculations needed to get values for Q and V.

Fig 6. A Spreadsheet for Calculation of Flow Rate and Average Velocity

Example #7: What would be the flow rate and average velocity of water flowing 8 inches deep in an 18 inch diameter corrugated metal storm drain pipe with a slope of 0.004? Solution: Using equations from chapter 2: r = D/2 = 18/2 inches = 9 inches = 0.75 ft h = y = 8 inches = 0.6667 ft θ = 2 arccos [ (r - h)/r ] = 2 arccos [ (9 - 8)/9 ] = 2.92 radians A = r2(θ - sinθ)/2 = (0.752)[2.92 - sin(2.92)]/2 = 0.759 ft2 P = rθ = (0.75)(2.92) = 2.19 ft Rh = A/P = 0.759/2.19 = 0.35 ft From Example #5: for y/D = 8/18 = 0.4444 and nfull = 0.014: n = 0.018 Now the Manning equation can be used to calculate Q and V: Q = (1.49/n)A(Rh2/3)S1/2 = (1.49/0.028)(0.759)(0.352/3)(0.0041/2) Q = 1.27 cfs

V = Q/A = 1.27/0.759 = 1.68 ft/sec

Now, we will turn to an example with more than half full pipe flow The calculations are the same as shown above for Examples #6 and #7, except for the use of the equations from Chapter 3 being used to calculate A, P, and Rh. Equation (3) will be used to calculate n/nfull, and the Manning equation calculations will be the same as in Examples #6 and #7.

Example #8: What would be the flow rate and average velocity of water flowing 12 inches deep in an 18 inch diameter corrugated metal storm drain pipe with a slope of 0.004?

Solution: Using equations from chapter 3: r = D/2 = 18/2 inches = 9 inches = 0.75 ft h = 2r - y = (2*9 – 12) inches = 6 inches = 0.5 ft θ = 2 arccos [ (r - h)/r ] = 2 arccos [ (9 - 6)/9 ] = 2.46 radians A = πr2 - r2(θ - sinθ)/2 = (0.752)[2.46 - sin(2.46)]/2 = 1.25 ft2 P = πr - rθ = π(0.75) - (0.75)(2.46) = 2.87 ft Rh = A/P = 1.25/2.87 = 0.44 ft For y/D = 12/18 = 0.6667, from Eqn (3): n/nfull = 1 + 0.666670.54 - 0.666671.20 = 1.189 With nfull = 0.014, from Table 1: n = (n/nfull)(nfull) = (1.189)(0.014) = 0.017 Now the Manning equation can be used to calculate Q and V: Q = (1.49/n)A(Rh2/3)S1/2 = (1.49/0.017)(1.25)(0.442/3)(0.0041/2) Q = 4.08 cfs

V = Q/A = 4.08/1.25 = 3.26 ft/sec

Section 7: Calculation of Required Pipe Diameter for a Target Value of y/D The equations used for calculations in this chapter are the same as those used in Chapter 6, for determining flow rate and average velocity for partially full pipe flow. In this case, however, an iterative solution is required. Excel’s “Goal Seek” tool is very convenient for finding iterative solutions and will be discussed as part of the presentation on spreadsheet calculations in this chapter. The inputs needed for determining required diameter are design flow rate, the pipe slope, the Manning roughness coefficient for full pipe flow, and a target value for y/D. The iterative solution for D proceeds by rearranging the Manning equation to the form: Q/(1.49*S1/2) = (A*Rh2/3)/n The left side of this equation can be calculated from the specified values of Q and S. The iterative solution consists of selecting a value of D and calculating a value for the right side of the equation. This needs to be repeated until the right side of the equation is equal to the left side of the equation. Although this sounds rather straightforward, each iteration requires quite a few calculations. After assuming a value for D, the following calculations are needed: y = (y/D)D; r = D/2; h = 2r – y; θ = 2 arccos[(r – h)r]; A = πr2 – r2(θ – sinθ)/2; P = 2π – rθ; Rh = A/P; n = nfull(n/nfull) = nfull [ 1 + (y/D)0.54 – (y/D)1.20 ] After completing all of these calculations, you can calculate the value of (A*Rh2/3)/n to compare with the target value for that quantity. The whole process needs to be repeated until the calculated value of (A*Rh2/3)/n is equal to the target value, to the desired degree of accuracy. This is rather cumbersome to do by hand, but it can be done quite conveniently with a spreadsheet as illustrated with Example #9 below.

Example #9: Determine the diameter required for steel pipe (nfull = 0.012), at a slope of 0.002, carrying 1.65 cfs at a target y/D = 0.8. Solution: The screenshot in Figure 7, on the next page, shows a spreadsheet solution to this example. The spreadsheet from which the screenshot in Figure 7 was extracted is available at: http://accessengineeringlibrary.com/browse/spreadsheet_S0009_Partially_Full_Pipe_Flow_Calc ulations.

The yellow cells are for the required inputs: design flow rate, Q; Manning roughness coefficient, nfull; the pipe slope, S; the target value for y/D, and an initial estimate for the pipe diameter, D, in inches. The spreadsheet calculates the pipe diameter and radius in feet and makes intermediate calculations necessary to calculates the value of Q/1.49*S1/2), which is the target value for (A*Rh2/3)/n. Instructions for use of the Excel “Goal Seek” tool are included on the spreadsheet. Carrying out the “Goal Seek” process results in the initial estimate for pipe diameter being replaced with the minimum required diameter. In this example, the minimum needed diameter is 12.4 inches. The minimum required diameter calculated by the spreadsheet will almost certainly not be a standard pipe diameter, so the user will need to select the next larger or next smaller standard pipe size and check on its suitability with this worksheet or the worksheet for calculating flow rate and average velocity or that for calculating normal depth. Tables with standard pipe size information are available at: Perry’s Chemical Engineers’ Handbook, 8th Ed, Table 10-22 (U.S. units) Piping Handbook, Seventh Ed., Table E2.1M (S.I. units)

Calculation of the required diameter for a target y/D less than 0.5 would be very similar, but the equations for h, q, A, P, Rh, and n would be those for less than half full flow, from Chapters 3 and 5. As noted in chapter 1, the constant in the Manning equation is 1.00 instead of 1.49 for S.I. units, so the rearranged form to use in the iterative process for determining required diameter for a target value of y/D is: Q/(1.00*S1/2) = (A*Rh2/3)/n The calculations for S.I. units are the same as those just illustrated for the example with U.S. units, using the following units: Q in m3/s, A in m2, P in m, Rh in m.

Fig 7. A Spreadsheet for Calculation of Required Pipe Diameter

Section 8: Calculation of Normal Depth The calculation of normal depth is similar to the calculations just described in Chapter 7 for determining the required pipe diameter to give a target value of y/D. The required inputs are pipe diameter, D; the Manning roughness, nfull; the pipe slope, S; and the design flow rate, Q. In this case the Manning equation is also rearranged to the form: Q/(1.49*S1/2) = (A*Rh2/3)/n In this case also, the value of the left side of the equation can be calculated and can be used as a target value for the right side of the equation in an iterative process to find the value of depth of flow, y, that will make the two sides of the equation equal, typically called the normal depth. This process is illustrated with Example #10 below. Example #10: Determine the depth of flow (normal depth) for water flowing at 18 cfs through a 48 inch diameter pipe with nfull = 0.011, and a slope of 0.0003 ft/ft. Solution: A screenshot of a spreadsheet to carry out this process is shown in Fig 8 below. The overall process is very similar to that presented and discussed in Chapter 7. The needed inputs are pipe diameter, D, Manning roughness, nfull, pipe slope, S, and flow rate, Q. The spreadsheet then calculates Q/(1.49*S1/2) = 697.5, which is thus the target value for (A*Rh2/3)/n, when y is equal to yo, the normal depth. Entering an initial estimate for the normal depth, yo, and then carrying out the “Goal Seek” process as detailed on the spreadsheet, leads to the solution: yo = 30.6 in. The spreadsheet from which the screenshot in Figure 8 was extracted is available at: http://accessengineeringlibrary.com/browse/spreadsheet_S0009_Partially_Full_Pipe_Flow_Calc ulations

Fig 8. A Spreadsheet for Calculation of Normal Depth – U.S. units

Calculation of normal depth for a pipe flowing less than half full is essentially the same as just illustrated for a pipe flowing more than half full, but the equations to calculate the hydraulic radius for less than half full pipes must be used. In the spreadsheet illustrated by the screenshots in Figures 8 and 9, IF statements are used to calculate the cross-sectional area of flow, A, and the wetted perimeter, P, so that the correct equations are used based on the value of y/D. Example #11 illustrates normal depth calculation using S.I. units, in a case with less than half full flow. Example #11: Determine the depth of flow (normal depth) for water flowing at 0.12 m3/s through a 900 mm diameter pipe with nfull = 0.011, and a slope of 0.0008 ft/ft. Solution: A screenshot of a spreadsheet to carry out this process is shown in Fig 9 below. The overall process is very similar to that presented and discussed in Section 9. The needed inputs are pipe diameter, D, Manning roughness, nfull, pipe slope, S, and flow rate, Q. The spreadsheet then calculates Q/(1.00*S1/2) = 4.24, which is thus the target value for (A*Rh2/3)/n, when y is equal to yo, the normal depth. Entering an initial estimate for the normal depth, yo, and then carrying out the “Goal Seek” process as detailed on the spreadsheet, leads to the solution: yo = 310 mm. The spreadsheet from which the screenshot in Figure 9 was extracted is available at: http://accessengineeringlibrary.com/browse/spreadsheet_S0009_Partially_Full_Pipe_Flow_Calc ulations

Fig 9. A Spreadsheet for Calculation of Normal Depth – S.I. units

Summary The following Manning equation calculations for partially full, gravity flow in pipes were presented, discussed, and illustrated with examples in this paper. a) Calculation of flow rate and average velocity for specified pipe diameter, depth of flow, Manning roughness coefficient, and pipe slope; b) Calculation of required pipe diameter for specified flow rate, Manning roughness coefficient, pipe slope, and target fraction y/D; c) Calculation of normal depth for specified flow rate, pipe diameter, Manning roughness coefficient, and pipe slope.

These calculations are somewhat more complicated than comparable calculations for traditional open channel shapes, such as trapezoidal or rectangular because of the following two factors: a)

The equations used to calculate A, P, and Rh are a bit more complicated.

b)

Different equations must be used for less-than-half full and more-than-half full flow.

c)

Values for the Manning roughness coefficient need to be calculated as a function of y/D.

The nature of the equations and the calculations for partially full pipe flow make the use of spreadsheets a convenient way to make the three types of calculation itemized above. Spreadsheet screenshots and discussion of their use for these calculations are included in this book for each of those types of partially full pipe flow calculations.

References 1. Lee, C.C. & Lin, S.D., Handbook of Environmental Engineering Calculations, 2nd Ed., New York, McGraw-Hill Book Company, 2007, Sec. 1.5.4.3. Partially Filled Conduits. 2. Goswami, I., Civil Engineering All-in-One PE Exam Guide: Breadth and Depth, 2nd Ed., New York: McGraw-Hill Book Company, 2012. Sec. 303.11. Normal Depth of Flow in Circular Open Channels. 3. Green, D.W. & Perry, R.H., Perry’s Chemical Engineers’ Handbook, 8th Ed., New York: McGraw-Hill Book Company, 2008. Sec. 6.1.4. Incompressible Flow in Pipes and Channels 4.

Avallone, E.A., Baumeister III, T. & Sadegh, A., Marks’ Standard Handbook for Mechanical Engineers, 11th Ed., New York, McGraw-Hill Book Company, 2006. Sec. 3.3.16. Open Channel Flow.

5. Mays, L.W., Hydraulic Design Handbook, New York, McGraw-Hill Book Company, 1999. Sec. 3.4 Uniform Flow 6. Ricketts, J.T., Loftin, M.K., & Merritt, F.S., Standard Handbook for Civil Engineers, New York, McGraw-Hill Book Company, 2004. Sec. 21.6.4 Manning Equation for Open Channels 7. Camp, T.R., “Design of Sewers to Facilitate Flow,” Sewage Works Journal, 18 (3), 1946 8. Bengtson, Harlan H., Uniform Open Channel Flow and The Manning Equation, an online, continuing education course for PDH credit.