Least-Cost and Most Efficient Channel Cross Sections Gerald 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 the least cost. In this study, a large amount of channel construction costs were reviewed and analyzed to derive the channel construction cost function as the sum of the costs for the land acquisition of the channel’s alignment, lining material for the channel’s cross section, and earth excavation for the channel’s depth. Case studies conducted in this technical note indicate that the differences between the least-cost and most efficient cross sections are closely related to the channel lining to land acquisition cost ratio. When the lining to land unit cost ratio vanishes, the difference between these two cross sections is diminished. As revealed by the cost data, the least-cost channel section tends to be deeper if the land cost is much higher than the lining cost. This trade-off was incorporated into the normalized equation to provide direct solutions to the least-cost channel cross section. The normalization of the least-cost equations allows this approach to be transferred to other regions when the local cost data are available. DOI: 10.1061/共ASCE兲0733-9437共2009兲135:2共248兲 CE Database subject headings: Costs; Cross sections; Stormwater management; Open channel flow; Canals; Floods.
Introduction Flood channels are vital components in a storm-water drainage system. A flood channel is designed with consideration of hydraulic efficiency, construction cost, aesthetics, safety, and maintenance. It is almost impossible that a natural waterway could remain unchanged during the urban development process. For example, the alignment of a natural waterway has to be modified according to the underground utility conflicts and easement availability. The design engineer needs to select a channel cross section that is efficient in hydraulic performance and economical in construction cost. Applying the concept of duality, the most efficient channel cross section can be obtained by minimizing the excavated channel cross-sectional area subject to a specified design flow or maximizing the flow capacity subject to a specified channel excavated area 共Guo 2004兲. However, when the channel construction cost is more complicated than the earth volume excavation, the least-cost channel cross section is different from the most efficient because different objective functions are used in the optimization process 共Guo and Hughes 1984兲. As suggested, the construction cost for a rectangular concrete channel can be analyzed using the channel width as the key factor 共USACE 1991兲. To consider the freeboard requirement for a parabolic channel, a power-law channel was also investigated to maximize the hydraulic efficiency 共Anwar and Clarke 2005兲. All these studies indicate that the most efficient channel cross section can be related to the channel width to flow depth ratio. The latest construction cost record indicates that the channel 1
Graduate Student, Dept. of Civil Engineering, Univ. of Colorado— Denver, Denver, Co 80217. E-mail:
[email protected] 2 Professor, Dept. of Civil Engineering, Univ. of Colorado—Denver, Denver, Co 80217. E-mail:
[email protected] Note. Discussion open until September 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this technical note was submitted for review and possible publication on November 15, 2007; approved on July 29, 2008. This technical note is part of the Journal of Irrigation and Drainage Engineering, Vol. 135, No. 2, April 1, 2009. ©ASCE, ISSN 0733-9437/2009/2-248–251/$25.00.
construction cost is mainly composed of costs for land acquisition for the channel alignment, lining material for the channel cross section, and excavated earth volume for the channel depth 共RS Means 2007兲. These three cost elements are directly related to the channel width and flow depth. This fact implies that both efficient and least-cost channel cross sections can be formulated and optimized by the channel width to depth ratio. This study presents an attempt to formulate both channel construction cost and hydraulic efficiency functions directly related to the channel width and depth. After an extensive cost data analysis, it was found that the least-cost solution is sensitive to the trade-off between channel lining and land acquisition costs. In general, the least-cost channel cross section is narrower than the most efficient, and the difference between these two cross sections is diminished when the lining cost is 20 times or higher than the land cost. Mathematically, these two cross sections become identical as the land cost vanishes.
Optimization of Channel Cross Section Fig. 1 illustrates a typical symmetric trapezoidal channel cross section. The wetted perimeter consists of the channel bottom width and the wetted widths along the side slopes. The excavated cross-sectional area includes the flow area for the design flow and the height of freeboard selected based on safety. With consideration of freeboard, the channel construction cost is no longer a linear function with respect to the excavated cross section. As reported, the assumed power cost function leads to a different channel cross section from the most efficient 共Guo and Hughes 1984兲. In this study, it is attempted to formulate the channel construction cost as a function of channel width to flow depth ratio. According to the published construction cost record, the total cost for channel construction is composed of three elements, 共1兲 cost for channel excavation; 共2兲 cost for cross-sectional surface lining; and 共3兲 cost for land acquisition 共RS Means 2007兲. Obviously, the total channel construction cost varies with respect to the channel cross-sectional geometry. In this study, the cost function for channel construction is derived as
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Table 1. Recommended Design Criteria for Channel Cross-Sectional Optimization Various types of channel lining
Design constraints
Grass lining Grass lining on erosive on cohesive soils soils
Maximum flow velocity 1.5 m / s 0.035 Maximum Manning’s n Maximum depth 1.5 m Maximum channel 0.6% longitudinal slope Maximum side slope 4H : 1V Minimum freeboard 0.3 m Note: NA= not applicable.
Riprap lining
Concrete lining
2.1 m / s 0.035 5m 0.6%
3.7 m / s 0.04 NA 1.0%
NA 0.014 NA NA
4H : 1V 0.3 m
2.5H : 1V 1.5H : 1V 0.6 m 0.6 m
Fig. 1. Typical trapezoidal channel cross section
C = c1关y共b + zy兲 + f共T + zf兲兴 + c2关2冑y + 共zy兲 + b + 2冑 f + 共zf兲 兴 2
2
2
2
L = 0 = c1共y 2 + f 2兲 + c2共y/共z + 1兲0.5 + f/共z + 1兲0.5兲 + c3共2y + 2f兲 z
共1兲
共7兲
where C = cost of one unit length of channel construction; c1 = per unit area cost for channel excavation; c2 = per unit length cost for channel lining; c3 = per unit length cost for land acquisition; y = depth of flow; f = freeboard height; z = preselected side slope expressed in a rise to run ratio as: z共H兲 : 1共V兲; T = top width; and b = width of channel bottom. As illustrated in Fig. 1, the channel sectional parameters can be selected to minimize the objective function defined in Eq. 共1兲 that is subject to a given design flow rate. Consider that uniform open channel flow can be described by Manning’s equation. The capacity of the channel for the specified design flow can be expressed as
Eqs. 共5兲–共7兲 are solved iteratively to determine the best b / y ratio for a specified cost ratio of c2 / c3. The ratio of c1 / c3 in relation to the value of c2 / c3 is found to be within a small range according to the cost data. This is why the ratio of c2 / c3 is carried forward to develop the cost factor. Details that describe this type of optimization can be found elsewhere 共Das 2000兲.
+ c3关b + 2z共y + f兲兴
Q⬘ =
Qn
k n 冑S 0
− A5/3 P−2/3 = 0
共2兲
where Q = design flow; kn = dimensional constant equal to 1.486 for 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. 共1兲 is minimized subject to the equality constraint in Eq. 共2兲 and the variables in Eq. 共3兲 to be greater than or equal to zero: 兵b,y,z其 艌 0
共3兲
where, b, y, and z = geometric variables held to positive values. To solve the constrained objective function, an unconstrained Lagrangian objective function, L, is minimized as L = C + 兵Q⬘其
共4兲
where = Lagrangian multiplier. To obtain the solution, the firstorder derivatives for the necessary condition of unconstrained minimization are applied as
L L L L = = = =0 b y z
Least-Cost Channel Sections For this case, the value of in Eq. 共4兲 reflects the shadow price, that is, how the total cost will vary with respect to the ratio: Q / S0 共Das 2006兲. Further, the least cost also depends on the type of channel linings. As recommended, the commonly used channel linings are: 共1兲 grass lining on erosive soils; 共2兲 grass lining on cohesive soils; 共3兲 riprap lining; and 共4兲 and concrete lining. In this study, it is suggested that Eq. 共5兲 is solved for a range of variables within the engineering practice. For instance, the channel side slope is preselected to be: 0, 1, 2, or 4, based on the soil stability. The optimization process observes the water flow and safety design criteria recommended by the Urban Storm Water Drainage Criteria Manual 共Urban Drainage Flood Control District 2001兲. Table 1 is the summary of the recommended design criterion used in this study. Before optimizing the b / y ratio, the longitudinal slope, S0, has to be predetermined by the allowable permissible flow velocity using a drop structure 共Guo 2004兲. Numerous cases were solved for Eqs. 共5兲–共7兲. The large database generated in this study was then analyzed to produce a functional relationship. According to Manning’s formula, the channel hydraulic efficiency can be related to the b / y ratio. The most efficient channel cross section has been formulated as 共Chow 1959; Guo and Hughes 1984兲
共5兲
From L / b = 0 the value of is obtained, and from L / = 0, the following are obtained:
L = 0 = c1共b + 2zy兲 + c2共2冑1 + z兲 + c3共2z兲 y
共6兲
b = 2共冑1 + z2 − z兲 y
共8兲
In this study, a similar cross-sectional function to Eq. 共8兲 was adopted for the regression study on the optimal channel sections generated from the results of Eqs. 共5兲–共7兲 as
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Fig. 2. Cost factor developed for least-cost channel design
b = 2共1 − R兲共冑1 + z2 − z兲 y
共9兲
where R = cost factor determined by channel lining to land cost ratio. Fig. 2 presents the comparison between the database and Eq. 共9兲. As shown in Fig. 2, there is a trade-off between channel lining and land costs. As expected, Eq. 共9兲 recommends a deep channel section if the land acquisition cost is much higher than the lining cost. Therefore, the channel lining to land cost ratio is the key factor in determination of the cost factor, R. As a dimensionless variable, the value of R is found to be insensitive to channel side slope. The ratio of c1 / c3 was computed along with c2 / c3 and was found to be less sensitive. For this reason, the cost factor is computed using c2 / c3 to increase the range and accuracy of the cost function. As illustrated in Fig. 2, the best-fitted formula for cost factor falls under two sets of equations. First, if c2 / c3 艋 4, then R = − 0.189 ln共c2/c3兲 + 0.41
共10兲
and, second, if 4 ⬍ c2 / c3 艋 20, then R = − 0.058 ln共c2/c3兲 + 0.21
共11兲
For values where c2 / c3 are greater than 20 the difference between the most efficient and least-cost channel cross sections becomes negligible. Also, when R = 0, the least cost is the same as the most efficient channel cross section. Eqs. 共10兲 and 共11兲 were derived to provide the best fit to the lining 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 the lining to land cost ratio. As the lining to land cost ratio becomes higher than 20, the value of R is diminished to become more insignificant.
Design Schematics An example is used to illustrate the application of Eqs. 共9兲–共11兲. A concrete channel is designed to carry a peak flow of 7.08 m3 / s on a slope of 0.05%. The objective is to minimize the construction cost by selecting a proper b / y ratio. For this case, a side slope of 1V : 2H is adopted or z = 2.0. The Manning’s roughness coefficient of 0.014 is recommended for hydraulic calculations. The unit costs at the project site are found to be $2.15/ m2 per linear length for earth excavation, $39.81/ m2 per linear length for concrete linings, and $12.91/ m2 per linear length for land acquisition. As c2 / c3 = 3.08 and is less than 4, Eq. 共10兲 is applied to this case 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 calculated using Eq. 共9兲 to be 0.379. Applying Manning’s equation to this 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 comparison among a range of b / y ratios varying from 0.01 to 3.00. As plotted in Fig. 3, the b / y ratio is identified for the minimum total cost.
Conclusions Many previous studies indicate that the most efficient channel cross section can be defined by the b / y ratio. In current practice, the most efficient channel cross section is not necessary to provide the least-cost cross section. In this study, the latest channel cost record was collected and analyzed to provide the cost function directly related to the channel cross-sectional geometry. Using the optimization approach, the channel least-cost function is formulated using a dimensionless cost factor and the b / y ratio. The b / y ratio for the least-cost channel section is always
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Fig. 3. Cost comparison of minimal cost to total cost
smaller than that derived for the most hydraulically efficient section. The difference decreases as the lining to land cost ratio increases. This fact can be easily visualized using a rectangular channel as an example. With z = 0, Eq. 共9兲 is reduced to b = 2共1 − R兲 y
共12兲
When R becomes vanished, Eq. 共9兲 is reduced to the conventional solution derived for the best rectangular channel cross section. The application of Eq. 共9兲 to design practice is useful to the engineer who is concerned with construction costs. Although previous studies had derived equations and methods for least-cost channel cross sections, this study provides a simple application derived from the real world construction cost data. The normalization of Eqs. 共9兲–共11兲 allows the engineer to transfer the cost data from other regions into a similar operation.
Notation The following symbols are used in this technical note: A ⫽ cross-section flow area 关L2兴; b ⫽ channel bottom width 关L兴; C ⫽ total construction cost 关$兴; c ⫽ unit cost 关$/L兴 or 关$ / L2兴; kn ⫽ constant depending on the units; n ⫽ Manning’s roughness; P ⫽ channel wetted perimeter 关L兴; Q ⫽ design flow 关L3 / T兴; Q⬘ ⫽ specified constant;
R S0 T Tf f y z
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
cost factor; longitudinal channel slope 关L/L兴; channel top width 关L兴; top width with freeboard 关L兴; freeboard height 关L兴; channel flow depth 关L兴; side slope 关L/L兴; and Lagrangian multiplier.
References Anwar, A. A., and Clarke, D. 共2005兲. “Design of hydraulically efficient power-law channels with freeboard.” J. Irrig. Drain. Eng., 131共6兲, 560–563. Chow, V. T. 共1959兲. “Open channel hydraulics.” Design of channels for uniform flow, McGraw-Hill, New York, 160–162. Das, A. 共2000兲. “Optimal channel cross section with composite roughness.” J. Irrig. Drain. Eng., 126共1兲, 68–72. Das, A. 共2006兲. “Optimal design of channel having horizontal bottom and parabolic sides.” J. Irrig. Drain. Eng., 133共2兲, 192–197. Guo, J. C. Y. 共2004兲. Urban flood channel design, Water Resources Publication, Littleton, Colo., 143–175. Guo, J. C. Y., and Hughes, W. 共1984兲. “Optimal channel cross section with freeboard.” J. Irrig. Drain. Eng., 110共3兲, 304–314. RS Means. 共2007兲. “Online construction estimating software with costworks from RS Means.” 具www.meanscostworks.com典 共Feb. 15, 2007兲. United States Army Corps of Engineers 共USACE兲. 共1991兲. “Hydraulic design of flood control channels.” Engineer manual No. 1110-2-1601, 2-1, Washington, D.C. Urban Drainage Flood Control District. 共2001兲. Urban storm water design criteria manual, Vol. 1, Denver, MD-27.
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