Channel Design to Minimize Lining Material Costs

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17566 DECEMBER 1982 1R4

CHANNEL DESIGN TO MINIMIZE LINING

MATERIAL COSTS

By Thomas 3. Trout,' M. ASCE

ABSTRACT: A direct algebraic technique is developed to determine open

chan-nel cross-sectional designs which minimize lining material costs when base and side wall unit costs are different. Solution graphs, which indicate both the optimal parameter combination and the costs of deviating from the optimal design, are presented. The graphs or analytical technique are also effective in designing any trapezoidal channels. Use of the technique shows that mod-erate deviations from optimal designs are not costly.

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The material cost of channel linings depends upon the volume of lining ma-terial used. On a per unit length basis, the mama-terial costs are a function of the lining thickness and perimeter length, which, in turn, depends upon the channel cross-sectional shape. The capacity of a channel constructed on a given slope will also be a function of the cross-sectional shape. Using this interrelationship, cross sections that minimize lining material costs can be determined.

If the channel is lined with one material of uniform thickness, the problem reduces to one of determining the best hydraulic section (1). However, if material or lining thickness is changed along the perimeter, the problem becomes much more complex. This paper will present a solution to the cost minimization prob-lem when the material cost per unit surface area of the base of trapezoidal (or rectangular) channels is different from the material cost of the sides.

This analysis does not deal with placement or other construction costs. If they can be evaluated in terms of surface area, they can be combined with material costs. Otherwise, they must be considered separately. The technique will thus be most useful when long channel sections are to be constructed, allowing con-struction procedures to be oriented toward minimizing material costs or when labor costs are low relative to material costs, such as is the case in developing countries.

'Agricultural Engr., U.S. Dept. of Agr., Agricultural Research Service, Snake River Conservation Research Center, Kimberly, Idaho 83341; formerly Research Asst. Prof., Dept. of Agricultural and Chemical Engrg., Colorado State Univ., Fort Collins, Colo.

Note.—Discussion open until May 1, 1983. To extend the closing date one month, a written request must be filed with the Manager of Technical and Professional Publications, ASCE. Manuscript was submitted for review for possible publication on January 4, 1982. This paper is part of the Journal of the Irrigation and Drainage Division, Proceedings of the American Society of Civil Engineers, ©ASCE, Vol. 108, No. IR4, December, 1982. ISSN 0044-7978/82/0004-0242/$01.00.

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1R4 CHANNEL DESIGN 249 channels to minimize material costs when bed and side lining costs are different. The method can also be used to evaluate added costs of deviating from optimal designs. Use of the technique shows that moderate deviations from optimal de-signs do not usually cause a significant increase in costs. Although the benefits gained by minimizing costs may not be large, in the light of the simplicity of the technique, the knowledge gained from the exercise is worthwhile.

APPENDIX I.—REFERENCES

1. Chow, V. T., Open Channel Hydraulics, McGraw-Hill Book Co., Inc., New York, N.Y., 1959, pp. 128 and 160.

2. Nicholson, W., Microeconomic Theory: Basic Principles and Extensions, The Dreydon Press, Inc., Hinsdale, Ill., 1972, pp. 27-33 and 58-61.

3. Streeter, V. L., Fluid Mechanics, 5th ed., McGraw-Hill Book Co., Inc., New York, N.Y., 1971, p. 599.

APPENDIX IL-- NOTATION

The following symbols are used in this paper:

A

B

B,, B'

b C Ca CS

D„

E

E'

F k

k1, k2, and k3

n p

Q

lb

t, 2

lea

= cross-sectional flow area; = channel bottom width;

• bottom width value at the graph intercept (D 0); • bottom corner width;

channel bottom material cost per unit area; • channel lining material cost per unit length; = base material cost per unit length;

= side material cost per unit length;

constant in Manning's Equation for conversion to U.S. Cus-tomary units (1.486);

• normal flow depth;

= depth value at the graph intercept (B - 0);

= channel side material cost per unit area; • channel wetted side length;

= freeboard side length allowance; = freeboard allowance;

• corner material cost per unit length; = constants defined by Eqs. 12, 13, and 14; = Manning roughness coefficient;

wetted perimeter length; = flow rate;

• hydraulic radius; • channel slope;

= thickness of channel bottom lining; = thickness of channel side lining;