Scattergraph Principles and Practice 03 paper 2008... · The Design Method, the Lanfear-Coll Method, ... The scattergraph is a graphical tool that provides insight into sewer performance

© 2005 ADS LLC. All Rights Reserved.

1

Scattergraph Principles and Practice Camp’s Varying Roughness Coefficient Applied to the Manning Equation

Kevin L. Enfinger, P.E. and James S. Schutzbach 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. Most design methods assume that the roughness coefficient is constant, but historical research has shown that it varies as a function of flow depth. The use of a constant or varying roughness coefficient is often left to the discretion of the design engineer.

The same consideration applies to scattergraph methods that correlate the Manning Equation to flow monitor data. The Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method have been previously reported using a constant roughness coefficient. A modifi-cation is presented in this paper to incorporate a varying roughness coefficient into these methods. The selection of a constant or varying roughness coefficient can impact sewer capacity estimates by over 20%. Examples of these methods using a constant and varying roughness coefficient are provided from flow monitor locations throughout the United States. Laboratory research by the authors is also provided and indicates that the use of a varying roughness coefficient provides a more accurate determination of sewer capacity.

KEY WORDS Flow Monitoring, Manning Equation, Scattergraph, Roughness Coefficient

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. The Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method have been previously reported using a constant roughness coefficient.2 A modification is presented to incorporate a varying roughness coefficient into these methods. The selection of a constant or varying roughness coefficient can impact sewer capacity estimates by over 20%.


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.3 However, historical research reported by Camp and others has shown that the roughness coefficient varies as a function of flow depth.4 This variation can be expressed in general terms as provided in Equation (2).

)d(n=n D f (2)

where: n = roughness coefficient nD = roughness coefficient at d = D d = flow depth, ft D = diameter, ft The varying roughness coefficient is incorporated into the Manning Equation by direct substitution as shown in Equation (3).

2/13/2

D

SR)d(n

486.1=v

f (3)


nD = roughness coefficient at d = D d = flow depth, ft D = diameter, ft R = hydraulic radius, ft S = slope of the energy gradient


3

Based on this revised assumption, the Manning Equation can be algebraically rearranged such that the constant parameters are consolidated into a single coefficient, defined as the hydraulic coefficient, and restated as provided in Equation (4). This expression is useful in subsequent discussions.

3/2R)d(

C486.1=v

f (4)


d = flow depth, ft C = hydraulic coefficient R = hydraulic radius, ft The varying roughness coefficient has been historically reported in a graphical format. However, this relationship can also be described in equation form. A fourth order polynomial approximation of Camp’s varying roughness coefficient is provided in Equation (5):

432 3.27(d/D)-7.79(d/D) + 6.86(d/D)- 2.30(d/D) + 1.04=)d(f (5) where: d = flow depth, ft D = diameter, ft Other equations have also been reported in the literature by various researchers, including Zaghloul, Wong and Zhou, and Akgiray.5, 6, 7, 8 These equations are mathematically interchangeable with Equation (5) in subsequent discussions. 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. The relationships described using a constant roughness coefficient ( - - - ) and a varying roughness coefficient (• • •) are provided for comparison.

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

FIGURE 1: Hydraulic Relationship of the Manning Equation

n = constant

n = nD f (d)

n/nD

1.0 1.2 1.4


4

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.9 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.10 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. These methods have been previously reported using a constant roughness coefficient.2 Modifications are presented in the following sections to incorporate a varying roughness coefficient into these methods.


5

Design Method The Design Method uses the Manning Equation with a specified roughness coefficient and pipe slope and has been previously reported using a constant roughness coefficient.2 A modification is described here to incorporate a varying roughness coefficient into this method. The Manning Equation is applied using this modification under the general assumptions shown in Figure 3.

FIGURE 3: General Assumptions of the Design Method

flow monitor

n = nD f (d) S = constant

uniform flow

The Design Method incorporates the Manning Equation as expressed in Equation (6) and the hydraulic radius as defined in Equation (7).

3/2DM

DM R)d(

C486.1=v

f (6)

P

A=RDM (7)

where: v = flow velocity, ft/s CDM = hydraulic coefficient d = flow depth, ft RDM = hydraulic radius, ft A = wetted cross-section, 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 (8).

2/1

DDM S

n

1=C (8)

where: CDM = hydraulic coefficient n = roughness coefficient S = pipe slope If the value of the roughness coefficient is known at a given depth, nD is calculated using Equation (2) and Equation (5). The Design Method is then used to generate a pipe


6

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 using a varying roughness coefficient is demonstrated in the following example.

EXAMPLE Flow monitor data are obtained from a 30-in sewer, as shown in thescattergraph below. A pipe curve has been previously constructed using theDesign Method with a constant roughness coefficient and the roughnesscoefficient (n) and pipe slope (S) provided below.2

Use the Design Method with a varying roughness coefficient to construct a pipecurve on the scattergraph and estimate the full-pipe capacity of this sewer.Compare this result to the full-pipe capacity determined with a constantroughness coefficient.

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

n = constant


7

EXAMPLE

Calculate nD assuming n = 0.013 at d = 9 inches nD = 0.010

Calculate CDM assuming nD = 0.010 and S = 0.45% CDM = 6.71

Calculate vDM for 0 < d < D

For a circular sewer,11

(a)

(b)

(c)

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

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

A = (D2/8)(- sin

P = D/2

Ad

D

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, where QDM= 4.91 ft2 x 5.61 ft/s = 27.5 ft3/s or 17.8 MGD, about 30% greater than the correspondingvalues determined using a constant roughness coefficient.2

The conditions observed within this sewer are effectively described by the ManningEquation using the Design Method with a varying roughness coefficient. Previous resultsusing a constant roughness coefficient are shown for comparison.2

Calculate QDM for d = D(d)

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

00

05

10

15

20

25

30

d A P RDM vDM

in ft2 ft ft ft/s

o

0.63

ft2/3

f (d) RDM2/3/f (d)

0.32

0.46

0.59

0.68

0.74

0.73

1.04

1.27

1.29

1.24

1.19

1.12

1.00

3.17

4.63

5.86

6.80

7.39

7.29

0

5

10

20

25

15

Flo

w D

ept

h (in

)

0

30


max

min

93 6

= 0.013n

= 0.45%S

= 5.16CDM

DM

n = constant

n = nD f (d)

= 0.010nD

= 0.45%S

= 6.71CDM


8

Lanfear-Coll Method The Lanfear-Coll Method uses a curve fitting technique to fit the Manning Equation to flow monitor data and has been previously reported using a constant roughness coefficient.2 A modification is described here to incorporate a varying roughness coefficient into this method. The Manning Equation is applied using this modification under the general assumptions shown in Figure 4.

FIGURE 4: General Assumptions of the Lanfear-Coll Method

flow monitor

n = nD f (d) 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 (9) and the hydraulic radius as defined in Equation (10).

3/2LC

LC R)d(

C486.1=v

f (9)

P

A=RLC (10)

where: v = flow velocity, ft/s CLC = hydraulic coefficient d = flow depth, ft RLC = hydraulic radius, ft A = wetted cross-section, 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 (9) using a varying roughness coefficient, as described in Figure 5. Regression results are characterized using the coefficient of determination.12


9

FIGURE 5: Regression Using the Lanfear-Coll Method

A

D

d

Restate Equation (9) as y = a + bx using directsubstitution, where:

= RLC2/3 / f (d)

= 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 / f (d) (ft2/3)

R2

Perform least squares regression.

CalculateRLC

2/3 / f (d)

Regressionv vs RLC

2/3 / f (d)

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 using a varying roughness coefficient is demonstrated in the following example.


10

EXAMPLE Flow monitor data are obtained from a 42-in sewer, as shown in thescattergraph below. Tabular data are provided on the following page. A pipecurve has been previously constructed using the Lanfear-Coll Method with aconstant roughness coefficient.2

Use the Lanfear-Coll Method with a varying roughness coefficient to construct apipe curve on the scattergraph and estimate the full-pipe capacity of this sewer.Compare this result to the full-pipe capacity determined with a constantroughness coefficient.

Flo

w D

epth

(in

)


0

min

max

0

42

35

28

21

14

7

153 6 9 12

LC

n = constant

= 6.26CLC

= 0.84R2


11

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

0.58

0.58

0.57

0.57

0.56

0.56

0.56

0.62

7.18

7.40

7.11

7.15

6.89

7.00

6.82

7.22

4.17

4.31

4.09

4.09

3.88

3.93

3.82

4.48

6.95

6.96

6.87

6.84

6.73

6.71

6.71

7.41

0.34

0.34

0.33

0.33

0.32

0.32

0.31

0.38

0.053

0.192

0.058

0.099

0.026

0.083

0.013

0.037

EXAMPLE

0.000

0.052

0.004

0.000

0.079

0.029

0.124

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

(b)

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 = 8.05 and R2 = 0.82.

01:45

02:00

11/01

11/01

0.56

0.55

6.71

6.71

3.74

3.71

6.67

6.61

0.31

0.31

0.002

0.010

0.213

0.213

(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

0.56

0.56

0.56

0.58

0.58

0.57

0.57

0.56

0.55

0.71 0.62

7.17

f (d) RLC2/3/f (d)

1.30

1.30

1.30

1.29

1.29

1.29

1.29

1.30

1.30

1.28

Calculate CLC and R2 based on a least squares regression

Calculate RLC2/3 /f (d)


12

EXAMPLE

The full-pipe capacity is calculated using the Continuity Equation, QLC = AvLC, where QLC= 9.62 ft2 x 10.94 ft/s = 105.2 ft3/s or 68.0 MGD, about 29% greater than thecorresponding values determined using a constant roughness coefficient.2

The conditions observed within this sewer are effectively described by the ManningEquation fitted to observed flow depth and velocity data using the Lanfear-Coll Methodwith a varying roughness coefficient. Previous results using a constant roughnesscoefficient are shown for comparison.2

Solution: Construct pipe curve and estimate sewer capacity

(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

04.76

06.95

08.79

10.20

11.09

00

07

14

21

28

35

42

d P RLC vLC

in ft ft ft/so

0.88 10.94

0.00

1.05

2.81

4.81

6.81

8.57

9.62

A

ft2

0.40

0.58

0.74

0.85

0.93

ft2/3

0.91

1.27

1.29

1.24

1.19

1.12

1.00

f (d)

1.04

Calculate QLC for d = D

RLC2/3/f (d)

(c) Calculate vLC for 0 < d < D

F

low

De

pth

(in)


0

min

max

0

42

35

28

21

14

7

153 6 9 12

LC

n = constant

n = nD f (d)

= 8.05CLC

= 0.82R2

= 6.26CLC

= 0.84R2


13

Stevens-Schutzbach Method The Stevens-Schutzbach Method uses an iterative curve fitting technique to fit the Manning Equation to flow monitor data and has been previously reported using a constant roughness coefficient.2 A modification is described here to incorporate a varying roughness coefficient into this method. The Manning Equation is applied using this modification under the general assumptions shown in Figure 6.

flow monitor

ddog

FIGURE 6: General Assumptions of the Stevens-Schutzbach Method

n = nD f (d)

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 modified Stevens-Schutzbach Method incorporates the Manning Equation as expressed in Equation (11) and the hydraulic radius as defined in Equation (12).

3/2SS

SS R)d(

C486.1=v

f (11)

P

A=R e

SS (12)

where: v = flow velocity, ft/s CSS = hydraulic coefficient d = flow depth, ft RSS = hydraulic radius, ft

Ae = effective wetted cross-section, 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 using a varying roughness coefficient, as described in Figure 7. The magnitude of the dead dog (ddog) is varied in successive iterations until the coefficient of determination is maximized.


14

FIGURE 7: Regression Using the Stevens-Schutzbach Method

Assumeddog = 0

Regressionv vs RSS

2/3 / f (d)

CalculateR2

R2 Max?

Increase ddog0 < ddog < dmin

No

Yes

d

v

CalculateRSS

2/3 / f (d)

e

Ae

A

D

d de

ddog

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

= RSS2/3 / f (d)

= 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.2

RSS =Ae

P

Iteration 2

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

Slope = 1.486CSS

Iteration n

R2 maximizedFlo

w V

elo

city

(ft/

s)

Iteration 1

(Hydraulic Radius)2/3 / f (d) (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 using a varying roughness coefficient is demonstrated in the following example.


15

EXAMPLE

Use the Stevens-Schutzbach Method with a varying roughness coefficient toconstruct a pipe curve on the scattergraph and estimate the full-pipe capacity ofthis sewer. Compare this result to the full-pipe capacity determined with aconstant roughness coefficient.

Flow monitor data are obtained from a 27-in sewer, as shown in thescattergraph below. Tabular data are provided on the following page. A pipecurve has been previously constructed using the Stevens-Schutzbach Methodwith a constant roughness coefficient.2

max

min

Flo

w D

epth

(in

)

0

27

9

18


0 41 2 3

ddog

ddog = 6.45-in

CSS = 3.31

R2 = 0.95

n = constant

SS


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

0.57

0.56

0.56

0.55

0.55

0.54

0.53

0.56

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

1.20

1.14

1.11

1.09

1.09

1.04

0.98

1.17

1.98

1.92

1.88

2.12

1.93

1.91

1.90

1.89

1.87

1.84

1.82

1.92

180

177

175

186

0.32

0.31

0.31

0.31

0.30

0.29

0.29

0.32

180

177

175

186

0.031

0.014

0.008

0.004

0.015

0.006

0.000

0.022

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

(b)

vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -


Based on this iteration, CSS = 2.30 and R2 = 0.61. R2 is not maximized.

Solution: Calculate the hydraulic coefficient - Iteration 1

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

0.54

0.53

1.88

1.80

1.01

0.96

1.83

1.82

0.29

0.28

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 ft2/3

dmin

0.56

0.55

0.55

0.58

0.58

0.57

0.57

0.55

0.55

0.55

0.54

0.53

0.57

0.56

0.56

0.55

0.54

0.53

0.58 0.56

10.35 1.86

(a)

f (d) RSS2/3/f (d)

1.24

1.25

1.25

1.23

1.24

1.24

1.24

1.25

1.25

1.24

Assume ddog = 0.00 in. Calculate RSS2/3/f (d)

Calculate CSS and R2 based on a least squares regression


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

0.53

0.53

0.52

0.52

0.51

0.50

0.50

0.53

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

1.12

1.07

1.04

1.02

1.02

0.97

0.92

1.10

1.80

1.73

1.69

1.93

1.94

1.92

1.91

1.90

1.87

1.84

1.82

1.93

171

168

167

177

0.28

0.28

0.27

0.27

0.26

0.25

0.25

0.28

180

177

175

186

0.028

0.012

0.007

0.004

0.015

0.007

0.001

0.019

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

(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

0.50

0.50

1.88

1.80

0.94

0.89

1.83

1.81

0.25

0.25

0.003

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 ft2/3

dmin

0.51

0.50

0.49

0.53

0.53

0.52

0.52

0.50

0.49

0.51

0.50

0.50

0.53

0.53

0.52

0.52

0.50

0.50

0.53 0.53

10.35 1.86

(a)

f (d) RSS2/3/f (d)

1.24

1.25

1.25

1.23

1.24

1.24

1.24

1.25

1.25

1.24




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

0.50

0.49

0.49

0.48

0.48

0.47

0.46

0.49

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

1.05

1.00

0.97

0.95

0.95

0.90

0.85

1.02

1.61

1.55

1.51

1.74

1.96

1.93

1.91

1.90

1.87

1.83

1.81

1.94

163

160

158

169

0.25

0.24

0.24

0.24

0.23

0.22

0.21

0.24

180

177

175

186

0.024

0.010

0.006

0.003

0.015

0.008

0.001

0.017

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

(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

0.46

0.46

1.88

1.80

0.87

0.83

1.82

1.80

0.22

0.21

0.003

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 ft2/3

dmin

0.46

0.45

0.44

0.48

0.47

0.47

0.47

0.44

0.44

0.48

0.47

0.46

0.50

0.49

0.49

0.48

0.46

0.46

0.48 0.49

10.35 1.86

(a)

f (d) RSS2/3/f (d)

1.24

1.25

1.25

1.23

1.24

1.24

1.24

1.25

1.25

1.24




19

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

0.35

0.34

0.34

0.33

0.32

0.31

0.30

0.35

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

8.33

8.07

7.90

7.80

7.47

7.13

6.92

8.18

0.74

0.69

0.67

0.65

0.64

0.60

0.56

0.72

0.90

0.84

0.81

1.02

2.02

1.97

1.94

1.92

1.86

1.79

1.75

1.99

127

124

122

134

0.12

0.12

0.11

0.11

0.10

0.10

0.09

0.12

180

177

175

186

0.008

0.003

0.002

0.001

0.017

0.016

0.008

0.006

EXAMPLE

135

133

131

130

1.04

1.00

0.97

0.95

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


vavg

CSS = xy x2

1.486

/R2 =

SSE

SYY1 -


Based on this iteration, CSS = 3.88 and R2 = 0.95. R2 is maximized.

Solution: Calculate the hydraulic coefficient - Iteration n

01:45

02:00

08/01

08/01

13.04

12.88

1.88

1.80

7.03

6.87

0.82

0.80

123

121

176

175

3.46

3.43

01:45

02:00

08/01

08/01

0.31

0.30

1.88

1.80

0.58

0.54

1.77

1.74

0.09

0.09

0.011

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 ft2/3

dmin

0.25

0.24

0.23

0.28

0.28

0.27

0.27

0.24

0.23

0.32

0.31

0.30

0.35

0.34

0.34

0.33

0.31

0.30

0.28 0.35

10.35 1.86

(a)

f (d) RSS2/3/f (d)

1.24

1.25

1.25

1.23

1.24

1.24

1.24

1.25

1.25

1.24


(b)


20

EXAMPLE

(c)

The full-pipe capacity is calculated using the Continuity Equation, QSS = AvSS, where QSS= 13.9 ft3/s or 9.0 MGD, about 19% greater than the corresponding value determined witha constant roughness coefficient.2

The conditions observed within this sewer are effectively described by the ManningEquation fitted to observed flow depth and velocity data using the Stevens-SchutzbachMethod with a varying roughness coefficient. Previous results using a constantroughness coefficient are shown for comparison.2

Solution: Construct pipe curve and estimate sewer capacity

(d)

000

078

113

141

167

193

219

0.00

0.00

0.00

0.24

0.66

1.16

1.71

0.00

1.53

2.21

2.77

3.28

3.78

4.30

0.00

0.00

0.09

0.20

0.31

0.00

0.00

0.88

1.56

2.14

00

03

06

09

12

15

18

d Ae P RSS vSS

in ft2 ft ft ft/s

o

0.40 2.63

00.00

00.00

00.00

02.99

05.99

08.99

11.99

de

in

000

000

000

078

112

141

167

e

o

247

282

360

2.27

2.81

3.32

4.86

5.54

7.07

0.47

0.51

3.02

3.36

21

24

27 0.47 3.48

14.99

17.99

20.99

193

219

247

0.00

0.24

0.66

1.16

1.71

2.27

2.82

A

ft2

3.32

3.73

3.98

1.22

1.29

1.29

1.26

1.22

1.19

1.15

1.09

1.00

f (d)

1.04


Calculate vSS for 0 < d < D

Calculate QSS for d = D

ft2/3

RSS2/3/f (d)

0.00

0.00

0.15

0.27

0.37

0.46

0.52

0.58

0.60

max

min

Flo

w D

ep

th (

in)

0

27

9

18


0 41 2 3

ddog

ddog = 6.45-in

CSS = 3.31

R2 = 0.95

ddog = 6.01-in

CSS = 3.88

R2 = 0.95

n = constant

n = nD f (d)

SS


21

Laboratory Investigation Laboratory investigations were previously reported to demonstrate the performance of the Design Method, the Lanfear-Coll Method, and the Stevens-Schutzbach Method under controlled conditions.2 The results of these investigations can also be used to compare the use of a constant and varying roughness coefficient. 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


22

depth is measured in the test pipe with a stainless steel ruler, and flow quantity is 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. Results and Discussion Flow depth and velocity data obtained using the 3.0-in downstream obstruction are plotted on a scattergraph and evaluated with respect to the Manning Equation using the Stevens-Schutzbach Method. This method is applied to the first 15 laboratory observations using both a constant and varying roughness coefficient as shown in Figure 10.


Flo

w D

epth

(in

)

0

max

min

2

4

6

8

30 1 2

n = nD f (d)

n = constant

FIGURE 10: Laboratory Observations (15 of 30)

Laboratory Data

Note that these observations are effectively described using either a constant or varying roughness coefficient. The sum of the squared error (SSE) for the Stevens-Schutzbach Method using a constant or varying roughness coefficient is 0.01 (ft/s)2, but which assumption provides the best estimate of actual sewer capacity? For a varying roughness coefficient, the projected full-pipe velocity is 1.94 ft/s, with an estimated sewer capacity of 0.424 MGD – about 20% greater than the corresponding values determined using a constant roughness coefficient.


23

To further test the two assumptions, the remaining 15 laboratory observations are added to the scattergraph and compared with the existing pipe curves from Figure 9, as shown in Figure 11.


Flo

w D

epth

(in

)

0

max

min

2

4

6

8

30 1 2

n = nD f (d)

n = constant

FIGURE 11: Laboratory Observations (30 of 30)

Laboratory Data

The SSE for these observations is 0.19 (ft/s)2 using a constant roughness coefficient, while the SSE is 0.01 (ft/s)2 using a varying roughness coefficient. The SSE for the constant roughness coefficient is 19 times greater than the SSE for the varying roughness coefficient. These results indicate that the varying roughness coefficient provides a more accurate projection of sewer capacity than the constant roughness coefficient under these test conditions.

Conclusion The scattergraph is a graphical tool that provides insight into sewer performance through a simple and intuitive display of flow monitor data. The Manning Equation is an important component of the scattergraph and can be applied using a variety of methods. 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. Modifications are presented to incorporate a varying roughness coefficient into these methods. The selection of a constant or varying roughness coefficient can impact sewer capacity estimates by over 20%. Laboratory results indicate that the use of a varying roughness coefficient provides a more accurate determination of sewer capacity.


24

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

d = flow depth, in or ftvQnRSCDAPR2

= flow velocity, ft/s= flow quantity, 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 Hal Kimbrough for his statistical guidance. Special thanks are also extended to Patrick Stevens and Paul Mitchell for their technical review.

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. Enfinger, K.L. and Kimbrough, H.R. (2004). “Scattergraph Principles and Practice – A

Comparison of Various Applications of the Manning Equation,” Proceedings of the Pipeline Division Specialty Conference; San Diego, CA; American Society of Civil Engineers: Reston, VA.

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

Wastewater, McGraw-Hill, Inc., New York, NY. 4. Camp, T.R. (1946). “Design of Sewers to Facilitate Flow,” Sewage Works Journal,

Volume 18, 3-16. 5. Zahgloul, N.A. (1997). “Unsteady Gradually Varied Flow in Circular Pipes with Variable

Roughness,” Advances in Engineering Software, Volume 28, 115-131. 6. Wong, T.S.W. and Zhou, M.C. (2003). “Kinematic Wave Parameters and Time of Travel

in Circular Channel Revisited,” Advances in Water Resources, Volume 26, 417-425.


25

7. Akgiray, Ömer (2004). “Simple Formulae for Velocity, Depth of Flow, and Slope Calculations in Partially Filled Circular Pipes.” Environmental Engineering Science, Volume 21, 371-385.

8. Akgiray, Ömer (2005). “Explicit Solutions of the Manning Equation for Partially Filled

Circular Pipes,” Canadian Journal of Civil Engineering, Volume 32, 490-499. 9. 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.

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

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

11. Butler, D. and Davies, J.W. (2000). Urban Drainage. E & FN Spon, London. 12. Walpole, R.E. and Myers, R.H. (1989). Probability and Statistics for Engineers and

Scientists, 4th edition, Macmillan Publishing Company, New York, NY.

Documents

Scattergraph Principles and Practice 03 paper 2008... · The Design Method, the Lanfear-Coll Method, ... The scattergraph is a graphical tool that provides insight into sewer performance