Scattergraph Principles and Practice...Scattergraph Principles and Practice A Comparison of Various Applications of the Manning Equation ... including the Design Method, the Lanfear-Coll

© 2004 ADS LLC. All Rights Reserved.

1

Scattergraph Principles and Practice A Comparison of Various Applications of the Manning Equation

Kevin L. Enfinger, P.E. and Hal R. Kimbrough ADS Environmental Services 4940 Research Drive Huntsville, Alabama 35805 www.adsenv.com/scattergraph

ABSTRACT The Manning Equation is an empirical formula commonly used to design sewer systems. This equation is also used to describe the performance of existing sewers by evaluating flow monitor data on a scattergraph using a variety of methods, including the Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method. The proper selection and application of these methods have a significant impact on the calculation of sewer capacity and the evaluation of sewer performance based on flow monitor data. Examples of each method are provided from flow monitor locations throughout the United States. Laboratory research by the authors is also provided to further explore the performance of these methods and provide guidelines for their proper application.

KEY WORDS Flow Monitoring, Manning Equation, Scattergraph, Sewer Capacity

Introduction The scattergraph is a graphical tool that provides insight into sewer performance through a simple and intuitive display of flow monitor data. The resulting patterns form characteristic signatures that reveal important information about conditions within a sewer and the impact that these conditions have on sewer capacity.1 The Manning Equation is an important component of the scattergraph and can be applied using a variety of methods. Proper selection and application of these methods have a significant impact on the calculation of sewer capacity and the evaluation of sewer performance based on flow monitor data. Therefore, the purpose of this paper is to provide an overview and comparison of three methods that use the Manning Equation to estimate sewer capacity from flow monitor data and provide guidelines for their proper application.


2

Manning Equation The Manning Equation is an empirical formula used to design sewer systems. The most common expression of this formula is provided in Equation (1).

2/13/2 SRn

486.1=v (1)

where: v = flow velocity, ft/s

n = roughness coefficient R = hydraulic radius, ft S = slope of the energy gradient Several assumptions are generally made with respect to the Manning Equation: the roughness coefficient is constant, and the slope of the energy gradient equals the slope of the pipe.2 Based on these assumptions, the Manning Equation can be algebraically rearranged such that these parameters are consolidated into a single coefficient, defined as the hydraulic coefficient, and restated as shown in Equation (2). This expression is useful in subsequent discussions.

3/2CR486.1=v (2)

where: v = flow velocity, ft/s C = hydraulic coefficient R = hydraulic radius, ft The relationship between flow depth and velocity described by the Manning Equation is depicted in Figure 1 as a pipe curve ( - - - ) and provides a convenient reference to evaluate flow monitor data.

0.0 1.4

Flow Velocity (v/vD)

Flo

w D

ept

h (d

/D)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0 1.2

Manning Equation

FIGURE 1: Hydraulic Relationship of the Manning Equation


3

Manning Methods The Manning Equation is also used to describe the performance of existing sewers by evaluating flow monitor data on a scattergraph, as shown in Figure 2. The Manning Equation is used to generate a pipe curve which is then compared to actual flow monitor data ( ). This data may agree or disagree with the Manning Equation, depending on actual conditions at the monitoring location. In either case, important information can be learned about the performance of a sewer and its effect on sewer capacity.3 For example, the flow monitor data shown in Figure 2 indicate that this sewer operates as expected up to a flow depth of about 15 inches. However, as backwater conditions develop, flow conditions become deeper and slower and are revealed on the scattergraph as a departure from the pipe curve, resulting in surcharge and overflow conditions at a much lower capacity than expected.4 Three manual confirmations ( ) are also shown and provide a means to evaluate the accuracy of the flow monitor.

Flo

w D

epth

(in

)

0

30

5

10

15

20

25

Flow Velocity (ft/s)

0 8

70

65

60

55

50

45

40

35

min

max

2 4 6

FIGURE 2: Scattergraph of Flow Depth and Velocity Data

Flow Monitor Data

Manning Equation

The Manning Equation is an important component of the scattergraph and can be applied using three different methods, defined as the Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method. The Design Method uses the Manning Equation to describe a relationship between flow depth and velocity using a specified roughness coefficient and pipe slope. This relationship is then compared with actual flow monitor data. The Lanfear-Coll Method and the Stevens-Schutzbach Method use curve fitting techniques to correlate the Manning Equation directly to such data. Each method may rely on assumptions different from design or as-built conditions. An overview and comparison of these methods are provided in the following sections.


4

Design Method The Design Method uses the Manning Equation with a specified roughness coefficient and pipe slope. The Manning Equation is applied using this method under the general assumptions shown in Figure 3.

FIGURE 3: General Assumptions of the Design Method

flow monitor

n = constant S = constant

uniform flow

The Design Method incorporates the Manning Equation as expressed in Equation (3) and the hydraulic radius as defined in Equation (4).

3/2DMDMDM RC486.1=v (3)

P

A=RDM (4)

where: vDM = flow velocity, ft/s CDM = hydraulic coefficient RDM = hydraulic radius, ft A = wetted area, ft2 P = wetted perimeter, ft The roughness coefficient and the pipe slope are specified based on design assumptions, as-built documentation, or field observations and are used to calculate the hydraulic coefficient as shown in Equation (5).

2/1DM S

n

1=C (5)

where: CDM = hydraulic coefficient n = roughness coefficient S = pipe slope The Design Method is then used to generate a pipe curve which is compared to actual flow monitor data on a scattergraph. If the data agree with the pipe curve, then this method can be used to estimate the full-pipe capacity of the sewer, assuming the assumptions of this method remain valid at the monitoring location from 0 ≤ d ≤ D. The application of the Design Method is demonstrated in the following example.


5

EXAMPLE Flow monitor data are obtained from a 30-in sewer, as shown in thescattergraph below. The roughness coefficient (n) and the slope (S) are alsoprovided, based on design documentation.

Use the Design Method to construct a pipe curve on the scattergraph andestimate the full-pipe capacity of this sewer.

0

5

10

20

25

15

Flo

w D

epth

(in

)

0

30


max

min

93 6

= 0.013n

= 0.45%S


6

EXAMPLE

000

096

141

180

219

264

360

0.00

0.54

1.43

2.45

3.48

4.37

4.91

0.00

2.10

3.08

3.93

4.78

5.75

7.85

0.26

0.47

0.63

0.73

0.76

3.09

4.61

5.61

6.20

6.39

00

05

10

15

20

25

30

d A P RDM vDM

in ft2 ft ft ft/s

o

0.63 5.61

= 2cos-1(1 - 2d/D)

A = (D2/8)(- sin

P = D/2

Ad

D

Calculate CDM assuming n = 0.013 and S = 0.45%

CDM = 5.16

Calculate vDM for 0 < d < D

For a circular sewer,5

(a)

(b)

Solution: Calculate the hydraulic coefficient, construct pipe curve, and estimate sewer capacity

These results provide the necessary information to construct a pipe curve on ascattergraph, as shown below.

The full-pipe capacity is calculated using the Continuity Equation, QDM = AvDM. Therefore,QDM = 4.91 ft2 x 5.61 ft/s = 27.5 ft3/s or 17.8 MGD.

The conditions observed within this sewer are effectively described by the ManningEquation using the Design Method.

Calculate QDM for d = D(c)

0.40

0.60

0.73

0.81

0.83

RDM2/3

ft2/3

0.73

0

5

10

20

25

15

Flo

w D

epth

(in

)

0

30


max

min

93 6

= 0.013n

= 0.45%S

= 5.16CDM

DM


7

Lanfear-Coll Method The Lanfear-Coll Method uses a curve fitting technique to fit the Manning Equation to flow monitor data.6 The Manning Equation is applied using this method under the general assumptions shown in Figure 4.

FIGURE 4: General Assumptions of the Lanfear-Coll Method

flow monitor

n = constant S = constant

uniform flow

This method is applicable to flow monitor data obtained under uniform flow conditions and incorporates the Manning Equation as expressed in Equation (6) and the hydraulic radius as defined in Equation (7).

3/2LCLCLC RC486.1=v (6)

P

A=RLC (7)

where: vLC = flow velocity, ft/s CLC = hydraulic coefficient RLC = hydraulic radius, ft A = wetted area, ft2 P = wetted perimeter, ft This method provides an implicit solution to the Manning Equation and requires no direct knowledge of the roughness coefficient or the slope of the energy gradient. Flow depth and velocity data are used to calculate the hydraulic coefficient based on a least squares regression of Equation (6), as described in Figure 5. Regression results are characterized using the coefficient of determination.7


8

FIGURE 5: Regression Using the Lanfear-Coll Method

A

D

d

Restate Equation (6) as y = a + bx using directsubstitution, where:

= RLC2/3

= vLC

= 1.486CLC

= 0

x

y

b

a

= 2cos-1(1 - 2d/D)

A = (D2/8)( - sin )

P = D/2

For a circular sewer,

RLC =A

P

Flo

w V

eloc

ity(f

t/s) Slope = 1.486CLC

(Hydraulic Radius)2/3 (ft2/3)

R2

Perform least squares regression.

CalculateRLC

2/3

Regressionv vs RLC

2/3

CalculateR2

d

v

CalculateCLC =

b1.486

Done

The Lanfear-Coll Method is then used to generate a pipe curve which is compared to actual flow monitor data on a scattergraph. If the data agree with the pipe curve, then this method can be used to estimate the full-pipe capacity of the sewer, assuming the assumptions of this method remain valid at the monitoring location from 0 ≤ d ≤ D. The application of the Lanfear-Coll Method is demonstrated in the following example.


9

EXAMPLE Flow monitor data are obtained from a 42-in sewer, as shown in the

scattergraph below. Tabular data are provided on the following page.

Use the Lanfear-Coll Method to construct a pipe curve on the scattergraph andestimate the full-pipe capacity of this sewer.

Flo

w D

ept

h (in

)


0

min

max

0

42

35

28

21

14

7

153 6 9 12


10

EXAMPLE

(b)

(a)

Solution: Calculate the hydraulic coefficient

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

11/01

11/01

11/01

11/01

11/01

11/01

11/01

11/30

13.99

14.03

13.71

13.59

13.22

13.16

13.14

15.64

7.18

7.40

7.11

7.15

6.89

7.00

6.82

7.22

2.59

2.58

2.57

3.26

137

136

136

150

2.80

2.82

2.73

2.70

141

141

139

139

4.17

4.16

4.16

4.59

4.31

4.31

4.26

4.24

vavg

01:45

02:00

11/01

11/01

13.01

12.81

6.71

6.71

2.54

2.48

135

134

4.13

4.10

d

in

v

ft/s

time

hh:mm

date

mm/dd

A

ft2o

P

ft

RLC

ft ft2/3

0.62

0.62

0.62

0.65

0.65

0.64

0.64

0.61

0.61

0.73

0.73

0.73

0.75

0.75

0.74

0.74

0.72

0.72

0.71 0.80

7.17

RLC2/3

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

11/01

11/01

11/01

11/01

11/01

11/01

11/01

11/30

0.75

0.75

0.74

0.74

0.73

0.73

0.73

0.80

7.18

7.40

7.11

7.15

6.89

7.00

6.82

7.22

5.39

5.57

5.28

5.29

5.02

5.09

4.95

5.75

6.98

6.99

6.91

6.88

6.78

6.76

6.75

7.40

0.56

0.57

0.55

0.55

0.53

0.53

0.53

0.63

0.038

0.164

0.040

0.075

0.013

0.058

0.004

0.033

0.000

0.048

0.005

0.001

0.084

0.032

0.130

0.002

x xy vLC

ft2/3 ft5/3/s ft/s

y

ft/s

time

hh:mm

date

mm/dd

x2

ft4/3

(vLC - v)2

(ft/s)2

(v - vavg)2

(ft/s)2

SSE SYY xy x2

CLC = xy x2

1.486

/R2 =

SSE

SYY1 -

For this example, a total of 2,880data points were used. Completecalculat ions are avai lable in aspreadsheet that accompanies thistechnical paper.

Based on the regression results, CLC = 6.26 and R2 = 0.84.

01:45

02:00

11/01

11/01

0.72

0.72

6.71

6.71

4.85

4.81

6.72

6.66

0.52

0.51

0.000

0.002

0.221

0.221

Calculate RLC2/3

Calculate CLC and R2 based on a least squares regression


11

EXAMPLE

The full-pipe capacity is calculated using the Continuity Equation, QLC = AvLC. Therefore,QLC = 9.62 ft2 x 8.50 ft/s = 81.8 ft3/s or 52.9 MGD.

The conditions observed within this sewer are effectively described by the ManningEquation fitted to observed flow depth and velocity data using the Lanfear-Coll Method.

Solution: Construct pipe curve and estimate sewer capacity

(d)


(c) Calculate vLC for 0 < d < D

Calculate QLC for d = D

000

096

141

180

219

264

360

00.00

02.94

04.31

05.50

06.69

08.05

11.00

0.36

0.65

0.88

1.02

1.06

4.69

6.99

8.50

9.41

9.69

00

07

14

21

28

35

42

d P RLC vLC

in ft ft ft/s

o

0.88 8.50

0.00

1.05

2.81

4.81

6.81

8.57

9.62

A

ft2

0.50

0.75

0.91

1.01

1.04

RLC2/3

ft2/3

0.91

Flo

w D

ep

th (

in)


0

min

max

0

42

35

28

21

14

7

153 6 9 12

LC

n = constant

= 6.26CLC

= 0.84R2


12

Stevens-Schutzbach Method The Stevens-Schutzbach Method uses an iterative curve fitting technique to fit the Manning Equation to flow monitor data.8 The Manning Equation is applied using this method under the general assumptions shown in Figure 6.

flow monitor

ddog

FIGURE 6: General Assumptions of the Stevens-Schutzbach Method

n = constant

non-uniform flow

S = constant < S0 This method is applicable to flow monitor data obtained under uniform flow conditions or non-uniform flow conditions resulting from a variety of downstream obstructions, or dead dogs. Examples include offset joints, debris, and other related conditions. The Stevens-Schutzbach Method incorporates the Manning Equation as expressed in Equation (8) and the hydraulic radius as defined in Equation (9).

3/2SSSSSS RC486.1=v (8)

P

A=R e

SS (9)

where: vSS = flow velocity, ft/s CSS = hydraulic coefficient RSS = hydraulic radius, ft

Ae = effective wetted area, ft2 P = wetted perimeter, ft

Note that the definition of the hydraulic radius is modified from the traditional definition and requires certain assumptions regarding the shape and magnitude of the dead dog. Based on these assumptions, flow depth and velocity data are used to calculate the hydraulic coefficient based on an iterative least squares regression method, as described in Figure 7. The magnitude of the dead dog (ddog) is varied in successive iterations until the coefficient of determination is maximized.


13

FIGURE 7: Regression Using the Stevens-Schutzbach Method

Assumeddog = 0

Regressionv vs RSS

2/3

CalculateR2

R2 Max?

Increase ddog0 < ddog < dmin

No

Yes

d

v

CalculateRSS

2/3

e

Ae

A

D

d de

ddog

Restate Equation (8) as y = a + bx usingdirect substitution, where:

= RSS2/3

= vSS

= 1.486CSS

= 0

x

y

b

a

AssumeObstruction

= 2cos-1(1 - 2d/D)

A = (D2/8)( - sin )

P = D/2

For a circular sewer,

de = d - ddog

e = 2cos-1(1 - 2de/D)

Ae = (D2/8)(e - sin e)

CalculateCSS =

b1.486

Done

Effect of a dead dog can beapproximated using variousassumptions. By default thismethod uses an offset joint.

Further discussion regardingthis assumption is available inthe literature.8

RSS =Ae

P

Iteration 2

(Hydraulic Radius)2/3 (ft2/3) (Hydraulic Radius)2/3 (ft2/3)

Slope = 1.486CSS

Iteration n

R2 maximizedFlo

w V

elo

city

(ft/

s)

Iteration 1

(Hydraulic Radius)2/3 (ft2/3)

Perform iterative least squares regression.

The Stevens-Schutzbach Method is then used to generate a pipe curve which is compared to actual flow monitor data on a scattergraph. If the data agree with the pipe curve, then this method can be used to estimate the full-pipe capacity of the sewer, assuming the assumptions of this method remain valid at the monitoring location from 0 ≤ d ≤ D. The application of the Stevens-Schutzbach Method is demonstrated in the following example.


14

EXAMPLE

Use the Stevens-Schutzbach Method to construct a pipe curve on thescattergraph and estimate the full-pipe capacity of this sewer.

Flow monitor data are obtained from a 27-in sewer, as shown in thescattergraph below. Tabular data are provided on the following page.

max

min

Flo

w D

ept

h (in

)

0

27

9

18


0 41 2 3


15

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

14.34

14.08

13.91

13.81

13.48

13.14

12.93

14.19

0.70

0.69

0.69

0.69

0.68

0.67

0.67

0.70

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

14.34

14.08

13.91

13.81

13.48

13.14

12.93

14.19

1.47

1.41

1.37

1.35

1.36

1.29

1.23

1.44

1.98

1.92

1.88

2.12

1.92

1.90

1.89

1.89

1.87

1.85

1.84

1.91

180

177

175

186

0.49

0.48

0.48

0.47

0.46

0.45

0.45

0.48

180

177

175

186

0.037

0.016

0.009

0.005

0.015

0.005

0.000

0.026

EXAMPLE

187

185

183

183

2.15

2.10

2.06

2.05

187

185

183

183

0.063

0.030

0.017

0.010

0.017

0.004

0.000

0.045

3.53

3.47

3.44

3.65

3.67

3.63

3.60

3.59

x xy vSS

ft2/3 ft5/3/s ft/s

y

ft/s

time

hh:mm

date

mm/dd

x2

ft4/3

(vSS - v)2

(ft/s)2

(v - vavg)2

(ft/s)2

SSE SYY xy x2

(a)

(b)

vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -


Based on this iteration, CSS = 1.85 and R2 = 0.50. R2 is not maximized.

01:45

02:00

08/01

08/01

13.04

12.88

1.88

1.80

13.04

12.88

1.90

1.87

176

175

176

175

3.46

3.43

01:45

02:00

08/01

08/01

0.67

0.67

1.88

1.80

1.26

1.20

1.84

1.83

0.45

0.45

0.001

0.001

0.000

0.003

d e

in o

v

ft/s

time

hh:mm

date

mm/dd

de

in

Ae

ft2o

P

ft

RSS

ft

RSS2/3

ft2/3

dmin

0.56

0.55

0.55

0.58

0.58

0.57

0.57

0.55

0.55

0.68

0.67

0.67

0.70

0.69

0.69

0.69

0.67

0.67

0.58 0.70

10.35 1.86

Solution: Calculate the hydraulic coefficient - Iteration 1

Assume ddog = 0.00 in. Calculate RSS2/3

Calculate CSS and R2 based on a least squares regression


16

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

14.34

14.08

13.91

13.81

13.48

13.14

12.93

14.19

0.66

0.65

0.65

0.65

0.64

0.63

0.62

0.65

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

13.34

13.08

12.91

12.81

12.48

12.14

11.93

13.19

1.39

1.32

1.29

1.26

1.27

1.21

1.15

1.35

1.80

1.73

1.69

1.93

1.93

1.91

1.90

1.89

1.87

1.85

1.83

1.92

171

168

167

177

0.43

0.42

0.42

0.42

0.41

0.40

0.39

0.43

180

177

175

186

0.033

0.014

0.008

0.004

0.014

0.006

0.000

0.023

EXAMPLE

179

176

175

174

1.96

1.91

1.88

1.86

187

185

183

183

0.063

0.030

0.017

0.010

0.017

0.004

0.000

0.045

3.53

3.47

3.44

3.65

3.67

3.63

3.60

3.59

x xy vSS

ft2/3 ft5/3/s ft/s

y

ft/s

time

hh:mm

date

mm/dd

x2

ft4/3

(vSS - v)2

(ft/s)2

(v - vavg)2

(ft/s)2

SSE SYY xy x2

(a)

(b)

vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -



01:45

02:00

08/01

08/01

13.04

12.88

1.88

1.80

12.04

11.88

1.71

1.69

168

166

176

175

3.46

3.43

01:45

02:00

08/01

08/01

0.63

0.62

1.88

1.80

1.18

1.12

1.84

1.83

0.39

0.39

0.002

0.001

0.000

0.003

d e

in o

v

ft/s

time

hh:mm

date

mm/dd

de

in

Ae

ft2o

P

ft

RSS

ft

RSS2/3

ft2/3

dmin

0.51

0.50

0.49

0.53

0.53

0.52

0.52

0.50

0.49

0.64

0.63

0.62

0.66

0.65

0.65

0.65

0.63

0.62

0.53 0.65

10.35 1.86





17

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

14.34

14.08

13.91

13.81

13.48

13.14

12.93

14.19

0.61

0.61

0.60

0.60

0.59

0.58

0.58

0.61

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

12.34

12.08

11.91

11.81

11.48

11.14

10.93

12.19

1.30

1.23

1.20

1.18

1.18

1.12

1.06

1.26

1.61

1.55

1.51

1.74

1.94

1.92

1.91

1.90

1.87

1.84

1.82

1.93

163

160

158

169

0.38

0.37

0.36

0.36

0.35

0.34

0.33

0.37

180

177

175

186

0.029

0.012

0.007

0.004

0.014

0.006

0.000

0.020

EXAMPLE

170

168

166

166

1.77

1.72

1.69

1.67

187

185

183

183

0.063

0.030

0.017

0.010

0.017

0.004

0.000

0.045

3.53

3.47

3.44

3.65

3.67

3.63

3.60

3.59

x xy vSS

ft2/3 ft5/3/s ft/s

y

ft/s

time

hh:mm

date

mm/dd

x2

ft4/3

(vSS - v)2

(ft/s)2

(v - vavg)2

(ft/s)2

SSE SYY xy x2

(a)

(b)

vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -



01:45

02:00

08/01

08/01

13.04

12.88

1.88

1.80

11.04

10.88

1.53

1.50

159

158

176

175

3.46

3.43

01:45

02:00

08/01

08/01

0.58

0.58

1.88

1.80

1.09

1.04

1.83

1.82

0.34

0.33

0.002

0.000

0.000

0.003

d e

in o

v

ft/s

time

hh:mm

date

mm/dd

de

in

Ae

ft2o

P

ft

RSS

ft

RSS2/3

ft2/3

dmin

0.46

0.45

0.44

0.48

0.47

0.47

0.47

0.44

0.44

0.59

0.58

0.58

0.61

0.61

0.60

0.60

0.58

0.58

0.48 0.61

10.35 1.86





18

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

00:00

00:15

00:30

00:45

01:00

01:15

01:30

23:45

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

08/01

08/01

08/01

08/01

08/01

08/01

08/01

08/21

14.34

14.08

13.91

13.81

13.48

13.14

12.93

14.19

0.41

0.40

0.40

0.39

0.38

0.37

0.36

0.41

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

2.11

2.03

1.99

1.96

1.99

1.92

1.84

2.07

7.89

7.63

7.46

7.36

7.03

6.69

6.48

7.74

0.87

0.82

0.79

0.77

0.75

0.70

0.66

0.84

0.82

0.77

0.73

0.94

2.02

1.98

1.94

1.93

1.86

1.80

1.76

2.00

123

119

117

130

0.17

0.16

0.16

0.15

0.14

0.13

0.13

0.16

180

177

175

186

0.008

0.003

0.002

0.001

0.016

0.015

0.007

0.006

EXAMPLE

131

129

128

126

0.97

0.92

0.90

0.88

187

185

183

183

0.063

0.030

0.017

0.010

0.017

0.004

0.000

0.045

3.53

3.47

3.44

3.65

3.67

3.63

3.60

3.59

x xy vSS

ft2/3 ft5/3/s ft/s

y

ft/s

time

hh:mm

date

mm/dd

x2

ft4/3

(vSS - v)2

(ft/s)2

(v - vavg)2

(ft/s)2

SSE SYY xy x2

(a)

(b)

vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -


Based on this iteration, CSS = 3.31 and R2 = 0.95. R2 is maximized.

01:45

02:00

08/01

08/01

13.04

12.88

1.88

1.80

6.59

6.43

0.75

0.73

118

117

176

175

3.46

3.43

01:45

02:00

08/01

08/01

0.36

0.36

1.88

1.80

0.68

0.64

1.78

1.75

0.13

0.13

0.010

0.003

0.000

0.003

d e

in o

v

ft/s

time

hh:mm

date

mm/dd

de

in

Ae

ft2o

P

ft

RSS

ft

RSS2/3

ft2/3

dmin

0.23

0.22

0.21

0.26

0.25

0.25

0.25

0.22

0.21

0.38

0.37

0.36

0.41

0.40

0.40

0.39

0.36

0.36

0.26 0.41

10.35 1.86

Solution: Calculate the hydraulic coefficient - Iteration n




19

EXAMPLE

(c)

000

078

113

141

167

193

219

0.00

0.00

0.00

0.19

0.59

1.08

1.62

0.00

1.53

2.21

2.77

3.28

3.78

4.30

0.00

0.00

0.07

0.18

0.29

0.00

0.00

0.83

1.57

2.13

00

03

06

09

12

15

18

d Ae P RSS vSS

in ft2 ft ft ft/s

o

0.38 2.57

The full-pipe capacity is calculated using the Continuity Equation, QSS = AvSS. Therefore,QSS = 3.98 ft2 x 2.93 ft/s = 11.7 ft3/s or 7.5 MGD.

The conditions observed within this sewer are effectively described by the ManningEquation fitted to observed flow depth and velocity data using the Stevens-SchutzbachMethod.

Solution: Construct pipe curve and estimate sewer capacity

(d)

00.00

00.00

00.00

02.55

05.55

08.55

11.55

de

in

000

000

000

072

108

137

163

e

o

247

282

360

2.19

2.74

3.25

4.86

5.54

7.07

0.45

0.49

2.89

3.07

21

24

27 0.46 2.93

14.55

17.55

20.55

189

215

243

0.00

0.24

0.66

1.16

1.71

2.27

2.82

A

ft2

3.32

3.73

3.98

Calculate vSS for 0 < d < D

Calculate QSS for d = D

0.00

0.00

0.17

0.32

0.43

RSS2/3

ft

0.52

0.59

0.62

0.60


max

min

Flo

w D

ept

h (in

)

0

27

9

18


0 41 2 3

ddog

ddog = 6.45-in

CSS = 3.31

R2 = 0.95

SS


20

Laboratory Investigation Laboratory investigations were designed to demonstrate the performance of these methods under controlled conditions and were performed using hydraulic testing facilities located at Accusonic Technologies in Falmouth, Massachusetts. Equipment and Methodology The laboratory equipment used during this investigation was designed and configured to simulate hydraulic conditions encountered in the urban sewer environment. The general arrangement of this equipment is provided in Figure 8.

FIGURE 8: Laboratory General Arrangement

1 2Influent Chamber Effluent Chamber

Electromagnetic Flow Meter

Wet WellPump

Test Pipe

Baffles

1

2

Monitoring Point.

Downstream Obstruction (Variable). See Figure 9 for Detail.

Manual Valve

A pump provides flow through a 6-in PVC force main to an influent chamber. A manual valve regulates the pump, and an electromagnetic flow meter measures the pump discharge. Flow passes through three consecutive baffles within the influent chamber, minimizing surface disturbances before entering an 8-in PVC test pipe. Uniform and non-uniform flow conditions are observed and measured at a monitoring point located within the test pipe. Flow conditions are controlled using one of three obstructions of known depth, as depicted in Figure 9, positioned a fixed distance downstream from the monitoring point. Following discharge from the test pipe to an effluent chamber, the flow is returned to a wet well for re-circulation by the pump.

FIGURE 9: Downstream Obstructions for Laboratory Investigation

0.0-in 1.5-in 3.0-in

After placing an obstruction within the test pipe, the pump is activated, and flow is introduced into the system. Once the system has reached equilibrium, flow depth and quantity measurements are obtained at three consecutive one-minute intervals. Flow depth is measured in the test pipe with a stainless steel ruler, and flow quantity is


21

measured in the force main with the electromagnetic flow meter. These measurements are then used to calculate flow velocity in the test pipe using the Continuity Equation. A total of 30 flow depth and quantity measurements were obtained at a variety of pump settings for each obstruction. Results and Discussion Flow depth and velocity data obtained during the laboratory investigations are plotted on scattergraphs and evaluated with respect to the Manning Equation using the Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method, as shown in Figure 10. The Design Method is applied using a roughness coefficient of 0.009 and a pipe slope of 0.72%. These values were selected based on the recommendation of the Uni-Bell PVC Pipe Association and laboratory measurements, respectively.9

FIGURE 10: Laboratory Results

Flo

w D

epth

(in)

0

8


0 4

max

min

2 31

LC

SS

DM

Flo

w D

epth

(in)

0

8


0 4

max

min

2 31

DM

Flo

w D

epth

(in)

8


0 4

max

min

2 31

DM

0

2

4

6

2

4

6

2

4

6

(b) Downstream Obstruction = 1.5 in

(a) Downstream Obstruction = 0.0 in

(c) Downstream Obstruction = 3.0 in

LC

SS

SS

LC

Laboratory Data

The laboratory observations demonstrate that these methods provide similar results under uniform flow conditions, as shown in Figure 10a. However, the Stevens-Schutzbach Method best describes the relationship between flow depth and velocity under non-uniform flow conditions resulting from various dead dogs, as shown in Figure 10b and Figure 10c.


22

Conclusion The scattergraph is a graphical tool that provides insight into sewer performance through a simple and intuitive display of flow monitor data. The resulting patterns form characteristic signatures that reveal important information about conditions within a sewer and the impact that these conditions have on sewer capacity. The Manning Equation is an important component of the scattergraph and can be applied using a variety of methods, including the Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method. Each method applies a specific set of assumptions to the Manning Equation, and an understanding of these assumptions is essential to effective application of these methods. Proper selection and application of these methods have a significant impact on the calculation of sewer capacity and the evaluation of sewer performance based on flow monitor data. Laboratory results indicate that these methods provide similar results under uniform flow conditions. However, the Stevens-Schutzbach Method best describes the relationship between flow depth and velocity under non-uniform flow conditions resulting from various dead dogs.

Symbols and Notation The following symbols and notation are used in this paper:

d = flow depth, in or ftvQnRSCDAPR2

= flow velocity, ft/s= flow rate, ft3/s or MGD= roughness coefficient= hydraulic radius, ft= slope of the energy gradient= hydraulic coefficient= diameter, in or ft= wetted area, ft2

= wetted perimeter, ft= coefficient of determination

= Design Method= Lanfear-Coll Method= Stevens-Schutzbach Method

= effective= dead dog

DM

LC

SS

e

dog

VARIABLES SUBSCRIPTS

= averageavg

= minimummin

Acknowledgement The authors acknowledge John Pangallo for motivating this work and Karl Boone for his support of laboratory research activities. Special thanks are also extended to Patrick Stevens, Jim Schutzbach, Peter Keefe, and Paul Mitchell for their insight and thoughtful review.


23

References 1. Enfinger, K.L. and Keefe, P.N. (2004). “Scattergraph Principles and Practice – Building a

Better View of Flow Monitor Data,” KY-TN Water Environment Association Water Professionals Conference; Nashville, TN.

2. Metcalf & Eddy, Inc. (1981). Wastewater Engineering: Collection and Pumping of

Wastewater, McGraw-Hill, New York, NY. 3. Stevens, P.L. (1997). “The Eight Types of Sewer Hydraulics,” Proceedings of the Water

Environment Federation Collection Systems Rehabilitation and O&M Specialty Conference; Kansas City, MO. Water Environment Federation: Alexandria, VA.

4. Stevens, P.L. and Sands, H.M. (1995). “Sanitary Sewer Overflows Leave Telltale Signs

in Depth-Velocity Scattergraphs,” Seminar Publication – National Conference on Sanitary Sewer Overflows; EPA/625/R-96/007; Washington, D.C.

5. Butler, D. and Davies, J.W. (2000). Urban Drainage. E & FN Spon, London. 6. Lanfear, K.J. and Coll, J.J. (March 1978). “Modifying Manning’s Equation for Flow Rate

Estimates,” Water and Sewage Works, 68-69. 7. Walpole, R.E. and Myers, R.H. (1989). Probability and Statistics for Engineers and

Scientists, 4th edition, Macmillan Publishing Company, New York, NY. 8. Stevens, P.L. and Schutzbach, J.S. (1998). “New Diagnostic Tools Improve the Accuracy

of the Manning Equation,” Proceedings of the Water Environment Federation Technical Exhibition and Conference; Orlando, FL. Water Environment Federation: Alexandria, VA.

9. Uni-Bell PVC Pipe Association (2001). Handbook of PVC Pipe: Design and Construction,

4th edition, Dallas, TX.

Documents

Scattergraph Principles and Practice...Scattergraph Principles and Practice A Comparison of Various Applications of the Manning Equation ... including the Design Method, the Lanfear-Coll