least-cost and most efficient channel cross sections
TRANSCRIPT
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Least-Cost and Most Efficient Channel Cross SectionsGerald E. Blackler1 and James C. Y. Guo2
Abstract: It has been a long-standing concern to decide if a channel should be designed to have the highest hydraulic efficiency or theleast cost. In this study, a large amount of channel construction costs were reviewed and analyzed to derive the channel construction costfunction as the sum of the costs for the land acquisition of the channel’s alignment, lining material for the channel’s cross section, andearth excavation for the channel’s depth. Case studies conducted in this technical note indicate that the differences between the least-costand most efficient cross sections are closely related to the channel lining to land acquisition cost ratio. When the lining to land unit costratio vanishes, the difference between these two cross sections is diminished. As revealed by the cost data, the least-cost channel sectiontends to be deeper if the land cost is much higher than the lining cost. This trade-off was incorporated into the normalized equation toprovide direct solutions to the least-cost channel cross section. The normalization of the least-cost equations allows this approach to betransferred to other regions when the local cost data are available.
DOI: 10.1061/�ASCE�0733-9437�2009�135:2�248�
CE Database subject headings: Costs; Cross sections; Stormwater management; Open channel flow; Canals; Floods.
Introduction
Flood channels are vital components in a storm-water drainagesystem. A flood channel is designed with consideration of hydrau-lic efficiency, construction cost, aesthetics, safety, and mainte-nance. It is almost impossible that a natural waterway couldremain unchanged during the urban development process. For ex-ample, the alignment of a natural waterway has to be modifiedaccording to the underground utility conflicts and easement avail-ability. The design engineer needs to select a channel cross sec-tion that is efficient in hydraulic performance and economical inconstruction cost. Applying the concept of duality, the most effi-cient channel cross section can be obtained by minimizing theexcavated channel cross-sectional area subject to a specified de-sign flow or maximizing the flow capacity subject to a specifiedchannel excavated area �Guo 2004�. However, when the channelconstruction cost is more complicated than the earth volume ex-cavation, the least-cost channel cross section is different from themost efficient because different objective functions are used in theoptimization process �Guo and Hughes 1984�. As suggested, theconstruction cost for a rectangular concrete channel can be ana-lyzed using the channel width as the key factor �USACE 1991�.To consider the freeboard requirement for a parabolic channel, apower-law channel was also investigated to maximize the hydrau-lic efficiency �Anwar and Clarke 2005�. All these studies indicatethat the most efficient channel cross section can be related to thechannel width to flow depth ratio.
The latest construction cost record indicates that the channel
1Graduate Student, Dept. of Civil Engineering, Univ. of Colorado—Denver, Denver, Co 80217. E-mail: [email protected]
2Professor, Dept. of Civil Engineering, Univ. of Colorado—Denver,Denver, Co 80217. E-mail: [email protected]
Note. Discussion open until September 1, 2009. Separate discussionsmust be submitted for individual papers. The manuscript for this technicalnote was submitted for review and possible publication on November 15,2007; approved on July 29, 2008. This technical note is part of theJournal of Irrigation and Drainage Engineering, Vol. 135, No. 2, April
1, 2009. ©ASCE, ISSN 0733-9437/2009/2-248–251/$25.00.248 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE
J. Irrig. Drain Eng. 200
construction cost is mainly composed of costs for land acquisitionfor the channel alignment, lining material for the channel crosssection, and excavated earth volume for the channel depth �RSMeans 2007�. These three cost elements are directly related to thechannel width and flow depth. This fact implies that both efficientand least-cost channel cross sections can be formulated and opti-mized by the channel width to depth ratio. This study presents anattempt to formulate both channel construction cost and hydraulicefficiency functions directly related to the channel width anddepth. After an extensive cost data analysis, it was found that theleast-cost solution is sensitive to the trade-off between channellining and land acquisition costs. In general, the least-cost channelcross section is narrower than the most efficient, and the differ-ence between these two cross sections is diminished when thelining cost is 20 times or higher than the land cost. Mathemati-cally, these two cross sections become identical as the land costvanishes.
Optimization of Channel Cross Section
Fig. 1 illustrates a typical symmetric trapezoidal channel crosssection. The wetted perimeter consists of the channel bottomwidth and the wetted widths along the side slopes. The excavatedcross-sectional area includes the flow area for the design flow andthe height of freeboard selected based on safety.
With consideration of freeboard, the channel construction costis no longer a linear function with respect to the excavated crosssection. As reported, the assumed power cost function leads to adifferent channel cross section from the most efficient �Guo andHughes 1984�. In this study, it is attempted to formulate the chan-nel construction cost as a function of channel width to flow depthratio. According to the published construction cost record, thetotal cost for channel construction is composed of three elements,�1� cost for channel excavation; �2� cost for cross-sectional sur-face lining; and �3� cost for land acquisition �RS Means 2007�.Obviously, the total channel construction cost varies with respectto the channel cross-sectional geometry. In this study, the cost
function for channel construction is derived as/ MARCH/APRIL 2009
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C = c1�y�b + zy� + f�T + zf�� + c2�2�y2 + �zy�2 + b + 2�f2 + �zf�2�
+ c3�b + 2z�y + f�� �1�
where C=cost of one unit length of channel construction; c1
=per unit area cost for channel excavation; c2=per unit lengthcost for channel lining; c3=per unit length cost for land acquisi-tion; y=depth of flow; f =freeboard height; z=preselected sideslope expressed in a rise to run ratio as: z�H� :1�V�; T=top width;and b=width of channel bottom.
As illustrated in Fig. 1, the channel sectional parameters canbe selected to minimize the objective function defined in Eq. �1�that is subject to a given design flow rate. Consider that uniformopen channel flow can be described by Manning’s equation. Thecapacity of the channel for the specified design flow can be ex-pressed as
Q� =Qn
kn�S0
− A5/3P−2/3 = 0 �2�
where Q=design flow; kn=dimensional constant equal to 1.486for English units or 1.0 for SI units; n=Manning’s roughness; A=cross-sectional flow area; P=wetted perimeter; and S0
= longitudinal channel slope. The objective function in Eq. �1� isminimized subject to the equality constraint in Eq. �2� and thevariables in Eq. �3� to be greater than or equal to zero:
�b,y,z� � 0 �3�
where, b, y, and z=geometric variables held to positive values. Tosolve the constrained objective function, an unconstrainedLagrangian objective function, L, is minimized as
L = C + ��Q�� �4�
where �=Lagrangian multiplier. To obtain the solution, the first-order derivatives for the necessary condition of unconstrainedminimization are applied as
�L
��=
�L
�b=
�L
�y=
�L
�z= 0 �5�
From �L /�b=0 the value of � is obtained, and from �L /��=0,the following are obtained:
�L= 0 = c1�b + 2zy� + c2�2�1 + z� + c3�2z� �6�
Fig. 1. Typical trapezoidal channel cross section
�y
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�L
�z= 0 = c1�y2 + f2� + c2�y/�z + 1�0.5 + f/�z + 1�0.5� + c3�2y + 2f�
�7�
Eqs. �5�–�7� are solved iteratively to determine the best b /y ratiofor a specified cost ratio of c2 /c3. The ratio of c1 /c3 in relation tothe value of c2 /c3 is found to be within a small range according tothe cost data. This is why the ratio of c2 /c3 is carried forward todevelop the cost factor. Details that describe this type of optimi-zation can be found elsewhere �Das 2000�.
Least-Cost Channel Sections
For this case, the value of � in Eq. �4� reflects the shadow price,that is, how the total cost will vary with respect to the ratio: Q /S0
�Das 2006�. Further, the least cost also depends on the type ofchannel linings. As recommended, the commonly used channellinings are: �1� grass lining on erosive soils; �2� grass lining oncohesive soils; �3� riprap lining; and �4� and concrete lining.
In this study, it is suggested that Eq. �5� is solved for a range ofvariables within the engineering practice. For instance, the chan-nel side slope is preselected to be: 0, 1, 2, or 4, based on the soilstability. The optimization process observes the water flow andsafety design criteria recommended by the Urban Storm WaterDrainage Criteria Manual �Urban Drainage Flood Control District2001�. Table 1 is the summary of the recommended design crite-rion used in this study.
Before optimizing the b /y ratio, the longitudinal slope, S0, hasto be predetermined by the allowable permissible flow velocityusing a drop structure �Guo 2004�. Numerous cases were solvedfor Eqs. �5�–�7�. The large database generated in this study wasthen analyzed to produce a functional relationship. According toManning’s formula, the channel hydraulic efficiency can be re-lated to the b /y ratio. The most efficient channel cross section hasbeen formulated as �Chow 1959; Guo and Hughes 1984�
b
y= 2��1 + z2 − z� �8�
In this study, a similar cross-sectional function to Eq. �8� wasadopted for the regression study on the optimal channel sections
Table 1. Recommended Design Criteria for Channel Cross-SectionalOptimization
Design constraints
Various types of channel lining
Grass liningon erosive
soils
Grass liningon cohesive
soilsRipraplining
Concretelining
Maximum flow velocity 1.5 m /s 2.1 m /s 3.7 m /s NA
Maximum Manning’s n 0.035 0.035 0.04 0.014
Maximum depth 1.5 m 5 m NA NA
Maximum channellongitudinal slope
0.6% 0.6% 1.0% NA
Maximum side slope 4H :1V 4H :1V 2.5H :1V 1.5H :1V
Minimum freeboard 0.3 m 0.3 m 0.6 m 0.6 m
Note: NA=not applicable.
generated from the results of Eqs. �5�–�7� as
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b
y= 2�1 − R���1 + z2 − z� �9�
where R=cost factor determined by channel lining to land costratio. Fig. 2 presents the comparison between the database andEq. �9�.
As shown in Fig. 2, there is a trade-off between channel liningand land costs. As expected, Eq. �9� recommends a deep channelsection if the land acquisition cost is much higher than the liningcost. Therefore, the channel lining to land cost ratio is the keyfactor in determination of the cost factor, R. As a dimensionlessvariable, the value of R is found to be insensitive to channel sideslope. The ratio of c1 /c3 was computed along with c2 /c3 and wasfound to be less sensitive. For this reason, the cost factor is com-puted using c2 /c3 to increase the range and accuracy of the costfunction. As illustrated in Fig. 2, the best-fitted formula for costfactor falls under two sets of equations. First, if c2 /c3�4, then
R = − 0.189 ln�c2/c3� + 0.41 �10�
and, second, if 4�c2 /c3�20, then
R = − 0.058 ln�c2/c3� + 0.21 �11�
For values where c2 /c3 are greater than 20 the difference betweenthe most efficient and least-cost channel cross sections becomesnegligible. Also, when R=0, the least cost is the same as the mostefficient channel cross section.
Eqs. �10� and �11� were derived to provide the best fit to thelining to land cost ratios ranging from 0.1 to 20. As shown in Fig.2, the value of R decreases with respect to the increase of thelining to land cost ratio. As the lining to land cost ratio becomeshigher than 20, the value of R is diminished to become more
Fig. 2. Cost factor develop
insignificant.
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Design Schematics
An example is used to illustrate the application of Eqs. �9�–�11�.A concrete channel is designed to carry a peak flow of 7.08 m3 /son a slope of 0.05%. The objective is to minimize the construc-tion cost by selecting a proper b /y ratio. For this case, a sideslope of 1V :2H is adopted or z=2.0. The Manning’s roughnesscoefficient of 0.014 is recommended for hydraulic calculations.The unit costs at the project site are found to be $2.15 /m2 perlinear length for earth excavation, $39.81 /m2 per linear length forconcrete linings, and $12.91 /m2 per linear length for land acqui-sition. As c2 /c3=3.08 and is less than 4, Eq. �10� is applied to thiscase and R for the least-cost channel section is calculated as R=0.197. The b /y ratio for the least-cost channel section is calcu-lated using Eq. �9� to be 0.379. Applying Manning’s equation tothis case, the normal depth is found to be y=1.51 m. As a result,the channel width is b=0.57 m for the least-cost channel section.The above-presented solutions can be verified by the cost com-parison among a range of b /y ratios varying from 0.01 to 3.00. Asplotted in Fig. 3, the b /y ratio is identified for the minimum totalcost.
Conclusions
Many previous studies indicate that the most efficient channelcross section can be defined by the b /y ratio. In current practice,the most efficient channel cross section is not necessary to pro-vide the least-cost cross section. In this study, the latest channelcost record was collected and analyzed to provide the cost func-tion directly related to the channel cross-sectional geometry.Using the optimization approach, the channel least-cost functionis formulated using a dimensionless cost factor and the b /y ratio.
r least-cost channel design
ed foThe b /y ratio for the least-cost channel section is always
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smaller than that derived for the most hydraulically efficient sec-tion. The difference decreases as the lining to land cost ratioincreases. This fact can be easily visualized using a rectangularchannel as an example. With z=0, Eq. �9� is reduced to
b
y= 2�1 − R� �12�
When R becomes vanished, Eq. �9� is reduced to the conventionalsolution derived for the best rectangular channel cross section.The application of Eq. �9� to design practice is useful to the en-gineer who is concerned with construction costs. Although previ-ous studies had derived equations and methods for least-costchannel cross sections, this study provides a simple applicationderived from the real world construction cost data. The normal-ization of Eqs. �9�–�11� allows the engineer to transfer the costdata from other regions into a similar operation.
Notation
The following symbols are used in this technical note:A � cross-section flow area �L2�;b � channel bottom width �L�;C � total construction cost �$�;c � unit cost �$/L� or �$ /L2�;
kn � constant depending on the units;n � Manning’s roughness;P � channel wetted perimeter �L�;Q � design flow �L3 /T�;
Q � specified constant;
Fig. 3. Cost comparison
�
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R � cost factor;S0 � longitudinal channel slope �L/L�;T � channel top width �L�;
Tf � top width with freeboard �L�;f � freeboard height �L�;y � channel flow depth �L�;z � side slope �L/L�; and� � Lagrangian multiplier.
References
Anwar, A. A., and Clarke, D. �2005�. “Design of hydraulically efficientpower-law channels with freeboard.” J. Irrig. Drain. Eng., 131�6�,560–563.
Chow, V. T. �1959�. “Open channel hydraulics.” Design of channels foruniform flow, McGraw-Hill, New York, 160–162.
Das, A. �2000�. “Optimal channel cross section with composite rough-ness.” J. Irrig. Drain. Eng., 126�1�, 68–72.
Das, A. �2006�. “Optimal design of channel having horizontal bottom andparabolic sides.” J. Irrig. Drain. Eng., 133�2�, 192–197.
Guo, J. C. Y. �2004�. Urban flood channel design, Water Resources Pub-lication, Littleton, Colo., 143–175.
Guo, J. C. Y., and Hughes, W. �1984�. “Optimal channel cross sectionwith freeboard.” J. Irrig. Drain. Eng., 110�3�, 304–314.
RS Means. �2007�. “Online construction estimating software with cost-works from RS Means.” �www.meanscostworks.com� �Feb. 15, 2007�.
United States Army Corps of Engineers �USACE�. �1991�. “Hydraulicdesign of flood control channels.” Engineer manual No. 1110-2-1601,2-1, Washington, D.C.
Urban Drainage Flood Control District. �2001�. Urban storm water de-
inimal cost to total cost
of msign criteria manual, Vol. 1, Denver, MD-27.
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